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Study of within herd variability in milk fat, protein and lactose content of bulk milks in British Colunbia… Williams, Christopher John 1973

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A STUDY OF WITHIN HERD VARIABILITY IN MILK FAT, PROTEIN AND LACTOSE CONTENT OF BULK MILKS IN BRITISH COLUMBIA AND FACTORS AFFECTING THE DESIGN OF HERD MILK SAMPLING PROGRAMS  by CHRISTOPHER JOHN WILLIAMS B.S.A., University of B r i t i s h Columbia, 1967 M.Sc, University of B r i t i s h Columbia, 1971  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Animal Science  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA May, 1973  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r  an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e  and  that  study.  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s .  be  granted by  the Head of my  I t i s understood t h a t copying or  of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be written  permission.  Department o f  vV  The U n i v e r s i t y of B r i t i s h Vancouver 8, Canada  Department or  Columbia  publication  allowed without  my  ABSTRACT  Three sets of data were used to estimate v a r i a t i o n , from a l l sources, associated with bulk milk sampling, and testing programs.  Three milk samples were taken from each  shipment of 26 herds from March 14, 1970 to A p r i l 24, 1971 (Experiment I ) . The set of three samples was handled as follows:  (1) one sample was used i n the formation of a  two-week composite;  (2) one sample was used i n the formation  of a one-week composite; and C3) one sample was analysed fresh.  Four milk samples were taken from each shipment  of 22 d i f f e r e n t herds from November 17 to December 16, 1971. Three of the four samples were analysed fresh i n duplicate (Experiment I I ) .  The fourth sample was divided into three  parts and each part was used i n the formation of a composite. Each composite was analysed i n duplicate a f t e r a two-week c o l l e c t i o n period on alternate days.  (Experiment I I I ) . A l l milk samples  Herd milk was shipped (8,894) were analysed  for milk f a t , protein and lactose using Infrared Milk Analysers. Estimates, obtained from Experiment I by the analyses of variance of a hierarchal model (herds, periods within herds and shipments within herds and periods), of within herd-period  (15 shipments per period) variances of percent  milk f a t , protein and lactose were;  0.01371 ± .00030, 0.00787  ± .00017 and 0.00548 ± .00012 r e s p e c t i v e l y . obtained from Experiment  Estimates were  II of within herd-period variance  and i t s components by the analyses of variance of a h i e r a r c h i a l model.  The estimates of these variances f o r percent, milk.  f a t , protein and lactose respectively were: period variance —  Cl) within herd-  0.01329 ± .00064, 0.00507 ± .00031 and  0.00483 ± .00017; (.2) b i o l o g i c a l (shipment to shipment) variance —  0.00607 ± .00061, 0.00340 ± .00029 and 0.00110  ± .00014; (3) sampling  (within shipment) variance 0.00094  ± .00027, -.00021 ± .00006 and -.00033 ± .00013; and (.4) testing  (.within sample) variance —  0.00628 ± .00029,  0.00167 ± 0.00006 and 0.00373 ± .00013.  Estimates of within  herd-period variance of percent protein from Experiment I were s i g n i f i c a n t l y d i f f e r e n t from estimates from  Experiment  II. Orthogonal polynomials were used to estimate the r e l a t i o n s h i p between the s e r i a l c o r r e l a t i o n s from Experiment  (calculated  I) of milk constituent percentage and the  number of shipments separating two shipments f o r which the correlations were calculated.  Only the l i n e a r term  was s i g n i f i c a n t for percent protein and lactose and accounted for 99.7 and 98.4 percent of the t o t a l sums bf squares for these two milk constituents r e s p e c t i v e l y .  Linear and  quadratic a f t e r l i n e a r were s i g n i f i c a n t f o r percent milk f a t s e r i a l c o r r e l a t i o n s and accounted for 98.4 and 1.3  iv percent of the t o t a l sums of squares r e s p e c t i v e l y . Strata within periods was f i t t e d as an e f f e c t (Experiments I and II) i n a h i e r a r c h a l model and was a s i g n i f i c a n t source of v a r i a t i o n .  The variances of  estimates of herd-period mean milk constituent percentages obtained from various simple and s t r a t i f i e d random sampling schemes were calculated.  S t r a t i f i c a t i o n resulted  i n a r e l a t i v e l y small reduction i n the variances of these estimates. Estimates of the variances associated with the formation of a composite  sample obtained from  Experiment  III by the analysis of variance and from Experiments II were near zero.  I and  The variance of estimates of herd-  period mean milk constituent percentages obtained from two two-week composites were 0.00368, 0.00110 and 0.00205 f o r percent milk f a t , protein and lactose r e s p e c t i v e l y .  It  was calculated that four random samples would estimate herd-period mean milk constituent percentages a t l e a s t as p r e c i s e l y as two two-week composite Two-week composite  samples.  samples underestimated  percent  milk f a t by 0.045 percent milk f a t and overestimated percent protein and lactose by 0.023 and 0.010 percent respectively compared to corresponding estimates based on the fresh analyses of samples drawn from each shipment.  V  Simple and multiple regression techniques  were used  in an attempt to p r e d i c t herd differences i n within herdperiod variance from the average amount of milk and percent milk f a t , protein and lactose.  shipped  In general,  large within herd-period variances of milk constituent percentages were s i g n i f i c a n t l y associated with small herd milk shipments and high l e v e l s of milk f a t and p r o t e i n . However, the proportion of the t o t a l sums of squares accounted f o r by the various regression equations was r e l a t i v e l y low; therefore the equations were not useful for predicting herd-period  variances.  Within herd-period variance of percent milk f a t was highest i n the spring and autumn; therefore sampling frequency may need to be greater a t some seasons than a t others.  Differences among herds i n within  herd-period  variance of milk constituent percentages were s i g n i f i c a n t ; therefore random sampling schemes may have to be modified to s u i t i n d i v i d u a l herds.  vi TABLE OF CONTENTS PAGE ABSTRACT .  . i i  TABLE OF CONTENTS  vi  LIST OF TABLES  ix  LIST OF FIGURES  xiv  ACKNOWLEDGEMENTS  xv i  INTRODUCTION  .  1  LITERATURE REVIEW  4  PART 1 - ESTIMATION OF POPULATION PARAMETERS  9  INTRODUCTION MATERIALS AND METHODS  9 . . . . . . . .  10  C o l l e c t i o n and Analyses of Milk Samples  10  The Problem and D e f i n i t i o n of Terms Used S t a t i s t i c a l Methods . . . . . .  12 . . . . .  RESULTS AND DISCUSSION Estimates of Within Herd-Period Variance and  17 29  Components  .  29  E f f e c t s of Strata  „  35  Within Strata Variance . . . V a r i a b i l i t y of Estimates from Various Sampling  44  Schemes . . . . . . .  55  Composite Sampling C a l c u l a t i o n of the C r i t e r i o n of P r e c i s i o n  . . . . .  59 67  vii PAGE  Composite Sampling  v e r s u s Random S a m p l i n g  CONCLUSIONS  . . . .  69  . . . . .  79  PART 2  82  INTRODUCTION  82  MATERIALS AND METHODS  85  Source  85  o f Data  S t a t i s t i c a l Methods  85  RESULTS AND DISCUSSION  94  P e r i o d E f f e c t s o n M i l k S h i p m e n t Weight! a n d Milk Constituent Percentages  . . . . . . . . . .  -94  Transformations  96  Regression Analyses  . . . . .  Within Herd-Period Variance o f Percent M i l k Fat Within Herd-Period Variance o f Percent Protein . . . . . Within Herd-Period  98 106  Variance of Percent  Lactose  . . . . . I l l  Conclusion of Regression Analyses  . . .  Herd and P e r i o d V a r i a t i o n Season V a r i a t i o n  96  . . .  . . . . . . . .  Herd V a r i a t i o n  117 119 128  D i s t r i b u t i o n of Within Herd-Period Variances A l l P o s s i b l e Samples f o r Seven Sampling Schemes - E x p e r i m e n t I M o n i t o r i n g Random S a m p l i n g  116  133 . . . . .  . . . . . . . . . . .  139 158  viii PAGE CONCLUSIONS LITERATURE CITED  164 . . . . .  168  ix  LIST OF TABLES TABLE 1.  2.  3.  4. 5A.  5B.  5C.  6A.  6B.  6C.  PAGE ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS EXPERIMENT I  30  ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS EXPERIMENT I I . . . . . . .  .  31  COMPONENTS OF WITHIN HERD-PERIOD VARIANCE (±S.E.) ESTIMATED FROM EXPERIMENT I I AND WITHIN HERD-PERIOD VARIANCE (±S.E.) ESTIMATED FROM EXPERIMENT I PERIODS ARE FIFTEEN CONSECUTIVE SHIPMENTS  .  32  . . . .  37  WITHIN HERD SERIAL CORRELATIONS FOR PERCENT MILK FAT, PROTEIN AND LACTOSE THE REDUCTION IN SUMS OF SQUARES DUE TO SUCCESSIVE TERMS IN THE POLYNOMIAL OF EQUATION 19. PERCENT MILK FAT SERIAL CORRELATIONS  38  THE REDUCTION IN SUMS OF SQUARES DUE TO SUCCESSIVE TERMS IN THE POLYNOMIAL OF EQUATION 19. PERCENT PROTEIN SERIAL CORRELATIONS .  39  THE REDUCTION IN SUMS OF SQUARES DUE TO SUCCESSIVE TERMS IN THE POLYNOMIAL OF EQUATION 19. PERCENT LACTOSE SERIAL CORRELATIONS . . . . . . . . . . . .  40  ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS EXPERIMENT I — TWO STRATA PER PERIOD . . . . . . . . . . . .  45  ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS. EXPERIMENT I — THREE STRATA PER PERIOD  46  ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS. EXPERIMENT I — FOUR STRATA PER PERIOD  47  X  TABLE 7A.  7B.  7C.  8.  9.  10.  11. 12A.  12B.  12C.  PAGE ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS.EXPERIMENT II — TWO STRATA PER PERIOD  48  ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS.EXPERIMENT II — THREE STRATA PER PERIOD  49  ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS. EXPERIMENT II — FOUR STRATA PER PERIOD  50  WITHIN HERD-PERIOD TOTAL VARIANCE FROM EXPERIMENT I AND BIOLOGICAL AND TOTAL VARIANCE FROM EXPERIMENT II WITH NO STRATA AND TWO, THREE AND FOUR STRATA FOR PERCENT MILK FAT, PROTEIN AND LACTOSE . .  51  PREDICTED VARIANCE AND 99% CONFIDENCE INTERVAL OF THE MEAN OF FRESH SAMPLES OF VARYING SIZES DRAWN FROM A PERIOD OF 15 SHIPMENTS FOR PERCENT MILK FAT, PROTEIN AND LACTOSE SIMPLE AND STRATIFIED RANDOM SAMPLING . . . . . . o . . . . .  57  ANALYSIS OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS EXPERIMENT I I I — ESTIMATE OF COMPOSITING VARIANCE . . .  60  ESTIMATES OF COMPOSITING AND TESTING VARIANCE EXPERIMENT I I I . . . . . . . . . . .  62  ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES FITTING HERDS AND PERIODS (MODEL 8) EXPERIMENT I FRESH SAMPLE ESTIMATES . . . . . .  64  ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES FITTING HERDS AND PERIODS (MODEL 8) EXPERIMENT I TWO-WEEK COMPOSITE ESTIMATES  65  ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES FITTING HERDS AND PERIODS (MODEL 8) EXPERIMENT I TWO ONE-WEEK CONPOSITE ESTIMATES . . . . .  66  xi TABLE  PAGE  13.  VARIANCE OF COMPOSITES  14.  VARIANCES OF HERD-PERIOD MEAN MILK CONSTITUENT PERCENT ESTIMATED BY TWO TWO-WEEK COMPOSITES PER PERIOD . . . . . . .  70  PAIRED t - T E S T OF DIFFERENCES BETWEEN THE FRESH ESTIMATE OF A TWO WEEK PERIOD MEAN AND BOTH KINDS OF COMPOSITE ESTIMATES . . .  71  ESTIMATES OF SAMPLE S I Z E REQUIRED I F THE VARIANCE OF THE MEAN IS TO EQUAL THE VARIANCE OF THE MEAN OF TWO TWO-WEEK COMPOSITES  76  TESTS OF NORMALITY OF THE DISTRIBUTION OF WITHIN HERD-PERIOD VARIANCES BEFORE AND AFTER LOGARITHMIC TRANSFORMATION  97  15.  16.  17.  18A.  18B.  1 9 A .  .  SIMPLE AND MULTIPLE LINEAR REGRESSION C O E F F I C I E N T S PERCENT MILK FAT WITH FOUR STRATA PER PERIOD S I M P L E  O F  T H E  ( S L R )  NO  A N D  M U L T I P L E  C O E F F I C I E N T S  L O G A R I T H M  V A R I A N C E M I L K  O F  O F  P E R C E N T  P E R C E N T  M I L K  T H E  L I N E A R  F O R  T H E  W I T H I N  P R O T E I N  F A T ,  O N  P R O T E I N  S I M P L E  A N D  S T R A T A  P E R  99  100  (MLR) R E G R E S S I O N  H E R D - P E R I O D K I L O G R A M S A N D  L A C T O S E  -  S T R A T A  107  M U L T I P L E  C O E F F I C I E N T S  2OA.  68  2  SIMPLE (SLR) AND MULTIPLE LINEAR (MLR) REGRESSION COEFFICIENTS FOR THE REGRESSION OF THE LOGARITHM OF THE WITHIN HERD-PERIOD VARIANCE OF PERCENT MILK FAT ON KILOGRAMS OF MILK, PERCENT MILK F A T , PROTEIN AND LACTOSE - NO STRATA  R E G R E S S I O N  1 9 B .  (xlO )  P E R C E N T  L I N E A R  R E G R E S S I O N  P R O T E I N  W I T H  F O U R  P E R I O D  SIMPLE (SLR) AND MULTIPLE LINEAR (MLR) REGRESSION COEFFICIENTS FOR THE REGRESSION OF THE LOGARITHM OF THE WITHIN HERD-PERIOD VARIANCE OF PERCENT LACTOSE ON KILOGRAMS MILK, PERCENT MILK FAT, PROTEIN AND LACTOSE NO STRATA  1 0 8  112  xii TABLE 20B.  21.  22.  PAGE SIMPLE AND MULTIPLE LINEAR REGRESSION COEFFICIENTS PERCENT LACTOSE WITH FOUR STRATA PER PERIOD  113  MAXIMUM VALUE OF a FOR THE PRECISION OF A RANDOM SAMPLE TO MEET THE SPECIFIED CRITERION  118  PERIOD AVERAGE WITHIN HERD-PERIOD VARIANCE Ca) OF PERCENT MILK FAT WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA  120  PERIOD AVERAGE WITHIN HERD-PERIOD VARIANCE (oh OF PERCENT PROTEIN WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA  124  PERIOD AVERAGE WITHIN HERD-PERIOD VARIANCE ( a ) OF PERCENT LACTOSE WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA . . . . . . . . .  126  HERD AVERAGE WITHIN HERD-PERIOD VARIANCE ( a ) OF PERCENT MILK FAT WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA o  129  HERD AVERAGE WITHIN HERD-PERIOD VARIANCE ( a ) OF PERCENT PROTEIN WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA  131  HERD AVERAGE WITHIN HERD-PERIOD VARIANCE OF PERCENT LACTOSE WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA . . . .  134  FREQUENCY DISTRIBUTION OF THE VARIANCE OF PERCENT MILK FAT CALCULATED WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA PER PERIOD  136  FREQUENCY DISTRIBUTION OF THE VARIANCE OF PERCENT PROTEIN CALCULATED WITHOUT STRATA AND WITH TWO, THREE AND FOUR STRATA PER PERIOD  140  w  w  23.  24.  w  25.  w  26.  w  27.  28.  29.  X l l l  TABLE 30.  31.  32A.  32B.  33A.  33B.  34.  PAGE FREQUENCY DISTRIBUTION OF THE VARIANCE OF PERCENT LACTOSE CALCULATED WITHOUT STRATA AND WITH TWO, THREE AND FOUR STRATA PER PERIOD • • • • • e o * « * e « « o o o o *  1^2  PERCENTAGE OF HERD-PERIOD SUBCLASSES PREDICTED TO MEET THE CRITERION OF PRECISION (TABLE 211 . . . . . . . .  144  FREQUENCY DISTRIBUTION OF THE ABSOLUTE DEVIATIONS OF ALL POSSIBLE SIMPLE RANDOM SAMPLES, WITH ONE TO FOUR OBSERVATIONS PER SAMPLE, FROM THE PERCENT MILK FAT FRESH MEAN  146  FREQUENCY DISTRIBUTION OF THE ABSOLUTE DEVIATIONS OF ALL POSSIBLE STRATIFIED RANDOM SAMPLES, WITH ONE OBSERVATION PER STRATA AND TWO TO FOUR STRATA, FROM THE PERCENT MILK FAT FRESH MEAN . . . . . . .  148  FREQUENCY DISTRIBUTION OF THE ABSOLUTE DEVIATIONS OF ALL POSSIBLE SIMPLE RANDOM SAMPLES, WITH ONE TO FOUR OBSERVATIONS PER SAMPLE, FROM THE PERCENT PROTEIN FRESH MEAN .  150  FREQUENCY DISTRIBUTION OF THE ABSOLUTE DEVIATIONS OF ALL POSSIBLE STRATIFIED RANDOM SAMPLES, WITH ONE OBSERVATION PER STRATA AND TWO TO FOUR STRATA, FROM THE PERCENT PROTEIN FRESH MEAN . . . . . . . .  152  99 PERCENT CONFIDENCE LIMITS OF THE DIFFERENCE BETWEEN TWO RANDOM MILK SAMPLES . . . . . .  160  . . .  xiv  LIST OF FIGURES FIGURE  PAGE  1.  S e r i a l correlations of percent milk fat, • r protein and lactose . . . . . . . . . . . . .  41  2.  Within herd-period b i o l o g i c a l and t o t a l variance estimated with no strata and with two, three and four s t r a t a per period f o r percent milk f a t protein and lactose . . . .  52  The number of samples required (n) f o r random sampling to equal the p r e c i s i o n of composite sampling f o r various r a t i o s of b i o l o g i c a l to t e s t i n g variance (r) calculated from equation 29 . . . . . . . .  78  3.  4.  5. 6. 7. 8.  9.  10.  11.  .  Period average milk constituent percentages and milk shipment weight f o r t h i r t e e n periods . . . . . . . . . .  95  Within herd-period variance of milk f a t percent f o r thirteen periods . . . . . . . .  122  Within herd-period variance of protein percent for thirteen periods . . . . . . . . . . . .  125  Within herd-period variance of lactose percent for thirteen periods . . . . . . . . . . . .  127  D i s t r i b u t i o n of the within herd-period v a r i ance of milk f a t percent (no s t r a t a and four strata)  138  D i s t r i b u t i o n of the within herd-period variance of protein percent (no s t r a t a and four strata)  141  D i s t r i b u t i o n of the within herd-period variance of lactose percent (no s t r a t a and four strata)  143  D i s t r i b u t i o n of absolute deviations of a l l possible single samples (n=l) from the fresh sample estimate-percent milk f a t and J P 3 T O t - 6 IT!  0  •  •  •  •  O  O  O  9  O  •  •  9  9  O  »  O  9  •  XV  FIGURE 12 .  13.  14.  PAGE D i s t r i b u t i o n of absolute deviations of a l l possible samples of size two (n=2) from the fresh sample estimate-percent milk f a t and protein D i s t r i b u t i o n of absolute deviations of a l l possible samples of s i z e three (n=3) from the fresh sample estimate-percent milk f a t and protein D i s t r i b u t i o n of absolute deviations of a l l possible samples of size four (n=4) from the fresh sample estimate-percent milk f a t and protein  155  155  157  xv i  ACKNOWLEDGEMENTS  The author wishes to thank Dr. R.G. Peterson, under whose supervision this study was conducted, f o r his assistance i n planning the project and i n analysing the r e s u l t s .  The author also thanks Dr. CW. Roberts and  Dr. J . Hodges f o r their suggestions and c r i t i c i s m s . Thanks are extended  to the personnel of the B r i t i s h  Columbia Department of A g r i c u l t u r e , Dairy Branch:  Mr.  T.C.T. Chao, Technical Director; Mr. G.D. Johnson, Officer-in-Charge, Dairy Laboratory; and Mr. E.N. Jenstad, Dairy S p e c i a l i s t f o r supervision of data c o l l e c t i o n and for the analyses of milk samples.  The author also  expresses his sincere thanks to his wife for her support and encouragement.  1  INTRODUCTION  The producer price f o r whole milk i s usually establ i s h e d per hundred pounds of milk of a given milk f a t (and/ or other milk constituent) percentage.  This basic price  is adjusted for deviations of the milk shipped by i n d i v i d u a l dairy farmers from the given percentage.  Therefore,  determining the percent composition of herd milk i s important i n paying producers accurately. At the present time, B r i t i s h Columbia producer milk p r i c e s are established each month by a p r i c i n g formula which includes a d i f f e r e n t i a l for deviations from the given percentage for milk f a t only.  The accounting period, i n  B r i t i s h Columbia, i s a calendar month and i t i s necessary to sample herd milk i n order to determine  the monthly  average percent milk f a t . In general three sampling schemes can be  proposed:  (1) drawing a sample from each shipment and forming a composite which i s tested after a c o l l e c t i o n period of several days (composite sampling);  (2) drawing a milk  sample from each shipment and testing the sample fresh (fresh sampling) and  (3) drawing a milk sample from  randomly selected shipments and testing fresh (random  2 sampling).  Other methods are also possible but have serious  drawbacks, for example: 1.  systematic s e l e c t i o n of shipments - such as sampling every fourth or f i f t h  shipment—can  lead to biased estimates; 2.  formation of a composite of samples from randomly selected shipments includes the disadvantages inherent i n both composite and random sampling. The f i r s t scheme i s currently used i n B r i t i s h Columbia  and the usual compositing period i s two weeks.  The main  disadvantages of composite sampling l i e i n the labor required to sample each shipment and to transfer the sample to a composite b o t t l e .  In addition the compositing procedures  and storage of the composites could introduce bias and/or v a r i a t i o n i n the test r e s u l t s . The second scheme removes the need for forming and maintaining a composite b o t t l e for each herd but i t requires the  same number of samples as the compositing method and  more laboratory analyses. of  the three schemes.  However, i t i s the most precise  The t h i r d scheme also removes the  need for compositing, i t requires fewer samples than either of  the f i r s t two methods and fewer laboratory analyses  than the second scheme but w i l l probably require more laboratory analyses than the f i r s t  method i f i t i s to be  3 as precise.  However, the advent of automatic milk analysers  has reduced the time and costs of milk analyses.  This  equipment can output the t e s t r e s u l t s on punched tape and thus f a c i l i t a t e computer handling of test information. The main costs of bulk milk sampling are due to the c o l l e c t i o n and handling of milk samples.  Estimates of herd-period  means from random samples contain v a r i a t i o n due to true differences between shipments;  t h i s i s not a source of  v a r i a t i o n i n estimates obtained from either of the f i r s t two schemes.  Therefore random sampling can not be as  precise as the second method but should y i e l d unbiased estimates of the true herd-period percent milk composition. The purpose of t h i s study was to estimate the v a r i a b i l i t y , from a l l sources, of estimates of percentages of milk f a t , protein and lactose i n bulk tank milk shipments and to consider ways of assessing the percent milk f a t , protein and lactose i n herd milk without composite Only a small amount of research has been done on  samples. sampling  and testing bulk tank milk and there i s a need f o r a thorough analysis of a l l sources of v a r i a t i o n based on more comprehensive data and longer time periods than has been done i n most reported studies.  Estimates of the variances  associated with bulk tank milk sampling and testing are needed i f the p r e c i s i o n of various sampling schemes i s to be compared.  4  LITERATURE REVIEW  In studies of the v a r i a b i l i t y of milk f a t and  total  solids content of bulk herd milk i n Scotland, O'Keeffe . [15,16] sampled each d a i l y bulk tank shipment of ten herds for twelve months.  The milk samples were analysed f o r  percent milk f a t by the Gerber method and for percent t o t a l s o l i d s by the Claesson milk testing machine.  From these  data he estimated the between d a i l y shipment within herdmonth variances as 0.0246 and 0.039 for percent milk f a t and percent t o t a l s o l i d s respectively; estimates of the within herd-year variances were 0.043 and 0.085 f o r the same two milk components r e s p e c t i v e l y . Morris et a l . [14] took biweekly milk samples from bulk milk shipments of 88 Minnesota herds for one year.  Their estimates of the  within herd-year standard deviations were; 0.227 f o r percent milk f a t , 0.181  for percent protein and 0.147  for  percent s o l i d s - n o t - f a t . Edwards and Donaldson [7] sampled d a i l y bulk tank milk shipments of thirty-two B r i t i s h herds f o r t h i r t e e n days.  The milk samples were analysed for percent milk  f a t by the Gerber method and for t o t a l s o l i d s by a gravimeter method.  The s o l i d s - n o t - f a t percentage  was  5 calculated by d i f f e r e n c e .  Their estimates of the between  shipment within herd variances for the thirteen day period were 0.0227, 0.0114, and 0.0235 for percent milk f a t , s o l i d s - n o t - f a t and t o t a l s o l i d s r e s p e c t i v e l y .  These  workers computed the difference between consecutive shipment tests and found that the majority of day-to-day differences were small, 83 percent of the differences were less than 0.19 percent milk f a t , but the largest difference was percent milk f a t .  In O'Keefee's [16] study the largest  day-to-day difference was and Donaldson [7]  0.63  1.0 percent milk f a t . Edwards  reported that the 95 percent confidence  i n t e r v a l of the difference between two randomly selected single milk samples was  ±.39 percent milk f a t . These workers  found that while there was a tendency for small herds to have greater between shipment v a r i a t i o n than large herds the differences among herds were not s i g n i f i c a n t (p_<.05) by the analysis of variance when the herds were placed into three groups on the basis of the amount of milk shipped. Herrmann and Anderson [11] i n a comprehensive study of milk f a t testing i n the U.S.A., sampled 49,117 milk shipments (two days production i n each shipment) from herds shipping to eleven d i f f e r e n t milk plants over a period of four months for most herds and over the remaining herds.  a one year period for  The milk samples were tested for  6  percent f a t by the Babcock method.  These workers estimated  that the within herd-month standard deviation of percent milk f a t was  0.146.  Boswell et a l . [3] sampled d a i l y milk  shipments from 86 herds throughout England and Wales f o r a period of one year.  The milk samples were analysed for  percent milk f a t , s o l i d s - n o t - f a t and t o t a l s o l i d s ; the average within herd-month standard deviations of these milk constituents were 0.16%, 0.081% and 0.20%  respectively.  Herrmann and Anderson [11] and Boswell et a l . [3] found that the within herd-month standard deviation of percent milk f a t was highest i n November; these two studies reported values of 0.177 for t h i s month.  percent and 0.19  percent r e s p e c t i v e l y  The lowest standard deviations occurred i n  the l a t e winter and early spring i n both studies; Herrmann and Anderson [11] reported February to be the lowest month CO.137%), while Boswell et a l . [3] found March to be the lowest month (0.14%). a secondary peak i n May  The study of Boswell et a l . [3] showed CO.17%).  O'Keeffe's  [16] study  showed the highest within herd-month variance of milk f a t percent i n May  CO.0458) with a second peak i n October  (0.0381); the lowest values occurred i n the winter months with January  (0.0085) being the lowest.  Boswell et a l . [3] reported that high within herdmonth variance of percent milk f a t was associated with small herds.  Herrmann and Anderson [11] used multiple  7 regression techniques to estimate the e f f e c t s of: of milk f a t ,  (2) amount of milk shipped,  (1) l e v e l  (3) the c o e f f i c -  ient of v a r i a t i o n of the amount of milk shipped and (4) the variance of environmental temperature  on the within  herd-month variance of percent milk f a t . The regression model accounted for a s i g n i f i c a n t t o t a l sums of squares.  (p<_.05) reduction i n the  Of the four independent  variables  used only the variance of environmental temperature  was  not a s i g n i f i c a n t  remain-  (p£.05) source of v a r i a t i o n .  The  ing independent variables were negatively associated with the herd-month variance of percent milk f a t . Herrmann and Anderson [11] found that composite  milk  samples underestimated percent milk f a t as compared to the percentage calculated from fresh milk samples. amount of bias was -.011 -.095  percent to 0.031  The  percent milk f a t but varied from  percent by milk plant; thus i n d i c a t i n g  that the amount of bias i n composite the handling of the samples.  samples depended on  Preston [17] also reported  that the percent milk f a t estimated from composite was  average  lower that the corresponding percentage  samples  calculated  from fresh samples. To estimate the components of within herd-period variance, O'Keeffe [16] drew t r i p l i c a t e milk samples from the d a i l y bulk shipments of eight herds for eight days. The milk samples were analysed i n duplicate f o r percent  8  milk f a t by the Milko-Tester Mark II and for percent t o t a l s o l i d s by the Claesson milk testing machine.  He estimated  the variances associated with sampling the bulk tank were 0.0017  and 0 . 0 0 3 2 for percent milk f a t and t o t a l s o l i d s  respectively. 0.0010  The variances associated with testing were  and 0 . 0 0 2 4 for percent of the same two milk  components r e s p e c t i v e l y ; however, O'Keeffe [ 1 6 ]  suggested  that the testing variance of percent milk f a t estimated i n t h i s study was much lower than i s usually encountered under p r a c t i c a l conditions.  In a study of bulk tank  methods, Dimick and Atherton [5] milk was  sampling  reported that bulk cooled  thoroughly mixed a f t e r three minutes of a g i t a t i o n  and therefore that the variance associated with sampling bulk tanks i s generally low i f sampling procedures are c a r e f u l l y followed; these r e s u l t s are supported by L i s k a et a l . [ 1 3 ] . fat  i n a review of automatic testing of milk for  and protein Green [ 1 0 ] reported estimates of the  standard deviations associated with testing milk samples on Infrared milk analysers (IRMA) under p r a c t i c a l laboratory conditions ranging from 0 . 0 6 to 0 . 0 9 for both percent milk f a t and percent protein.  Biggs [2]  reported that the  standard deviation between duplicates on IRMA equipment was  0 . 0 3 or less for a l l three milk components.  PART 1 ESTIMATION OF POPULATION PARAMETERS 1.  INTRODUCTION  The design of a sampling scheme to meet a s p e c i f i e d p r e c i s i o n requires knowledge of the appropriate population variances.  The purpose of Part 1 of t h i s thesis was  to  estimate the variances associated with sampling bulk tank milk shipments under various sampling schemes. sources of v a r i a t i o n are assumed;  Two main  (1) v a r i a t i o n between  the true percent composition of shipments ( i . e . sampling variance i n the s t a t i s t i c a l sense) and  (2) v a r i a t i o n  associated with the various procedures of estimation. These estimates are used to p r e d i c t standard errors of herd-period mean milk constituent percentages  estimated  under d i f f e r e n t sampling schemes and to determine  the  number of samples needed i f estimates obtained by random sampling are to equal the p r e c i s i o n of estimates obtained by composite  sampling.  10  1.  MATERIALS AND METHODS  C o l l e c t i o n and Analyses of Milk Samples Three sets of data were c o l l e c t e d for t h i s studyby drawing samples from bulk milk shipments of Fraser Valley dairy herds.  As herd milk was  shipped on alternate  days, each sample represented two days herd milk production Ofour m i l k i n g s ) . A l l milk samples were analysed by the B r i t i s h Columbia Department of Agriculture (BCDA), Dairy Branch Laboratory for percent milk f a t , protein and lactose using an Infrared Milk Analyser  (IRMA).  Milk  samples were taken by regular Tank Milk Receivers who  used  the following procedure; bulk milk was agitated for f i v e minutes and then a 100 ml. sample was drawn by taking 20 ml. of milk from each corner and from the middle of each tank.  This procedure conforms to the regulations  governing sampling of bulk milk and i s supposed to be followed by a l l Tank Milk Receivers when drawing a milk sample.  The weight of milk i n the shipment was  at the time of sampling.  recorded  The samples were maintained  on i c e u n t i l received at the milk plant.  11 Experiment I.  Three milk samples were taken from  each shipment of twenty-six herds, a l l shipping to the same milk plant, for a period of approximately thirteen months (March 14, 1970 to A p r i l 24, 1971).  Three herds stopped  shipping during the experimental period.  The set of three  milk samples was handled as follows: 1.  one sample was used i n the formation of a twoweek composite of seven fresh samples;  2.  the second sample was used i n the formation of a one-week composite of either three or four fresh samples;  3.  the t h i r d sample was analysed f r e s h .  Mercuric chloride and potassium dichromate were used a s perservatives f o r the composite samples.  The composites  were formed i n the plant receiving the milk.  The t o t a l  numbers of samples analysed were; 4,701 fresh, 697 twoweek and 1,334 one-week. Experiment I I .  Four milk samples were taken from  each bulk shipment of twenty-two d i f f e r e n t herds, a l l shipping to the same milk plant Ca d i f f e r e n t plant than the herds i n Experiment I ) , f o r a period of one month, from November 17 to December 16, 1971. were sampled per herd.  F i f t e e n shipments  Three of the four samples were  12 analysed fresh i n duplicate, with duplicates randomly assigned to analysers (1,910 analyses).  The fourth  sample was used i n Experiment I I I . Experiment I I I . Experiment II was  The fourth sample c o l l e c t e d i n  was divided into three parts and each part  used i n the formation of a composite.  Each composite  was analysed i n duplicate a f t e r a two-week c o l l e c t i o n period  (252 analyses).  Potassium  dichromate was used as  a preservative for these composites.  This set of  composites was accumulated i n the BCDA, Dairy Branch Laboratory. The t o t a l number of observations for a l l three  experiments was 8,894. The Problem and D e f i n i t i o n of Terms Used The purpose of t h i s study was  to design a random  sampling scheme to estimate, with a l e v e l of p r e c i s i o n acceptable to the dairy industry, the percent milk f a t , protein and lactose i n milk shipped by farmers during an accounting period.  Accounting periods i n B r i t i s h Columbia  are currently one month long and milk i s usually shipped on alternate days therefore a period i n t h i s study  (.unless  otherwise specified) was defined as f i f t e e n consecutive shipments.  Each herd-period of f i f t e e n shipments was  considered to be a f i n i t e population of shipments drawn  13 from an i n f i n i t e population of such herd-period populations. The word "sample" was used both i n i t s s t a t i s t i c a l sense and also to r e f e r to a small quantity of milk removed from a shipment of milk for analysis.  The meaning intended  should be clear from the context i n which the word used.  was  The term "milk constituent" was used to r e f e r to the  three main milk constituents (milk f a t , protein and lactose) only. The p r e c i s i o n of an estimate may be considered acceptable i f differences between estimates of herd-period means can be mainly a t t r i b u t e d to true differences associated with herd, period or herd-period e f f e c t s and only to a small degree be attributed to the vagaries of sampling.  I t was  assumed i n t h i s study that the p r e c i s i o n of the compositing sampling method most commonly used  (two composites  of  seven or eight shipments i n each period) was acceptable to the industry and that i n t e r e s t i n a random sampling scheme was motivated by a desire to reduce the cost involved i n sampling every shipment and i n building and storing composite  samples.  i n t h i s study was  Therefore the c r i t e r i o n of p r e c i s i o n that a random sampling scheme should  estimate herd-period means with a standard error equal to or less than the standard error of a composite  estimate.  The sample s i z e i n a sampling scheme i n which three t r a i t s are measured and the same p r e c i s i o n i s required for  14 each t r a i t ever, be  i s determined  the c r i t e r i o n  different  by  of p r e c i s i o n  the  the sampling  criterion  milk  estimated  was a  on  For  the p r e c i s i o n o f  sampling  scheme was  was  met  fat  c o n t e n t may  in  for this milk  which case  useful  to the  included  in this  two  deemed t o be  period variance  constituent.  sampling  main components o f  a herd  in  of whose by  the  in this  i f the  P r o t e i n or  study  f a t and criterion  solids-not-  included i n future milk pricing variability  formulae  would  be  and  solids-not-fat,  were  study. scheme t o meet t h e  estimates  and  specified  the v a r i a n c e of  the w i t h i n  constituent percent.  The  herd-period within  2 (a ) m e a s u r e s t h e v a r i a b i l i t y w •*  the m i l k c o n s t i t u e n t percentages from  adequate  r e q u i r e s the e s t i m a t i o n o f  v a r i a n c e of m i l k  milk  used  the o n l y e s t i m a t e  the main emphasis  design of a sampling  c u r r e n t composite  of  However,  industry, therefore, percent protein  the  criterion  satisfy  estimates of percent milk  estimates of  lactose,  The  be  could  scheme i s t o  assumed t o have b e e n a c c e p t e d  this reason  could  t h e r e f o r e the  t h e r e f o r e the p r e c i s i o n was  How-  study  constituent currently  f a t estimates  c o u l d be  industry.  i n this  f o r a l l three milk c o n s t i t u e n t s .  establishing milk prices;  precision  trait.  most p r e c i s e l y  scheme i f t h e  f a t i s the o n l y m i l k  percent milk  used  f o r e a c h m i l k c o n s t i t u e n t and  c o n s t i t u e n t w h i c h i s now determine  t h e most v a r i a b l e  of  f o r a g i v e n p e r i o d and  of  herdestimates  each shipment o f can  be w r i t t e n ,  15 a f t e r O'Keeffe [15]:  2  <T w  =  2  2  2  a' + a + a* a s t  (1  2  where  b i o l o g i c a l variance—measures to  the v a r i a b i l i t y due  true differences among shipments i n milk  constituent percentages; a  2 s  sampling variance—measures the v a r i a b i l i t y among milk samples taken from the same shipment;  2 o"  t  testing variance—measures  the v a r i a b i l i t y among  r e s u l t s of analyses (done a t d i f f e r e n t times on d i f f e r e n t analysers) on the same sample. Sampling and testing variances are due to procedures of estimation and may be combined: a  2  a  =  a + a s t 2  2  C2)  2 where  aa  i s the within shipment v a r i a n c e — d u e to both sampling and t e s t i n g .  B i o l o g i c a l variance can be a t t r i b u t e d mainly t o : 1.  day to day v a r i a b i l i t y i n both quantity and composition of milk produced by i n d i v i d u a l cowst h i s factor would give r i s e to random shipment to shipment  2.  fluctuations;  removals from or additions to the milking herd;  16 3.  changes with  i n routines  milking  and/or p e r s o n n e l  and h a n d l i n g  c h a n g e s may o c c u r  associated  the h e r d — t h e s e  r e g u l a r l y and g i v e  r i s e to  cyclic fluctuations; 4.  consistent directional percentage of a milk due  trends  w o u l d be e x p e c t e d  positive milk  to give  to i t i n time;  a function of their  diminishing  to a  important  o f any  I f this  i n another  their correlation  distance  as t h e d i s t a n c e  correlations).  expected  rise  i n one s h i p m e n t w i t h t h e  o f t h e same c o n s t i t u e n t  shipment c l o s e being  seasons—this  c o r r e l a t i o n between t h e p e r c e n t  constituent  percent  time i n t h e  c o n s t i t u e n t , w h i c h may be  t o the i n f l u e n c e o f changing  factor  a p a r t and  increases  (serial  factor i s relatively  the b i o l o g i c a l  variance  would be  t o be l o w e r i n a s h o r t p e r i o d I n which case d i v i s i o n  than i n a  long  period.  into  s u b - p e r i o d s o r s t r a t a and randomly s e l e c t i n g  shipments f o r sampling fied the  random s a m p l i n g ) w o u l d b e e x p e c t e d standard  Each milk composite;  of periods  from each s t r a t a  error of the estimated  mean a s c o m p a r e d t o s i m p l e  a  across  random  shipment i n a p e r i o d  therefore  the variance  between composite v a r i a b i l i t y ,  (stratito reduce  herd-period sampling.  i s sampled  to form  of the estimated  i s due t o w i t h i n  mean,  shipment  v a r i a b i l i t y and to v a r i a b i l i t y introduced by the procedures associated with the formation of a composite variance).  B i o l o g i c a l variance—which  (compositing  measures the  v a r i a b i l i t y among true shipment v a l u e s — i s not a component of the variance of composite estimates of a herd-period mean.  The e f f e c t s of the procedures associated with the  formation of a composite sample could lead to consistent over- or under-estimation  (bias) of herd-period means by  composite samples.  S t a t i s t i c a l Methods Estimation of within herd-period variance.  The  fresh sample data of Experiment I were used to estimate the within herd-period variances of percent milk f a t , p r o t e i n " and lactose using the analysis of variance.  The l i n e a r  mathematical model assumed was:  y  where  +  h.  i  +  (3)  U ) + w.k ( i j )  p . ,. v  3  the observed milk consistuent percent of the kth shipment i n the j * * period of the i 1  herd ; the general mean ; the e f f e c t associated with the i * - * 2  1  herd,  2  N(0,ov) , a, i s the variance among herd means;  18 >;j ( i )  *  t  1 S  f f  e  e  c  ° f t h e J*"* p e r i o d  t  h e r d , NtO,a period  i n the  1  2 2 ), a i s the variance p p  means w i t h i n  i ^  1  among  herds; 4* V*  w  strata  k(ij)  t  the  shipment w i t h i n the 2 h e r d , UtO,o^)  Ti h  jth  period  and i  the  within  herd-period  t  per period  were e s t i m a t e d  fat, protein This  variance  significant  from Experiment using  four  I for  the analysis herd-  among s t r a t a means a n d  strata variance.  be e x p e c t e d  and  partitioned the within  I f s t r a t a were  source o f v a r i a t i o n then  sampling would  i s  f o r two, three  into; the variance  residual within  2  variance.  and lactose  analysis  a  f  strata variances  variance.  period  effect of the k  i e  The w i t h i n  percent milk of  *  stratified  t o be w o r t h w h i l e .  a  random  The l i n e a r  m a t h e m a t i c a l model assumed was;  v  i j k l  =  *  +  i  h  Pj(i)  +  +  s  k(ij)  t  +  w  s  l ( i j k )  ( 4 )  where y^jkl  the observed milk it*  shipment  1  period s  t  k(ij)  t  h  e  e  f  f  period  i n the k  of the i e  c  t  o  f  constituent  t  of the i  1  k  n  strata  i n the j  t  h  strata i n the j t *  " h e r d , N(0,a  ), a St  variance and  among s t r a t a means w i t h i n  2 2 ( a . - t w o s t r a t a , a ., st s t 2 and a ,, - f o u r s t r a t a ) ;  herds  strata  S t  t  h  herd;  n  the  t  percent of the  1  i sthe SL  periods three  19 the e f f e c t of the l ^ shipment i n the f c  ws l ( i j k )  -th s t r a t a , j*-* period and i th herd, 1  2  1 : 1 : 1  2  N(0,o" ), o i s the ' ws ' ws  within strata  2  2  variance (a - two s t r a t a , a , - three ws ws 2  strata and a , , - four strata) ; ws 11  and the remaining symbols have been defined i n equation 3. Estimation of the components of within herd-period variance.  The r e p l i c a t e d  sampling and testing data  c o l l e c t e d for one period i n Experiment II was used to estimate the components, given i n equation 1, of the within herd-period variance.  The l i n e a r mathematical model  assumed was:  y. i -^ijkl  =  u + h.i + d .u/ C . ii) + s,k,( i•j ) • \ +t,,.., lCijk)  (5)  x  where ijkl  the observed milk constituent percent of the 1 ^  test on the k  shipment of the i * - * the general mean;  h.  the e f f e c t of the i  sample from the j t *  herd;  1  u  t n  t  herd, N(0,a );  h  2  the e f f e c t of the j*-* shipment of the 1  l  j Ci)  2  i  t  h  2  herd, N(0,a^),  i s the b i o l o g i c a l  variance; *kCij)  the e f f e c t of the k sample from the j 2 2 shipment of the i ^ * herd, N(0,a ) , a i s s s the sampling variance; t n  1  t  n  1  t. ,.  .  the e f f e c t of the 1  t h  t e s t on the k  t h  sample from the j t * shipment of the i t * 1  2  1  2  herd, N ( 0 , a l , a t  i s the t e s t i n g  variance. Mean squares were set equal to t h e i r expectations and the r e s u l t i n g equations were solved to obtain estimates of the components of the within herd-period variance. The within strata b i o l o g i c a l variances f o r two, three and four strata per period were estimated from Experiment  II f o r percent milk f a t , protein and lactose.  The l i n e a r mathematical model assumed was: v.... •^ljklm  =u^  + h. + s t . ,., + ds, ,. .. + t ,, ., , I D d) k ( i j ) mCijkl) x  (6)  v  '  where ^ijklm  t  *  observed milk constituent percent of  ie  the mt* t e s t on the i t * sample from the 1  1  kth shipment i n the j t * s t r a t a and the 1  it* stj ^  herd;  1  the e f f e c t of the j t * s t r a t a of the i t h 1  2  2  herd, N(.0,a . ) , a  i s the variance among 2  s t r a t a means within herds  d s  k(ij)  t  (a . - two s t r a t a , st ' 2 2 a - three strata and a .,, - four strata) st. st the kth *rh 2 shipment i2n the 2 s t r a t a and i * - herd, N ( 0 , a ) , o^ i s h  e  e  f  f  e  c  t  o  f  1 1  ds  s  the within s t r a t a b i o l o g i c a l variance 2 2 (a, - two s t r a t a , a, , - three s t r a t a ds ' ds' 2 and a . ,, - four s t r a t a ) ; 51  and the remaining symbols have been defined i n equation  5.  The difference between estimates of within herdperiod variances  (.both with and without strata) obtained  from Experiments I and II were tested by a two-tailed F-test. Estimation of compositing  variances.  The data of  Experiment I I I (.triplicate composites and duplicate tests) were used to estimate compositing  variances for percent  milk f a t , p r o t e i n and lactose using the analysis of variance.  The  y. ., , •'ijkl  l i n e a r mathematical model assumed  =  was:  \x + h. + g. ,. . + c, ,. .. + t, ,. ., »  i  H  ^ 3 Ci)  kCi})  (7)  lU;jk)  '  where i s the observed m i l k c o n s t i t u e n t p e r c e n t o f  ijkl  the l ^ h t e s t on the k*-* composite  i n the  1  compositing p e r i o d of the i * - * H h g  1  herd;  the g e n e r a l mean; the e f f e c t of the i  ±  j Ci)  t  *  1 S  e  f f  the i  t  e  c  t  o f the j***  t  *  1S  e  ^  e  herd, N ( 0 , a ) ;  h  1  2  compositing p e r i o d i n  herd, N ( 0 , a ) ,  h  2  among compositing °kCij)  t  c  t  °f ^ t  ie  a  2  i s the variance  period means within herds; k*** composite i n the 1  • i_  compositing  period of the i  herd,  j*-*  1  N(0,a ), a  i s the compositing variance;  the e f f e c t of the l composite the i ^  r  i n the j t *  t e s t on the k  n  1  compositing period and  herd, NC0,a ), 2  a , i s the testing 2  variance. The number of degrees of freedom associated with estimates of compositing variances from Experiment I I I were r e l a t i v e l y small.  Also composites were formed by the  s t a f f of the BCDA, Dairy Branch Laboratory; usually composites are formed i n the laboratories of the milk plants to which the herd milk i s shipped (as was the case i n Experiment  I ) . For these reasons estimates of compositing  variances of percent milk f a t , protein and lactose were obtained from Experiment  I by an i n d i r e c t method using  estimates of sampling and testing variances from II.  Experiment  The l i n e a r mathematical model assumed was:  y + h. + p! + r . . 1  j  ID  (8)  where the mean milk constituent percent of the ith h  e r c  j f  o  r  the j t h period, periods i n  this analysis were seven consecutive shipments (two weeks); h.  x  the e f f e c t of the i th  the e f f e c t of the j 2  seven shipment 2  period, N(0,a p , ) , ap , i s the variance among period means; r. ij  the j o i n t e f f e c t of the i, th herd and the j  r  period which includes the i n t e r a c t i o n  between the ±*-h herd and the j*"* period 1  (hp . .) and the random error ( e i - ) . k  Model 8 does not y i e l d d i r e c t estimates of the variance of composite formation nor of the variance of a composite estimate.  To estimate these variances three estimates of  the mean percent composition of seven shipments of milk were used as dependent v a r i a b l e s i n model ( 8 ) .  These were:  (a) the mean of seven fresh samples weighted by the weight of milk i n each shipment; (b) the two-week composite estimate;  (c) the mean of two one-week composites,  weighted  by the amount of milk represented by each composite.  The  difference between the r e s i d u a l v a r i a t i o n of the fresh sample mean and the r e s i d u a l s of the two kinds of composites were equated to t h e i r expectations i n order to solve for the desired estimates.  The expectations of the within  herd-period variances of the three estimates of the mean percent composition for a two week period can be w r i t t e n :  24  xf  n. s  o = xc 2  2  + i + ol n. s t  (10)  2  c  a  n. t  CT  2 .j n,+n2 n a x2c . = o% + -^v- as + 2c 2 n. 0  2  where  0  2  2  2 n.  ^ t  • (11)  2 a ^  the within herd-period variance of the mean of seven shipments each sampled and tested once;  a  c2  the within herd-period variance of a twoweek composite;  2 a  the within herd-period variance of the weighted  9  mean of two one-week composites; 2 2 o* and o"2 c  C  the variances associated with the formation of a two-week composite and two one-week composites r e s p e c t i v e l y ;  n^ and n2  the number of shipments i n each of the two one-week composites (n^ = 3 and n2 = 4);  n.  the number of shipment i n a two-week composite, (n. = n^ + xi^ = 7);  2 a  2 and a.  defined i n equation 1.  Equations  9, 10 and 11 represent the expectations  of the random error of the r e s i d u a l mean square a r i s i n g from r^j  i n model 8; therefore, the expectations of the r e s i d u a l  25  mean square for each kind of sampling may be written as follows: 1  2  .  =  a , ,... + -=<s + —^2 ph(f) n. s n. t  a  =  a , . + a + -^a + a phCc) c n. s t  rf  2  rc  _2  .  .2 a .  2  1 2  2  2  (12) '  v  (13)  2  2  2  2  2  n  1  + n  2  n•  2  n  l  + n  2  2  2  n•  where 2  o" £ r  i s the r e s i d u a l mean square of the mean of seven shipments;  2  a  i s the r e s i d u a l mean square of two-week composites;  2  a" 2 r  C  i s the r e s i d u a l mean square of the mean of two one-week composites;  2 a  ph(f) i s the variance associated with the herdperiod i n t e r a c t i o n e f f e c t f o r fresh sampling; 2  °*phCc) i s the variance associated with the herdperiod i n t e r a c t i o n e f f e c t f o r two-week composite sampling; ph(2c) 2 a , »„ . i s the variance associated with the herdperiod i n t e r a c t i o n e f f e c t f o r two oneweek composite sampling; 2  2  o" and o" defined i n equation 1. s  t  The remaining symbols and the c o e f f i c i e n t s a s s o c i ated with variances have been defined i n equations 9, 10 and 11. If the i n t e r a c t i o n variance i s assumed to be equal for a l l three kinds of sampling then the following equations hold:  For two-week composites:  a  2  rc  - a rf 2  =  a + % a c 7 t 2  (15)  2  by rearrangement and s u b s t i t u t i o n i n equation 10; 2 °xc  =  a  2 2 1 2 rc - rf 1 s a  a  +  +  1  2  T °t  <> 16  ,,^  v  For two one-week composites:  a  2  r2c  - a = rf . 2  a  2  2c  +  i49|  a  (17)  2  t  and i t follows that:  a  2  x2c  =  cr - a + % a + \ a r2c rf 7 s 7 t 2  2  2  (18)  2  2 The estimates of sampling  (a ) and t e s t i n g  2 (a )  variances obtained from the analysis of the data of Experiment I I were used to solve equations 15 to 18 for  the variance associated with the formation of composites 2  2  (a Ca  or a  ) and for the variance of a composite  9  2  estimate  2  or o , ) of the period mean. Serial correlations. The s e r i a l c o r r e l a t i o n s , r , u  of y.. with y.., xj  J  J  Xj+U  were calculated on a within herd basis;  where y.. i s the observed milk constituent percent of the j  t  n  shipment of the i ^  herd and u varies from 1 to 14.  The s e r i a l c o r r e l a t i o n c o e f f i c i e n t s were p l o t t e d on a correlogram versus u.  The r e l a t i o n s h i p between the s e r i a l  correlations and u was estimated by f i t t i n g a f i f t h degree orthogonal polynomial, a f t e r Snedecor and Cochran [18]. The mathematical model assumed was: 5  ru  =  bo +  .L  = 1  b. x u  C19)  1  where r^  i s the within herd s e r i a l c o r r e l a t i o n coefficient;  b  Q  u  the population mean when u equals zero; i s the number of shipments separating the two shipments for which r  u  was calculated  Cu = 1,14} ; b^  i s the regression c o e f f i c i e n t of r  on u . 1  u  The graph of the equation which included only those powers i n u which produced a s i g n i f i c a n t reduction i n the sums  28  of squares was plotted on the correlogram. Other s t a t i s t i c a l methods. used on the data of Experiment  Paired t - t e s t s were  I to t e s t for bias i n composite  estimates of herd-period (seven shipment periods) mean milk constituent percentage; the estimates obtained from each of the two kinds of composites were compared with the corresponding estimates obtained from fresh tests of milk samples taken from each shipment (fresh sample estimates).  Fresh  sample estimates were assumed to be the best unbiased estimates. The l e v e l of s i g n i f i c a n c e was  0 . 0 5 for a l l s t a t i s t i c a l  tests. Standard errors of estimates of components of variance were calculated by the method of Anderson and Bancroft [ 1 ] .  Standard errors of l i n e a r combinations  variances were computed after Welch [ 2 0 ] .  of  1. RESULTS AND DISCUSSION  Estimates of Within Herd-period Variance and Components The analysis of variance tables, showing the expectations of mean squares, of h i e r a r c h a l models 3 and 5 used for the analysis of Experiments Tables 1 and 2 r e s p e c t i v e l y .  I and II are presented i n  The estimates from  Experiment  II of b i o l o g i c a l , sampling and testing variances f o r a l l three milk constituents are reported i n Table 3.  The  within herd-period variances estimated from Experiment I. are shown i n column 5 of Table 3. Sampling variance.  The estimate of the sampling  variance f o r percent milk f a t was 0.00094 ± .00027 which was 7.1 percent of the t o t a l within herd-period variance. The estimates of sampling variance f o r percent protein and lactose were small and negative.  These r e s u l t s  indicated that drawing a milk sample, by the method used i n the current study, was not an important source of v a r i a t i o n f o r any of the three milk constituents. These findings agree with those of Dimick and Atherton [5] and Liska et a l . {13] who found that sampling variance was low when bulk milk was properly agitated p r i o r to taking a sample.  TABLE 1 ANALYSES  Source Herds  OF VARIANCE OF MILK CONSTITUENT PERCENTAGES BULK MILKS EXPERIMENT I DF  (h)  SS  a  MS  a  F  1127.81 180.47 37.70  45.11253 7.21873 1.50787  99.09* 38.85* 9.50*  Periods (p)/h  289  131.58 53.70 45.88  0.45528 .18581 .15877  33.20* 23.60* 29.00*  Shipments/p&h  4188  57.43 32.98 22.93  .01371 .00787 .00548  Total  4502  three values l i s t e d f o r each source lactose respectively.  * significant  source  of v a r i a t i o n .  EMS  a  25  the and  OF HERD  a  2 w  2 2 p  2 w  2 1 p  2 3 h  2  w  of v a r i a t i o n are f o r percent milk  fat, protein  TABLE 2 ANALYSES  OF VARIANCE OF MILK CONSTITUENT PERCENTAGES BULK MILKS EXPERIMENT I I „a F  DF  ss  21  158.53 51.53 10.35  7.54901 2.45382 0.49296  171.00* 113.80* 50.06*  Shipments (d)/h  300  13.24 6.47 2.95  0.04416 .02156 .00985  5.40* 15.15* 2.95*  Samples  633  5.17 0.90 2.11  .00817 .00142 .00333  1.30* .77 .84  955  6.00 1.76 3.82  .00628 .00184 .00400  Source Herds  (h)  Tests/h,  (s)/h&d  d&s  Total  a  MS  a  OF HERD  EMS  2  2  2  2  c~+k,a +k-,a*+k.at t I s 3 d 4 h  2 t  2 I s  2 2 d  oj+k..a t I s 2  1909  the t h r e e v a l u e s l i s t e d f o r each and l a c t o s e r e s p e c t i v e l y .  source o f v a r i a t i o n a r e f o r percent m i l k  f a t , protein  * significant k  x  = 2  source o f v a r i a t i o n . k  2  = 5.93  k  3  = 5.95  k  4  = 86.81 LO  TABLE 3 COMPONENTS OF WITHIN HERD-PERIOD VARIANCE (±S.E.) ESTIMATED FROM EXPERIMENT I I AND WITHIN HERD-PERIOD VARIANCE. (±S.E.) ESTIMATED FROM EXPERIMENT I PERIODS ARE F I F T E E N CONSECUTIVE SHIPMENTS  Variance  (xlO )  Milk Constituent  Biological (1)  Sampling (2)  Testing (3)  % Milk f a t  0.607±.061  0.094±.027  0.628±.029  1.329±.064  1.371+.030  Total (4)  a  Total (5)  b  %  Protein  .340±.029  -.021±.006  c  •167±.006  0.507±.031  0.787±.017  %  Lactose  .110±.014  -.033±.013  c  .373±.013  .483±.017  .5481.012  di  b  total  o f columns o n e t o t h r e e ;  within herd-period variance  2 2 2 2 i . e . a = a , + a + a. w d s t  2 (a ) e s t i m a t e d  from experiment one.  when e s t i m a t e s o f s a m p l i n g v a r i a n c e were n e g a t i v e t h e t e s t i n g v a r i a n c e was e s t i m a t e d w i t h i n c r e a s e d d e g r e e s o f f r e e d o m b y c o m b i n i n g t h e sums o f s q u a r e s f o r s a m p l i n g a n d testing.  to  33 Testing  variance.  and  their  and  l a c t o s e w e r e ; 0.00628 ±  0.00373 ± testing  standard  Estimates  errors  testing  for percent milk  [6]  (Table 3).  variances  fat, protein  .00029, 0.00167 ±  .00013 r e s p e c t i v e l y  v a r i a n c e by Dunn  of  .00006  Estimates  f o r the p e r c e n t of  and of  the  same  t h r e e m i l k c o n s t i t u e n t s ( t h e a n a l y s e s were p e r f o r m e d same l a b o r a t o r y a s  the  0.00612, 0.00631 a n d automatic  milk  variances with  analyses  i n the c u r r e n t study)  0.00505 r e s p e c t i v e l y .  a n a l y s e r s , Green  [10]  IRMA. u n d e r p r a c t i c a l  study  Estimates  for percent milk  fat closely  estimate current and  of  agreed  testing  s t u d y was  testing  f a t and  t h e r a n g e g i v e n by G r e e n milk  of  [10]  reported that  with  the e s t i m a t e  [10].  by  [6] and  the e s t i m a t e  machine to machine v a r i a t i o n represented  the  total  and  in  the  by Dunn  [6]  The  testing  The  difference variance may  from time to  laboratory conditions. c u r r e n t study  [6].  c u r r e n t study  that testing variance varies  as d e f i n e d i n the  thus  i n the  in  for percent  by Dunn  the estimate  the e s t i m a t e o f p e r c e n t p r o t e i n  under p r a c t i c a l  fell  variance for percent protein  between  indicate  milk  ( T a b l e 3)  b e l o w t h e r a n g e r e p o r t e d by G r e e n  Dunn  testing  v a r i a n c e i n the c u r r e n t  the e s t i m a t e  s m a l l e r than  of  laboratory conditions  lactose and  were;  In a review  were i n t h e r a n g e 0.0036 t o 0.0081 f o r a l l t h r e e constituents.  i n the  time  Testing variance  i n c l u d e d ; sample p r e p a r a t i o n , machine p r e c i s i o n  variance associated with  and testing  34 and sample handling procedures.  The difference between  estimates of testing variance i n the current study and the variance between duplicates on IRMA of 0.0009 reported by Biggs  [2] may be attributed to the contribution of the  factors l i s t e d above other than machine p r e c i s i o n . The testing variances were 47.2, 32.9 and 77.2 percent (calculated from Table 3) of the t o t a l within herdperiod variance f o r percent milk f a t , protein and lactose respectively.  Therefore testing was an important source  of v a r i a t i o n and consequently the number of determinations would have an important bearing on the variance of estimated period mean milk constituent percent.  The variance of the  mean percent lactose would depend mainly on the number of determinations and would be r e l a t i v e l y independent of the sampling scheme. B i o l o g i c a l variance.  Estimates of b i o l o g i c a l  variances and t h e i r standard errors were 0.00607 ±.00061, 0.00340 ± .00029 and 0.00110 + .00014 f o r percent milk f a t , protein and lactose r e s p e c t i v e l y .  The b i o l o g i c a l  variances were 45.7, 67.1 and 22.8 percent of the t o t a l within herd-period variance f o r percent milk f a t , protein and lactose r e s p e c t i v e l y . Within herd-period variance.  Estimates of the  t o t a l within herd-period variance are shown i n columns four (Experiment III and f i v e  (Experiment I) of Table 3.  These estimates from the two experiments were not s i g n i f i cantly d i f f e r e n t , by two-tailed F-tests, for percent milk f a t or percent lactose.  However, differences between  the estimates were s i g n i f i c a n t f o r percent p r o t e i n .  The  components of within herd-period variance were defined to be; b i o l o g i c a l , sampling and testing variances (equation 1).  Sampling variance was concluded to be very small  (Table 3) for a l l three milk constituents. Therefore differences between the estimates of within herd-period variance obtained from Experiments  I and II can mainly  be a t t r i b u t e d to differences i n b i o l o g i c a l variance and/or i n testing variance i n the two experiments.  If biological  variances d i f f e r i n the population, then random sampling schemes may  have to be modified f o r d i f f e r e n t herds and/or  d i f f e r e n t periods.  V a r i a b i l i t y of testing variance would  mean that the variance of estimates of herd-period mean milk constituent percentages cannot be accurately predicted for any sampling scheme.  E f f e c t s of Strata Two  shipments of milk which are close together i n  time can be expected to be more s i m i l a r i n milk constituent percent than two shipments which are more widely separated. (Materials and Methods}.  36  Serial correlation. correlations, r  u  The sets of product moment  f o r pairs of shipments u shipments apart  were calculated f o r values of u from one to fourteen on a within herd basis f o r twenty-three herds of Experiment I across thirteen periods f o r percent milk f a t , p r o t e i n and lactose.  The r e s u l t s are presented i n Table 4 and a  correlogram shown i n Figure 1. The sets of product moment c o r r e l a t i o n s were f i t t e d to equation 19.  The values of u  1  ( i = 1,5) i n equation 19  were replaced by orthogonal polynomial c o e f f i c i e n t s from Fisher and Yates  [8]. The reduction i n sums of squares was  tested as each successive term was added.  As the objective  was to f i n d the polynomial of lowest degree that was a good f i t , c a l c u l a t i o n s were stopped when two successive additions were both non-significant (Tables 5A, 5B and 5C f o r percent milk f a t , protein and lactose r e s p e c t i v e l y ) .  The c o e f f i c i e n t s  i n the r e s u l t i n g polynonomial equations were transformed to y i e l d equations expressed i n terms of u. and graphs of these equations are shown  These equations i n Figure 1.  Cochran [4] has shown that when a s e r i a l c o r r e l a t i o n e x i s t s i n a population the standard error of the mean of a sample i s reduced by using either s t r a t i f i e d random or systematic sampling techniques.  Cochran [4] also showed  that when the correlogram i s a s t r a i g h t l i n e the variance of systematic sampling was equal to the variance of a s t r a t i f i e d random sample, provided that there was no  37  TABLE 4 WITHIN HERD SERIAL CORRELATIONS FOR PERCENT MILK FAT, PROTEIN AND LACTOSE 3,  Number of Shipments Apart (u)  S e r i a l Correlations  Number of Paired Values  Milk Fat  Protein  Lactose  1 2  0.826  0.736  4227  .777  .693  0.749 .724  3 4  .742  .656 .604  .684  4168  .648  4145  .685 .663 .632  ,590 .548  .633 .631  4120 4099  .518  .597  4084  .603 .572  .479 .432  .590 .564  4057 4037  10 11 12 13  .560 .541  .396  4020 3988  .519 .502  .366 .346 .302  .531 .528  14  .478  .263  .505 .465 .438  3969 3955 3933  5 6 7 8 9  .707  the s e r i a l c o r r e l a t i o n , r within herd b a s i s .  u  of  with y ^  4196  + u  computed on a  38 TABLE 5A THE REDUCTION IN SUMS OF SQUARES DUE TO SUCCESSIVE TERMS IN THE POLYNOMIAL OF EQUATION 19. PERCENT MILK EAT SERIAL CORRELATIONS  Source Total  DF  SS  13  0 .1513275  1  .1490156  Deviations from Linear  12  .0023119  Reduction to Quadratic  1  .0019158  11  .0003961  1  .0001011  10  .0002950  Reduction to Quartic  1  .0000218  Deviations from Quartic  9  .0002732  Reduction to Linear  Deviations from Quadratic Reduction to Cubic Deviations from Cubic  s i g n i f i c a n t reduction of sums of squares.  MS  0.000193  F  773.5*  .0000360  53.2*  .0000295  3.4  .0000304  0.7  39  TABLE 5 B  THE REDUCTION IN SUMS OF SQUARES DUE TO SUCCESSIVE TERMS IN THE POLYNOMIAL OF EQUATION 19. PERCENT PROTEIN SERIAL CORRELATIONS Source  . DF  SS  13  0.2913929  1  Deviations from Linear  12  .2906398  Reduction to Quadratic  1  .0000859  11  .0006672  1  .0000107  10  .0006565  Total Reducation to Linear  Deviations from Quadratic Reduction to Cubic Deviations from Cubic  .0007531  *  s i g n i f i c a n t reduction i n sums of squares.  MS  F  0.0000628 4631.7* .0000607  1.4  .0000657  0.2  40 TABLE 5C THE REDUCTION IN SUMS OF SQUARES DUE TO SUCCESSIVE TERMS IN THE POLYNOMIAL OF EQUATION 19. PERCENT LACTOSE SERIAL CORRELATIONS Source Total  DF  SS  13  0.1130053  1  .1112329  Deviations from Linear  12  .0017724  Reduction to Quadratic  1  .0000135  11  .0017589  1  .0005543  10  .0012046  Reduction to Linear  Deviations from Quadratic Reduction to Cubic Deviations from Cubic  s i g n i f i c a n t reduction i n sums of squares.  MS  0.0001477  F  753.1*  .0001599  0.1  .0001205  4.6  Figure 1  S e r i a l correlations of percent milk f a t , protein and lactose  periodic f l u c t u a t i o n i n the population.  However, when the  correlogram i s concave upward he reported that the. variance of systematic sampling was less than the variance of s t r a t i f i e d random sampling.  When periodic v a r i a t i o n e x i s t  i n a population then the variance of systematic samples and the amount of bias i n estimates provided by systematic samples depend on the r e l a t i o n s h i p between the sampling frequency and the period of the f l u c t u a t i o n s .  Therefore,  when fluctuations of unknown or v a r i a b l e period may e x i s t i n a population, s t r a t i f i e d random sampling i s to be preferred to systematic sampling.  Cyclic fluctuations i n  milk constituent percentages may be present i n the population currently under study  (Material and Methods).  The period of these f l u c t u a t i o n s may d i f f e r between herds and also vary from time to time within a herd.  Thus  estimates of herd-period mean milk constituent percentages obtained by systematic sampling techniques could be biased; therefore, the use of systematic sampling was r e j e c t e d i n the current study. The r e l a t i o n s h i p s between the s e r i a l correlations of percent p r o t e i n and percent lactose and u were estimated as l i n e a r .  The reduction i n sums of squares due to the  l i n e a r f i t was 99.7 percent and 98.4 percent  (calculated  from Tables 5B and 5C respectively) of the t o t a l sums of squares of the s e r i a l c o r r e l a t i o n s of percent p r o t e i n  43  and percent lactose r e s p e c t i v e l y .  The r e l a t i o n s h i p between  the s e r i a l correlations of percent milk f a t and u contained a s i g n i f i c a n t contribution due to the quadratic term i n the equation.  The reduction i n sums of squares due f i t t i n g both  linear and quadratic terms was 99.7 percent of the t o t a l sums of squares of the s e r i a l correlations of percent milk f a t ; the reduction due to f i t t i n g the l i n e a r term only was 98.4 percent of the t o t a l sums of squares Table 5A).  (calculated from  Therefore, although the graph describing the  r e l a t i o n s h i p between the s e r i a l c o r r e l a t i o n s of percent milk f a t and u was concave upwards, the departures from a l i n e a r r e l a t i o n s h i p were r e l a t i v e l y small. These r e s u l t s indicated that, f o r a l l milk constituents, the s e r i a l correlations decreased r e g u l a r l y as"" u increased. Therefore, the variance of estimates of herd-period mean milk constituent percentages obtained by s t r a t i f i e d random sampling would be expected to be smaller than the variance of estimates obtained by simple random sampling.  The variance of the  estimates would be expected to be lowest, f o r s t r a t i f i e d random sampling schemes, when one observation i s taken from each strata and when a l l strata are of equal s i z e , Cochran 14].  44 Within Strata Variance Estimates of the within strata variance were obtained from Experiment I using s t a t i s t i c a l model 4.  Estimates of  the within strata b i o l o g i c a l variance were obtained from Experiment II using s t a t i s t i c a l model 6. s t r a t i f i c a t i o n of f i f t e e n shipment periods used:  Three l e v e l s of (one month) were  (a) two s t r a t a , one of seven and one of eight ship-  ments; (b) three s t r a t a of f i v e shipments each; and (c) four s t r a t a with four shipments i n three s t r a t a and three shipments i n the fourth stratum!  The analysis of variance table  showing expectations of mean squares of Experiments I with two, three and four s t r a t a are presented i n Tables 6A, 6B and 6C r e s p e c t i v e l y . The r e s u l t s for Experiment II are presented i n Tables 7A to  7C.  Table 8 shows the b i o l o g i c a l  variance f o r Experiment I I and the within herd-period variance f o r both Experiments f o r a l l three milk constituents and f o r two, three and four s t r a t a .  The e f f e c t of s t r a t a  was a s i g n i f i c a n t source of v a r i a t i o n i n a l l analyses. Therefore f i t t i n g s t r a t a reduced  the magnitude of the within  herd-period variance (within strata variance),  The r e s u l t s  were p l o t t e d i n Figure 2 f o r percent milk f a t , protein and lactose f o r both experiments.  The values p l o t t e d i n  Figure 2 f o r no s t r a t a were from Table 3.  TABLE 6A ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS EXPERIMENT I — TWO STRATA PER PERIOD a Source DF SS MS EMS 2 2 2 2 25 1127.81 Herds (h) a +k„a .+k a +k,a' 45.11253 99.09* ws 4 s t 5 p 6 h 180.47 7.21873 38.85* 37.70 1.50787 9.50* a  a  F  4  Periods (p)/h  289  131.58 53.70 45.88  0.45528 .18581 .15877  8.60* 7.39* 10.. 05*  Strata  315  16.68 7.92 4.98  .05296 .02515 .01580  5.03* 3.89* 3.41*  Shipments/h,p & s t 3873  40.75 25.06 17.95  .01052 .00647 .00464  (st)/h & p  Total  c  2 , 2 , 2 ws 2 s t 3 p  a  2  2  ws Vst  a  +  2  ws  4502  the t h r e e v a l u e s l i s t e d f o r each source o f v a r i a t i o n a r e f o r p e r c e n t m i l k f a t , p r o t e i n and l a c t o s e r e s p e c t i v e l y . *  s i g n i f i c a n t source o f v a r i a t i o n , k, = 7.08 1  k  2  0  = 7.22  k- = 14.29 3  k„ = 7.24 4  k  5  c  = 14.33  k. = 172.88 6  TABLE 6B ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS. EXPERIMENT I — THREE STRATA PER PERIOD  Source Herds  ss  DF 25  (h)  a  1127.81 180.47 ' 37.70  MS  F  a  45.11253 7.21873 1.50787  99.09* 38.85* 9.50*  Periods (p)/h  289  131.58 53.70 45.88  0.45528 .18581 .15877  11.74* 10.21* 14.02*  Strata  629  24.38 11.45 7.21  .03876 .01821 .01132  4.18* 3.01* 2.55*  3559  33.04 21.53 15.81  .00928 .00605 .00444  (st)/h & p  Shipments/h/p  & st  2 2 2 2 ws' Vst' 5VVh  a  a  a  +  ws'  2 ws'  a  + k  + k  +k  2 st' a  + k  3 p a  , 2 l st' a  , ws' 2  4502  Total  t h e t h r e e values l i s t e d f o r each source and l a c t o s e r e s p e c t i v e l y .  a  significant k, = 4.75 1  EMS  a  source k  2  0  of v a r i a t i o n are f o r percent milk  of variation.  = 4.81  k- = 14.29 3  k. = 4.82 4  k  5  c  = 14.33  k  c  6  172.88  f a t , protein  TABLE 6C ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS. EXPERIMENT I ~ FOUR STRATA PER PERIOD  Source  SS  DF  MS  a  F  3  EMS  a  25  1127.81 180.47 37.70  45.11253 7.21873 1.50787  99.09* 38.85* 9.50*  a  Periods (p)/h  289  131.58 53.70 45.88  0.45528 .18581 .15877  15.7 4* 11.15* 16.37*  a  Strata  944  27.31 15.73 9.15  .02893 .01666 .00970  3.12* 3.13* 2.28*  Shipments/h,p & st  3244  30.12 17.25 13.78  .00928 .00532 .00425  Total  4502  Herds  (h)  (st)/h & P  the t h r e e v a l u e s l i s t e d f o r each source nd l a c t o s e r e s p e c t i v e l y . significant k  x  =  3.55  source k  2  =  2  ws  2  ws  2 ws  a  ,  +  k  4 sf  1  '  +  k  2  a  1  •  +  k  l  a  ,  a  k  3  = 14.29  k  4  s f '  3.66  k  c  ='14.33  k  = 172.88  c  O  t  + k  c  3 p 0  s f i  of variation are f o r percent milk  =  1  ws' i  of variation.  3.66  ,+k ,a +k a? 5 p 6 h 2  •  1  f a t , protein  TABLE 7A ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS EXPERIMENT I I — TWO STRATA PER PERIOD  Source  DF  ss  a  MS  _a F  3  EMS  Herds (h)  21  158.53 51.53 10.35  7.54901 2.45382 0.49296  46.30* 21.33* 16.82*  Strata  22  3.59 2.53 0.64  .16304 .11505 .02931  4.69* 8.12* 3.53*  278  9.66 3.94 2.31  .03474 .01416 .00831  4.25* 9.95* 2.49*  633  5.17 0.90 2.11  .00817 .00142 .00334  1.30* .77 .84  955  6.00 1.76 3.82  .00628 .00184 .00400  (st)/h  Shipments  Samples  (ds)/h & s t  ( s ) / h , s t & ds  Tests/h,st,ds & s  Total  2 , 2 , 2 , 2 , 2 t I s 3 ds 5 s t 6 h 2 , 2 , 2 , 2 t I s 3 ds 4 s t 2 t  2 I s  2 2 ds  2 2 c:+k,cr t I s  1909  t h e three v a l u e s l i s t e d f o r each source o f v a r i a t i o n a r e f o r p e r c e n t m i l k f a t , p r o t e i n and l a c t o s e r e s p e c t i v e l y . *  a  s i g n i f i c a n t source o f v a r i a t i o n . k  x  =2  k  2  = 5.93  k =5.95 3  k = 43.16 4  k  5  =  4  3  .66  k = 86.81 g  TABLE 7B ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS EXPERIMENT I I — THREE STRATA PER PERIOD  DF  ss  Herds (h)  21  158.53 51.53 10.35  7.54901 2.45382 0.49296  75.54* 25.17* 4.97*  Strata  44  4.40 4.29 1.36  .09993 .09748 .03094  2.89* 11.45* 4.97*  256  8.85 2.18 1.59  .03456 .00851 .00622  4.23* 5.98* 1.86*  633  5.17 0.90 2.11  .00817 .00142 .00334  1.30* 0.77 0.84  955  6.00 1.76 3.82  .00628 .00184 .00400  Source  (st)/h  Shipments  Samples  (ds/h & s t  ( s ) / h , s t & ds  Tests/h,st,ds & s  Total  a  MS  F  a  EMS  a  2 2 2 V lV 3 ds' k  k  a  t  + k  l s  a  t  + k  l s  a  a  a  + k  + k  + k  3 ds' a  2 5 st' a  + k  + k  2 6 h a  4 st' a  2 ds' a  aj+k-.a t I s 2  <  1909  t h e t h r e e values l i s t e d f o r each source o f v a r i a t i o n a r e f o r p e r c e n t m i l k f a t , p r o t e i n and l a c t o s e r e s p e c t i v e l y .  a  s i g n i f i c a n t source o f v a r i a t i o n . k, = 2 1  k  2 0  = 5.93  k, = 5.95 3  k. = 28.88 4  k  5 c  = 29.06  k, = 86.81 6  ^  TABLE 7C ANALYSES  OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS. EXPERIMENT I I — FOUR STRATA PER PERIOD  DF  SS'  21  158.53 51.53 10.35  7.54901 2.45382 0.49296  80.68 38.43* 19.02*  66  6.18 4.21 1.71  .09357 .06386 .02592  3.10* 6.63* 4.88*  (ds)/h & s t  234  7.07 2.25 1,24  .03020 .00963 .00531  3.70* 6.77* 1.57*  ( s ) / h , s t & ds  633  5.17 0.90 2.11  .00817 .00142 .00334  1.30* 0.77 0.84  955  6.00 1.76 3.82  .00628 .00184 .00400  Source Herds  (h)  Strata  (st)/h  Shipment  Samples  Tests/h,st,ds  & s  Total  a  th@  and  1909  three  values  lactose  significant k  x  = 2  k  2  l i l t e d  fea?  ©aeh  §©u£@©  MS  EMS  o\+k,a +k_c , ,+k,a ., ,+k,a t I s 3 d s " 5 s t " 6 h 2  2  2  2  J  a  t  +  a  t  +  k  k  l  l  k  l  s  a  a  s  2 <  +  +  +  3  k  k  2  a  a  d s "  +  k  4  a  s f  d s "  2 a  ©f v a r i a t i e n a r © f © E p t r e t n t  s  milk  fat,  g^ettin  respectively. source = 5.93  of V a r i a t i o n . k  3  = 5,95  k  4  = 21.50  k  5  = 22.31  k  g  = 86.81  o  51 TABLE 8 WITHIN HERD-PERIOD TOTAL BIOLOGICAL AND TOTAL WITH NO STRATA AND FOR PERCENT MILK  VARIANCE FROM EXPERIMENT I AND VARIANCE FROM EXPERIMENT II TWO, THREE AND FOUR STRATA FAT, PROTEIN AND LACTOSE .  _2 (Components of Variance (xlO ) Experiment II Experiment I  Number of Strata  Biological A.  Total  Total  Milk fat: percent 0.607+.061  1.329± .064  1.371±.030  Two  .448+.051  1.170±.054  1.052±.024  Three  .445+.052  1.167±.056  0.928±.022"  Four  .372±.048  1.094± .049  .928±.023  0.340+.029  0.507±.031  0.787±.017  Two  .215±.020  .382± .021  •647±.015  Three  •119±.013  .286±.014  .605±.014  Four  .139+.015  •306±.016  .532±.013  0.110+.014  0.483±.017  0 .548±.012  Two  •084±.012  .457±.016  .464±.011  Three  .049±.010  .422±.014  .444±.011  Four  .033±.009  •406±.014  •425±.011  None  B.  Protein percent  None  C.  Lactose percent  None  52  o A  1.5  • n  2 w 2 w 2 d  from Experiment  I.  from Experiment  II  from Experiment  II  1.0 . -P cd  4-1  CN I  o  rH  r* rH •H  0.5  s  CU  cn  td -P  0.0  CO  o H  (U  1.0  •p  a  CD d  -p  -P U)  c: o o  •H  CD •P  0.5  0  U Pi  M  rH -r(  0.0  e  MH  1.0  0  CD  O  c: •rl  rd  CD 01  rd  •P  >  0  0.5  O  rd  a  0.0  T"  0 Figure  2  2 Number o f  3 strata  • 1  4  W i t h i n h e r d - p e r i o d b i o l o g i c a l and t o t a l v a r i a n c e e s t i m a t e d w i t h no s t r a t a and w i t h two, t h r e e and four s t r a t a per p e r i o d f o r percent m i l k f a t p r o t e i n and l a c t o s e  53 Within strata variance of percent milk f a t . estimates of within herd-period v a r i a n c e — n o  The  strata—of  percent milk f a t obtained from the two experiments were not s i g n i f i c a n t l y d i f f e r e n t by an F-test.  However the  estimates of within herd-period variance of percent milk f a t calculated within strata from Experiment  (within strata variance)  I were s i g n i f i c a n t l y lower than the  estimates from Experiment  II.  S t r a t i f i c a t i o n i s expected  to reduce the e f f e c t of time trends on the magnitude of the within herd-period variance (Materials and Methods). Differences between estimates of herd-period variance obtained from the two experiments  can be a t t r i b u t e d to  either differences between b i o l o g i c a l variances or to differences between testing variances i n the two sets of data.  S t r a t i f i c a t i o n would be expected to reduce  b i o l o g i c a l variance only:  testing variance (within  shipment variation) would not be a l t e r e d by s t r a t i f i c a t i o n . Therefore, the-results indicated that time trends (averaged across thirteen periods) i n Experiment  I may have  been a more important source of v a r i a t i o n than time trends i n the single period i n Experiment  II.  Thus d i r e c t i o n a l  changes i n milk constituent percentages may i n some periods (seasons) than i n others.  be greater In which case,  unless s t r a t i f i c a t i o n can e f f e c t i v e l y s t a b i l i z e within herd-period variances, i t may  be necessary to take more  54 milk samples i n some seasons than i n other seasons i f the same l e v e l of p r e c i s i o n i s to be achieved f o r a l l seasons. Within s t r a t a variances of percent protein .  The  estimates of the within herd-period variances, with and without s t r a t i f i c a t i o n , f o r percent protein obtained from Experiment  I were a l l s i g n i f i c a n t l y lower than the  corresponding estimates obtained from Experiment 8 and Figure 2} by an F-test.  II (Table  The difference between the  estimates was r e l a t i v e l y constant f o r a l l l e v e l s of stratification.  The r e s u l t s indicated, by use of the same  reasoning that was applied to the r e s u l t s f o r the within herd-period variance of milk f a t percentage, that testing variance was d i f f e r e n t between the two data sets.  If  testing variance changes from time to time then predictions of the variance of herd-period mean milk constituent percentages cannot be made accurately.  However, v a r i a b i l i t y  of testing variance would a f f e c t the p r e c i s i o n of a l l milk sampling schemes; although, the magnitude of the change i n p r e c i s i o n may not be the same f o r a l l schemes.  The  difference between b i o l o g i c a l variance and within herdperiod variance was assumed to be equal to t e s t i n g (and sampling!variance,  equation 1 .  I f b i o l o g i c a l variances  were approximately the same i n the two data sets, then testing variance i n Experiment  I could be approximated  by the difference between b i o l o g i c a l variance estimated  55 from Experiment Experiment  I.  II and within herd-period variance from The estimate of testing variance f o r  percent protein i n Experiment of these differences was  I obtained by the average  0.00440 (calculated from  Table 8). Within strata variance of percent lactose.  The  differences between the estimates of within herd-period variances, with and without s t r a t i f i c a t i o n , f o r percent lactose obtained from Experiments  I and II were non-  significant.  V a r i a b i l i t y of Estimates from Various Sampling Schemes 2  The variance of the mean (a-) estimated by drawing a simple random sample of n shipments from a period of N shipments can be w r i t t e n : ax 2  =  i ta (l-£)+ a + n d N s 2  2  a] t  (20)  2  where the symbols have been defined i n equation 1.  The  variance of the mean estimated by a s t r a t i f i e d random sample can be written:  (21)  where VL . VI.  the weight a t t a c h e d t o the i  1  s t r a t a and i s  equal to the number o f shipments i n the i s t r a t a d i v i d e d by the t o t a l number o f shipments i n the p e r i o d ; m  the number o f s t r a t a i n the p e r i o d ;  n^  the number o f o b s e r v a t i o n s drawn from the i * " *  1  strata; the number of shipments i n the i * - *  1  strata;  2  the w i t h i n s t r a t a b i o l o g i c a l v a r i a n c e , which i s assumed equal f o r a l l s t r a t a . With e q u a l s t r a t a s i z e and equal number o f o b s e r v a t i o n s per stratum,  af x  e q u a t i o n 2 1 reduces tos  =  4 n m  [a  2  ds  Cl " 2 l E ) + a  N  2  3  + a ) 2  t  (22)  where n' The  i s the number o f o b s e r v a t i o n s p e r stratum.  other symbols remain as p r e v i o u s l y d e f i n e d . The e s t i m a t e s o f the v a r i a n c e s o b t a i n e d from t h e  a n a l y s e s o f Experiment I I were s u b s t i t u t e d i n t o the a p p r o p r i a t e equations and  to calculate  the v a r i a n c e o f the mean  the 99% c o n f i d e n c e i n t e r v a l about the mean f o r v a r i o u s  sampling  schemes (Table 9 ) . The r e s u l t s  showed t h a t the  r e d u c t i o n i n the c o n f i d e n c e l i m i t s by s t r a t i f i c a t i o n was  57 TABLE 9 PREDICTED VARIANCE AND 99% CONFIDENCE INTERVAL OF THE MEAN OF FRESH SAMPLES OF VARYING SIZES DRAWN FROM A PERIOD OF 15 SHIPMENTS FOR PERCENT MILK FAT, PROTEIN AND LACTOSE SIMPLE AND STRATIFIED RANDOM SAMPLING Number  3  % Milk Fat 2b 99%CL O"-  % Protein 2b o99%CL  C  X  Simple random sampling  A. 1 2 3 4 5 6 7 8 9 15 B. 2 4 6 8 3 6 9 4 8  X  C  % Lactose b o99%CL 2  X  + .180 + .124 + .099 + .083 + .072 + .064 + .058 + .052 + .047 + .027  0 .475 .234 .154 .113 .089 .073 .062 .053 .046 .025  ±.178 ±.125 ±.104 ±.087 ±.077 ±.070 ±.064 ±.059 ±.056 ±.041  S t r a t i f i e d random sampling Two Strata 0.558 + .193 0.178 + .109 .264 .082 + .074 + .133 .182 + .110 .05 0 + .057 .126 ± .092 .034 ± .047 Three Strata  0 .224 .109 .071 .052  ±.122 ± .085 ±.069 ±.059  1.289 0.624 .403 .292 .225 .181 .149 .126 .107 .048  0.359 .165 .100 0.254 .114  ± .293 ±.203 + .164 ±.139 ±.122 + .111 ±.100 ±.092 + .084 ±.057  0.484 .231 .146 .104 .079 .062 .050 .041 .034 .011  C  ± .154 ± .105 ± .082  0.087 .036 .024  + .076 + .049 + .040  0 .137 .067 .044  ±.096 ±.067 ± .054  + .130 ±.087  Four Strata 0 .068 ± .067 .030 ±.044  0 .101 .049  ± .082 ± .057  number of samples per period f o r both simple and s t r a t i f i e d random sampling  a  variance of the mean xlO 99% confidence i n t e r v a l of the mean.  58  r e l a t i v e l y small and diminished as n increased. duction i n the confidence  i n t e r v a l s was  The  greatest for  percent protein and l e a s t for percent lactose.  These  r e s u l t s can be a t t r i b u t e d mainly to two f a c t o r s . b i o l o g i c a l variance  re-  Firstly,  (between shipment variation) was  the  only component of the within herd-period variance that could be expected to be reduced by s t r a t i f i c a t i o n ; sampling and t e s t i n g variances not be a l t e r e d .  (within shipment variation) would  Therefore, s t r a t i f i c a t i o n would be  expected to reduce the confidence  i n t e r v a l s to a greater  extent for those milk constituents for which b i o l o g i c a l variance was variance.  a major component of the within  herd-period  Secondly, the f i n i t e population c o r r e c t i o n  factor applied only to the b i o l o g i c a l variance therefore the contribution of b i o l o g i c a l variance to the  standard  error of the mean would be reduced more r a p i d l y as sample s i z e increased than the contribution of sampling and testing variances.  Thus for r e l a t i v e l y large n the  contribution of b i o l o g i c a l variance to the standard error would be small and therefore the e f f e c t of any i n the magnitude of b i o l o g i c a l variance by  reduction  stratification  on the standard error would diminish as n increased. S t r a t i f i c a t i o n could s t i l l be worthwhile i f i t resulted in a reduction i n the frequency of large deviations from the true mean by eliminating the p r o b a b i l i t y of drawing  a l l observations from either the beginning or the end of a period.  Although large deviations may occur with  r e l a t i v e l y low frequency their occurrance could be of concern to the i n d i v i d u a l milk producer as h i s payment f o r the period's milk shipments are based on the r e s u l t s of the estimate of the mean percent milk f a t .  Composite  Sampling  Variance of composites—Experiment  III.  The c r i t e r i o n  of p r e c i s i o n i n the current study was that a random sampling scheme should estimate herd period means at l e a s t as prec i s e l y as two-week composite  sampling.  Experiment  I I I was  designed to provide estimates of the standard error of herdperiod means estimated by the mean of two two-week composites; one of seven shipments and one of eight shipments. Each shipment of milk was sampled i n the formation of composites;  therefore, the variance of a composite  estimate was e n t i r e l y a t t r i b u t a b l e to procedures of estimation; of a composite  C l ) sampling, sample.  (2) testing and C3) formation  The data of Experiment  I I I were  analysed by s t a t i s t i c a l model 7 to obtain estimates of the variance associated with the formation of composite  samples.  The analyses of variance table showing the expectation of mean squares i s presented i n Table 10.  Compositing was  not a s i g n i f i c a n t source of v a r i a t i o n f o r any of the three  TABLE 10 ANALYSIS OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS EXPERIMENT I I I : ESTIMATE OF COMPOSITING VARIANCE  Source  MS  DF  ss  Herds Ch)  20  20.7105 7.2451 1.3535  1.03553 0.36225 .06768  48.40* 22.08* 6.93*  Periods Cg)/h  21  0.4493 .3445 .2050  .02140 .01641 .00976  15.41* 3.08* 4.53*  Composites (c)/h & g  84  .1167 .4474 .1812  .00139 .00533 .00216  1.14 0.91 .47  Tests/h,g & c  126  .1530 ,7343 .5808  .00121 .00583 .00461  Total  251  a  a  F  EMS  a  2 2 2 2 cf+2a^+6a^+12a^ t e g h a +2a +6a t e g 2  2  2  the three values l i s t e d for each source of v a r i a t i o n are f o r percent milk f a t , protein and lactose respectively. s i g n i f i c a n t source of v a r i a t i o n .  o  61 milk  constituents studied.  testing  variance  e x p e c t a t i o n s and s o l v i n g  formation for  t h e mean s q u a r e s the r e s u l t i n g  of a composite  s a m p l e was  f a t , a n d was  (-.001226  obtained that  of testing  Experiment  significantly  support  large i n absolute  I I I (Table 11).  lower  from  estimates F - t e s t s showed  variance f o r percent i n Experiment  significantly  higher.  I I I then i n  l a c t o s e were n o t  the c o n c l u s i o n s , based  testing  These  significantly results  on t h e c o m p a r i s o n o f t h e e s t i -  mates o f w i t h i n h e r d - p e r i o d v a r i a n c e s o b t a i n e d  from  I and I I , t h a t t h e t e s t i n g v a r i a n c e f o r p e r c e n t vary  III  indicated  a l s o may variances  from  vary from  time  to time.  that testing from  time  The r e s u l t s  from  Experiments  protein  Experiment  variance f o r percent milk f a t  to time.  Experiment  milk  The e s t i m a t e s o f  b e t w e e n t h e two e x p e r i m e n t s .  may  ±  f o r percent  variance obtained  variance f o r percent  different  The e s t i m a t e  .00013)  (-.000251  I I ; the estimate of percent p r o t e i n  v a r i a n c e was testing  (0.000087 ±  I I ( T a b l e 3) w e r e compared w i t h  from  Experiment  low  relatively  the estimate of t e s t i n g  f a t was  equations.  ± .000332).  Estimates Experiment  to their  low a n d n e g a t i v e  f o r percent protein.  l a c t o s e w h i l e n e g a t i v e was value  and  estimate of the variance a s s o c i a t e d with the  percent milk  .00054 6)  of compositing  ( T a b l e 11) f o r a l l t h r e e m i l k c o n s t i t u e n t s  were o b t a i n e d by e q u a t i n g  The  Estimates  The e s t i m a t e s  I I were b a s e d  of  on a n a l y s e s  testing done  62  TABLE 11 ESTIMATES OF COMPOSITING AND TESTING VARIANCE EXPERIMENT I I I  Variance (xlO Milk Constituent  Compositing  Testing  % Milk f a t  0.0087 ± .0130  0.1214 ± .0152  % Protein  -.0251 ± .0546  .5627 ± .0547  % Lactose  -.1226 ± .0332  .3629 ± .0352  63 over a period of one month. i n Experiment  The analyses of the composite  I I I were done on two days (two weeks apart)  in the same month as the analyses f o r Experiment I I . Therefore, testing variances would appear to be subject to considerable short-term f l u c t u a t i o n s . Variance of composites  —  from the analyses of Experiment formation of composites v a r i a t i o n of composite percent composition.  Experiment  I.  The r e s u l t s  I I I indicated that the  i s not an important source of sample estimates of the period mean  However, as the number of degrees of  freedom associated with these estimates was r e l a t i v e l y low the data from Experiment model 8.  I were analysed by s t a t i s t i c a l  The residuals from these analyses were equated  to t h e i r expectations to y i e l d estimates of the variance of composite  formation and the variance of a composite  estimate as shown i n equations 9 to 18 ( S t a t i s t i c a l Methods) . The analysis of variance tables (model 8) f o r fresh sample, two-week composite and two one-week  composite  estimates are presented i n Tables 12A, 12B and 12C respectively.  The estimates of sampling and testing variance  (Table 3) and the r e s i d u a l mean squares  (Tables 12A, 12B,  and 12C) were used to solve equations 15 through 18 f o r the variance of composite estimates and the variance  64  TABLE 12A ANALYSIS OF VARIANCE OF MILK CONSTITUENT PERCENTAGES FITTING HERDS AND PERIODS (MODEL 8) EXPERIMENT I FRESH SAMPLE EXTIMATES  Source  MS a  Fa  DF  ss  Herds  25  151.938 23.982 4.955  6.07751 0.95930 .19821  393.3* 110.1* 55.3*  Periods  25  11.579 4.158 5.241  .46316 .16632 .20962  30.0* 19.1* 58.5*  Residual  564  8.714 4.915 2.023  .01545 .00871 .00359  Total  614  a  the t h r e e v a l u e s l i s t e d f o r each source of v a r i a t i o n are for p e r c e n t m i l k f a t , p r o t e i n and l a c t o s e r e s p e c t i v e l y * significant  source of  variation.  65 TABLE 12B ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES FITTING HERDS AND PERIODS (MODEL 8) EXPERIMENT I TWO-WEEK COMPOSITE ESTIMATES DF  ss  Herds  25  139.792 23.808 4.304  5.59169 0.95233 .17216  256.9* 89.1* 24.7*  Periods  25  11.190 5.958 4.623  .44760 .23831 .18492  20.6* 22.3* 26.5*  Residual  564  12.275 6.024 3.931  0.02176 0.01068 .00697  Total  614  Source  a  MS  a  F  a  the three values l i s t e d f o r each source of v a r i a t i o n are for percent milk f a t , protein and lactose r e s p e c t i v e l y . s i g n i f i c a n t source of v a r i a t i o n .  66  TABLE 12C ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES FITTING HERDS AND PERIODS (MODEL 8) EXPERIMENT I TWO ONE-WEEK COMPOSITE ESTIMATES DF  ss  Herds  25  143.551 25.356 5.138  5.74205 0.97424 0.20552  311.6* 96.4* 39.6*  Periods  25  9.601 4.791 4.513  0.38404 0.19164 0.18052  20.8* 20.0* 34.8*  Residual  564  10.394 5.700 2.924  0.01843 0.01011 0.00518  Total  614  Source  a  MS  a  F  a  the three values l i s t e d f o r each source of v a r i a t i o n are for percent milk f a t , protein and lactose r e s p e c t i v e l y . s i g n i f i c a n t source of v a r i a t i o n .  associated with the formation of a composite both types of composite  sample f o r  samples used i n Experiment  I . The  estimates of compositing variances f o r a l l three milk constituents (Table 13) were smaller, or only s l i g h t l y larger, than their standard errors f o r both two-week composites  and two one-week composites.  These r e s u l t s  support the conclusions, based on Experiment the formation of a composite of v a r i a t i o n .  I I I , that  i s not an important source  I f sampling and compositing variances  are both small then testing was the most important source of v a r i a t i o n of composite  estimates  (equation 10).  C a l c u l a t i o n of the C r i t e r i o n of P r e c i s i o n The estimates of the variance of a composite  for a  two-week compositing period (Table 13) were used to c a l c u l a t e 2 the variance of the mean of two two-week composites  (a- ).  The equation can be w r i t t e n : m  n. 2 „ - i - o i=l N i  4  =  m  i s the number of composite  0  x c  2  (23)  2  x  c  where samples i n a period  (month) ; n. l  i s the number of shipments i n the i * * ing period;  n  composit-  68  TABLE 13 VARIANCE OF COMPOSITES  Two-week Milk Constituent % Milk f a t  CxlO" ) 2  Two One-week 2 x2c C  xc  c  °2c  0.735±.151  0.093±.195  0.401±.137  0.067±.140  % Protein  .221±.078  .054±.081  .164±.076  .079±.077  % Lactose  .410± .048  •018±.041  .212±.035  .022±.036  variance of a seven shipment composite Cfor a two week period). variance associated with the formation of a two week composite. variance of the mean (for a two week period) of two composites of three or four shipments each. variance associated with the formation of two one-week composites.  N  i s the number of shipments i n a period (month) ;  a _ 2  c  variance of the i ^ h composite,  (which i s defined  in equation 10 f o r two-week composites). The variance of the mean of two two-week composite are presented i n Table 14.  These values were the  maximum variances of the means of random samples allowable i f the c r i t e r i o n of p r e c i s i o n was to be met.  Composite Sampling versus Random . Accuracy of composites. t e s t the accuracy of composite  Sampling Experiment  sampling.  I was used to Three estimates  of herd mean milk constituent percentages f o r each twoweek period were obtained from Experiment  I.  The best  unbiased estimate of each herd two-week mean was considered to be the mean of the fresh samples, weighted by the amount of milk i n each shipment (fresh sample estimates). Paired t-tests were used to t e s t differences between fresh sample estimates and means estimated by; value of a two-week composite  (a) the observed  and (b) the mean of two one-  week composites weighted by the amount of milk represented by each  composite. The r e s u l t s  (Table 15) indicated that percent milk  f a t was s i g n i f i c a n t l y underestimated by both types of composites;  the difference between fresh and both types  70 TABLE 14 VARIANCES OF HERD-PERIOD MEAN MILK CONSTITUENT PERCENT ESTIMATED BY TWO TWO-WEEK COMPOSITES PER PERIOD  Milk Constituent  _2 Variance (xlO ) of Estimates of Herd-Period Means  %  Milk f a t  0.3675  %  Protein  0.1105  %  Lactose  0.2050  TABLE 15 PAIRED t-TEST OF DIFFERENCES BETWEEN THE F R E S H ESTIMATE OF A TWO WEEK PERIOD MEAN AND BOTH KINDS OF COMPOSITE ESTIMATES 3  F r e s h v s . Two-week Milk Constituent  Diff.  S.D.  b  Fresh t  Diff.  v s . Two One Week S.D.  b  t  % Milk f a t  -.045  0.113  -10.6*  -.045  0.080  % Protein  0.023  .085  7.0*  0.003  .060  1.3  % Lactose  .010  .064  4.2*  .010  .057  4.3*  DF  69 4  mean o f f r e s h s a m p l e s b  mean d i f f e r e n c e :  from seven c o n s e c u t i v e  c o m p o s i t e s minus f r e s h ,  significant difference.  6 56  shipments.  -14.5*  of composite  estimates was  0.045 percent milk f a t .  protein was overestimated by two-week composites difference was composite  Percent  (the  0.025 percent p r o t e i n ) , but two one-week  estimates were not s i g n i f i c a n t l y d i f f e r e n t from  fresh estimates.  Both types of composites  percent lactose by 0.010  percent lactose.  overestimated Herrmann  and Anderson [11] and Preston [17] also reported that percent milk f a t was  lower i n composite  samples than  fresh samples. Estimation of sample s i z e . the  current study was  The c r i t e r i o n used i n  that the standard error of an  estimate from a random sample should be at l e a s t as low as the standard error of the estimate from composite currently i n use.  samples  The number of random samples (n) required  to give a predetermined variance of the mean can be found by rearranging equation 20 to y i e l d ;  n  =  a / (a- + a ) w ' x d N 2  2  (24)  where 2 a-  i s the predetermined variance of the mean;  N  i s the number of shipments i n the period;  and the remaining symbols have been defined i n equation 1. This equation also holds for s t r a t i f i e d random sampling i f the s t r a t a s i z e are equal, the sampling f r a c t i o n i s the  same f o r a l l strata and a, i s defined as the within strata d b i o l o g i c a l variance. 2 The appropriate predetermined  variances (a-) i n the  current study f o r each milk constituent were the variances of the mean milk constituent percentages, f o r a period of fifteen  (N) shipments, estimated by two two-week composite  samples (Table 14). The numbers of random samples required per month f o r each milk constituent were calculated from 2 equation 24 by using these variances as a- and the X  2 estimates of b i o l o g i c a l variances (o^) 2  and within herd-  period variances (a^) from Experiment  I I (Table 3). The  r e s u l t s of the c a l c u l a t i o n s are presented i n Table 16. The c r i t e r i o n s p e c i f i e d that random sampling should be a t l e a s t as precise as composite  sampling; therefore, as the  number of samples has to be a whole number, the values i n Table 16 should be increased to the next whole number. This c a l c u l a t i o n showed that four simple random samples per period would be predicted to estimate herd-period mean milk constituent percentages with a variance less than the variance of current estimates which are based on two two-week composites per period. The variance of composite  estimates i s due e n t i r e l y  to procedures of estimation; i . e . compositing, and t e s t i n g .  sampling  The proceeding analyses indicate that  testing variance i s the most important  (Tables 3 and 11)  74  of the three.  The variance of estimates based on random  sampling are due to both procedures of estimation (sampling and testing) and to b i o l o g i c a l or day to day differences of true shipment means.  Therefore the number of random  samples required to give an estimate of the mean with a p r e c i s i o n equal to that of composite  samples depends on  the r e l a t i o n s h i p between b i o l o g i c a l variance and procedural variance.  An expression defining t h i s r e l a t i o n s h i p  can be derived by equating the expectations of the variance of a composite  estimate to the expectations of the variance  of the mean of a random sample of n  shipments.  The variance of the mean of m composites  collected  over a period of N shipments with each composite representing the same number of shipments (N/m) can be w r i t t e n :  axc 2  =  i  m  (a + a ) + c t 2  (25)  2  N  Equating t h i s equation to equation 20 and rearranging yields:  n  =  m(r + 1)/{1 + ^ a  [ a + a (| - 1)]+ 2  2  (26)  a  where n  i s the number of random samples required to give an estimate of the mean with p r e c i s i o n equal to that from a composite  scheme;  N  i s the number of shipments i n the period f o r which an estimate of the mean i s desired;  m  i s the number of compositing periods i n the current composite scheme;  a  2  i s the variance associated with the formation  c  of a composite; r  i s the r a t i o  2  2  a, / a ; d a  and the remaining symbols have been defined i n equations 1 and 2. The term:  g2  C  S  a  N  i n the denominator of equation 26 reduces to zero i f :  a  =  2  (1 - | )  a\  (28)  and i s near zero i f both a2 and a2 are small r e l a t i v e to c s 2  a , that i s , i f the main procedural source of v a r i a t i o n t  is testing.  I f the term shown i n formula 27 can be  assumed to be very close to or equal to zero then equation 26 reduces t o : n  =  mCr + 1) / (1 + g£)  (29)  76  TABLE 16 ESTIMATES OF SAMPLE SIZE REQUIRED IF THE VARIANCE OF THE MEAN IS TO EQUAL THE VARIANCE OF THE MEAN OF TWO TWO-WEEK COMPOSITES  Method  a  3  Percent Milk f a t  Percent Protein  Percent Lactose  A  3.26  3.81  2.27  B  3.31  4.78  2.49  Method.  A  Calculated from formula 24, with  experiment two. calculated from formula 29.  2  2  and o ^ estimated from  77  If  2  ac  is  large  the use of the s i m p l i f i e d e q u a t i o n w i l l  2 tend to o v e r e s t i m a t e  n.  If  ag  is  large  then n w i l l  be u n d e r e s t i m a t e d .  The advantage of u s i n g e q u a t i o n 29  r a t h e r than e q u a t i o n 24 i s  t h a t o n l y the r a t i o of  biological  sampling v a r i a n c e need to be known  to t e s t i n g p l u s  or e s t i m a t e d  i n o r d e r to c a l c u l a t e  the number of random  samples needed to r e p l a c e a c o m p o s i t i n g scheme w i t h a random sampling scheme of e q u a l p r e c i s i o n . of  Estimates  sample s i z e c a l c u l a t e d by e q u a t i o n 2 9 are  presented  i n T a b l e 16 and agree w e l l w i t h those c a l c u l a t e d by e q u a t i o n 24. The number of samples r e q u i r e d were graphed  (Figure 3) versus  (n i n formula 29)  the r a t i o of b i o l o g i c a l  to t e s t i n g variance  (r i n e q u a t i o n 29) f o r two,  and  per p e r i o d .  four composites  to f i n d  The graph can be used  the number of samples r e q u i r e d  values of r for three compositing  three  (n)  schemes.  for  various  2  T"  T"  1  2  T  3 Ratio  Figure  3  T  4  T  5  l  I  6  of b i o l o g i c a l  8 to t e s t i n g  9  10  variances  11  12  13  —r  —I  14  15  (r)  The number o f s a m p l e s r e q u i r e d (n) f o r random s a m p l i n g t o e q u a l t h e p r e c i s i o n of composite sampling f o r v a r i o u s r a t i o s of b i o l o g i c a l to t e s t i n g v a r i a n c e (r) c a l c u l a t e d f r o m e q u a t i o n 29 CO  79  1. CONCLUSIONS  Estimates of sampling variance f o r a l l three milk constituents were very small r e l a t i v e to the t o t a l within herd-period variance of milk constituent percentages. From these r e s u l t s i t can be concluded that the method of sampling bulk milk used i n t h i s study introduced l i t t l e v a r i a t i o n into estimates of milk constituent percentages of bulk milk. Estimates of compositing variances were also small for a l l three milk constituents. As both compositing and sampling variances were concluded to be small, then it  follows  i n composite  that testing i s the main source of v a r i a t i o n sample estimates of herd-period mean milk  constituent percentages. Estimates of testing variances f o r percent milk f a t and percent protein obtained from Experiment  II were  s i g n i f i c a n t l y d i f f e r e n t from the corresponding estimates obtained from Experiment  I I I . Therefore i t can be con-  cluded that testing variances vary from time to time i n the laboratory.  I f t h i s conclusion i s true then s t a t i s -  t i c a l l y v a l i d predictions of the variances of estimates, obtained from any sampling scheme, of herd-period mean milk constituent percentages cannot be made.  However,  80  p r a c t i c a l considerations demand that reasonable l i m i t s be placed on the magnitudes of testing variances so that the v a r i a b i l i t y of estimates obtained from various sampling schemes can be at l e a s t approximated.  More  data would be required to estimate the amount of v a r i a t i o n i n testing variances. The analyses indicated that b i o l o g i c a l variances may vary from time to time or from herd to herd.  Varia-  t i o n i n b i o l o g i c a l variances would mean that a random sampling scheme may have to be modified f o r d i f f e r e n t seasons or herds.  However, i t may be possible to associate  differences i n b i o l o g i c a l variances with seasons or with variables associated with herds  (e.g. quantity of milk  shipped) and thereby simplify the modification of random sampling schemes to s u i t d i f f e r e n t herds or seasons. Two-week composite  samples were concluded to y i e l d  biased estimates of a l l three milk constituent percentages. One-week composites were biased estimates of percent milk f a t and lactose.  Random sampling would be expected to  y i e l d unbiased estimates.  Therefore, i t was concluded  that estimates of herd-period mean milk constituent percentages obtained from sampling four randomly selected shipments would be at least as precise as, and more accurate (unbiased) than, estimates obtained from two twoweek composites.  I t was also concluded that s t r a t i f i e d  random sampling  (with  one sample per strata) would  reduce the v a r i a b i l i t y of these estimates. testing and b i o l o g i c a l clusion apply to the  As both  variances may vary, these conaverage  condition and may  be v a l i d for a l l herds or periods.  not  82  PART 2 2.  INTRODUCTION  Estimates of population variances associated with bulk milk sampling and testing were obtained i n Part 1. These estimates were used to p r e d i c t the v a r i a b i l i t y of estimates of herd-period mean milk constituent percentages under various sampling schemes.  Part 2 of t h i s thesis  investigated some of the p r a c t i c a l problems associated with random sampling schemes. The r e s u l t s presented i n Part 1 indicated that the variance of estimates of herd-period mean milk constituent percentages obtained by random sampling would be expected to be no greater than the variance of estimates obtained by composite  sampling i f four random milk samples were taken  each period.  However, the variances of estimates obtained  by each of these two sampling schemes were a t t r i b u t e d to d i f f e r e n t sources. composite  The variance of estimates obtained by  sampling was a t t r i b u t e d to procedures of  estimation (sampling, testing and compositing).  The  variance of estimates obtained by random sampling was a t t r i b u t e d to true differences among shipments (biol o g i c a l variance) and to procedures of estimation (sampling and t e s t i n g ) .  83  The magnitude of variances associated with procedures of estimation may vary from time to time (Part 1 ) but would be expected to be e s s e n t i a l l y the same for  a l l herds  at  a  given  time.  The  magnitude of  b i o l o g i c a l variance, however, i s not necessarily the same for a l l herds.  Therefore, the variance of estimates  obtained by composite sampling would be s i m i l a r f o r a l l herds; but the variance of estimates obtained from random sampling could d i f f e r among herds.  Thus, although a  p a r t i c u l a r random sampling scheme may, on the average, meet a s p e c i f i e d acceptable l e v e l of p r e c i s i o n , estimates of herd-period mean milk constituent percentages obtained by t h i s scheme could be much more variable f o r some herds than others.  Each estimate i s economically important  to the i n d i v i d u a l producer; therefore, i d e a l l y , the variance of estimates should be the same f o r a l l herds. I f the v a r i a b i l i t y of estimates obtained by random sampling i s to be approximately equal f o r a l l herds then d i f f e r e n t sampling schemes may be necessary f o r some herds. For the above reasons the data of Experiment I were used to estimate i f herds d i f f e r e d i n within herdperiod variances of milk constituent percentages and i f these differences  ( i f any) were large enough to warrant  different  sampling  The  of  if  data  herd-period  schemes  Experiment 2 variances  for certain were also  herds.  used  can be predicted  to  estimate  from e a s i l y measured variables associated with herds. Within herd-period variances of milk constituent percentages may d i f f e r among seasons.  These differences  may be a t t r i b u t e d to changes i n e i t h e r b i o l o g i c a l variance or testing variance  (Part 1).  If biological  variance i s higher at c e r t a i n seasons, sampling frequency should be increased i n these seasons.  On t h i s basis  sampling schemes may need to be modified not only f o r c e r t a i n herds but also for c e r t a i n periods. Mistakes i n sample i d e n t i f i c a t i o n , analyses etc., can occasionally be made; therefore, t e s t r e s u l t s should be systematically checked for gross e r r o r s . volved i n checking  Factors i n -  the r e s u l t s from a random sampling  scheme are discussed i n Part 2 .  85  2.  MATERIALS AND METHODS  Source of Data The f r e s h sample data c o l l e c t e d i n Experiment I (defined i n Part 1 ) from the twenty-three  herds that  shipped milk throughout the thirteen periods of Experiment I were used for the analyses i n Part 2 . A l l periods i n Part 2 were of f i f t e e n consecutive milk shipments (approximately  one month).  S t a t i s t i c a l Methods The average amount of milk i n each shipment i n each of the thirteen periods was c a l c u l a t e d (the arithmetic average of a l l shipments).  The mean milk constituent  percent f o r each period was also c a l c u l a t e d (the average of fresh samples each weighted by the amount of milk i t represented). 2  The within herd-period variances  (o^) of percent  milk f a t , protein and lactose were calculated f o r each herd-period subclass for twenty-three periods.  herds and t h i r t e e n  The within herd-period variances of percent  milk f a t , protein and lactose were also c a l c u l a t e d with two,  three and four s t r a t a per period by d i v i d i n g the  sums of squares pooled over the s t r a t a by the pooled  86 degrees of freedom.  Frequency d i s t r i b u t i o n s of these  variances were constructed.  Herd and period mean  within  herd-period variances of percent milk f a t , protein and lactose were calculated f o r each herd and each period by d i v i d i n g the pooled sums of squares (pooled over periods for herd means and pooled over herds f o r period means) by the pooled degrees of freedom. Regression analyses.  Simple and multiple regression  techniques were used to estimate the e f f e c t s of herd size (measured by average milk shipment weight) and milk constituent percentages  on the within herd-period variance  of a l l three milk constituents.  The sampling  distribution  of estimates of variances are not expected to be normal; a logarithmic transformation i s expected d i s t r i b u t i o n , Snedecor and Cochran [18] .  to y i e l d a normal Therefore the  d i s t r i b u t i o n s of the within herd-period variances and the natural logarithm (log ) of these variances were both e  tested f o r skewness and k u r t o s i s by the method of Snedecor and Cochran [18]. The regressions were f i t t e d  overall,  within herds and within periods. Simple l i n e a r regressions of the l o g  e  of the within  herd-period variances of percent milk f a t , protein and lactose were f i t t e d on each of four independent variables which are defined as follows:  87 M^j  F .ID.  t h e mean w e i g h t  (kg.) o f m i l k  of  and j  the i*"*  1  herd  t h e mean p e r c e n t herd  and j  t  n  t  milk  i n each  shipment  period;  n  f a t associated  with  th the i  period; th  P .ID.  t h e mean p e r c e n t  protein associated  with  the i  t h e mean p e r c e n t l a c t o s e a s s o c i a t e d th herd and j period.  with  the i  herd  j  t  n  period; th  L .ID.  The  overall  simple  linear  regression  model  assumed  was:  y. . •'l]  =  b  0 A  + b,X. . + e. . 1 i j i j  (30)  where y^j  the natural logarithm period percent  variance  o f each milk  of the i  bp  the population  b^  the simple  of the within  th  herd  constituent  th and j period;  m e a n w h e n X^.. e q u a l s  regression  herd-  coefficient  zero;  o f y^. on  X. . ; ID X.. ID  was s e t e q u a l  t o M.., D 1  F . . , P.. a n d L . . i n D ID ID 1  turn; e..  xj  t h e random  error,  2  N(0,a ) . e  88  The within herd simple  linear regression model  assumed was:  y. . = b 2  13  0 n  + h. + b,X. . + e. . l 1 i] ij  (31)  where bp  the population mean when equal  frequencies  e x i s t i n a l l subclasses and when X^.. equals zero; th h^  the e f f e c t associated with the i  herd;  b^  the within subclass simple regression coe f f i c i e n t of y.. on X..;  e. .  the random error N(0,a ) ; 2  and the remaining symbols were defined i n equation 30. The within period simple l i n e a r regression model assumed was: y. . = b + p. + b,X. . + e. . i] 0 1 13 13 J  (32)  rt  where Pj e. . 13  y^j  th the e f f e c t associated with the j period; 2 the random e r r o r , N(0,o* ) ; e defined i n equation 30;  and the remaining symbols were defined i n equation 31. M u l t i p l e regressions of the l o g  e  of within herd  period variances of percent milk f a t , protein and lactose  89 were f i t t e d on the f o u r independent v a r i a b l e s . The o v e r a l l m u l t i p l e r e g r e s s i o n model assumed was:  y.. = b * xi  0  n  + b,M.. + b~F.. + b,P.. + b.L.. + e.. 1 in 2 i i 3 in 4 i i IT  (33)  where b  the p o p u l a t i o n mean when M.., F..  n  L^j b^  a l l equal  P.. and  zero;  the p a r t i a l r e g r e s s i o n c o e f f i c i e n t o f y ^ j on  >_ z  M. . ; ID the p a r t i a l r e g r e s s i o n c o e f f i c i e n t o f y.. i]  >2  F. .; ID the p a r t i a l r e g r e s s i o n c o e f f i c i e n t o f y ^ on  on  P. . ; 3-D  >  the p a r t i a l r e g r e s s i o n c o e f f i c i e n t o f y^..o n  4  L. . ; ID e. . i]  the random e r r o r  2 N (0, a ) ; e  y.. was d e f i n e d i n e q u a t i o n 30 and M.., F.., P.. ^i] ^ ID ID ID and L.. were d e f i n e d on page 87. ID The w i t h i n herd m u l t i p l e r e g r e s s i o n model assumed was:  y. . = b + h. + b,M.. + b F . . + b P . . + b.L.. + e.. •*13 0 1 1 ID 2 13 3 i j 4 13 13 n  0  0  where bg  the p o p u l a t i o n mean when equal exist  i n a l l subclasses  frequencies  and when M.., F... ID ID  (34)  P.. and L.. a l l equal zero; ID ID was the e f f e c t associated with the it h herd; M  h. 1  was the within subclass p a r t i a l regression c o e f f i c i e n t of y ^ j on b  2  ..;  was the within subclass p a r t i a l regression c o e f f i c i e n t of y.. on F..;  ID  3 b^  was the within subclass p a r t i a l regression c o e f f i c i e n t of y. . on P. .; iD ID was the within subclass p a r t i a l regression  e.. ID y^j  c o e f f i c i e n t of y . . on L. .; iD ID 2 was the random error, N(0,a ); ® was defined i n equation 30;  J  a  and M . F . . , P.. and L.. were defined on page 87. ID ID J-D ^D The within period multiple regression model assumed was:  y. . = b. + p. + b M. . + b F . . + b-,P. . + b.L. . + e. . •'lD 0 ] 1 ID !D 3 ID 4 ID I D r  n  2  0  (35)  where p^  th was the e f f e c t associated with the j period;  e.. xj  was the random error,' N(0,a ' e ); '  y^j  was defined i n equation 30;  2  v  and the remaining symbols were defined i n equation 34. Simple and p a r t i a l regression c o e f f i c i e n t s were tested for s i g n i f i c a n c e by t - t e s t s .  Differences among adjusted herd and period means were tested by F-tests of the reduction i n the r e s i d u a l sums of squares of the o v e r a l l regressions obtained  Cboth simple and multiple)  by f i t t i n g the within  regression models.  subclass  The F-value i s calculated, a f t e r  Freese [9] as follows: F S _ i  , '  = V  S  S  E  " ' / MSE' s - 1  (36)  S S E  where SSE  the r e s i d u a l sums of squares from the o v e r a l l regression models, (equation 30 f o r simple and equation 33 for multiple regressions);  SSE  1  the r e s i d u a l sums of squares from the within subclass regression models (equations 31 and 32 f o r simple and equations 34 and 35 for multiple regressions);  MSE*  r e s i d u a l mean square from within  subclass  regression models; s  the number of subclasses  i n the within sub-  class regression models; v  the number of degrees of freedom associated with the error mean squares i n the within subclass regression models;  The above F-test i s i d e n t i c a l to the F-test of the main e f f e c t s i n the analysis of covariance classification.  i n the one-way  92  Within period regressions measured the extent to which herd differences i n the within herd-period variance of milk constituent percent can be a t t r i b u t e d to herd differences i n the independent v a r i a b l e s .  The within herd  regressions measured the extent to which changes i n the value of the independent variables i n a herd were associated with changes i n the within herd-period variance of milk constituent percent. A l l possible samples f o r seven sampling schemes were computer generated from the data of Experiment I . Frequency d i s t r i b u t i o n s of the absolute value of the deviation of each sample from the fresh mean were constructed f o r percent milk f a t and percent protein.  The fresh  mean was the mean of a l l fresh samples, weighted by the amount of milk i n the shipment, i n a period.  The seven  sampling schemes were for one to four random milk samples per period drawn without s t r a t i f i c a t i o n and with s t r a t i f i cation f o r those schemes with more than one milk sample per period.  The schemes were:  1.  One shipment sampled per period.  2.  Two shipments sampled per period; Ca} No s t r a t a Cb) Two s t r a t a Cone of seven and one of eight shipments1.  3.  Three shipments sampled per period:  93  (a) No s t r a t a Cb) Three strata 4.  (of f i v e shipments each).  Four shipments sampled per period; (a) No s t r a t a (b) Four strata  (three of four shipments and one  of three shipments).  94 2  - RESULTS AND DISCUSSION  Period E f f e c t s on Milk Shipment Weight and Milk Constituent Percentages Figure 4 shows the mean milk shipment weight and mean milk constituent percent f o r each of the thirteen periods used i n t h i s study. Percent milk f a t . Percent milk f a t dropped i n the spring, remained a t a r e l a t i v e l y low l e v e l through the summer and climbed slowly to i t s peak value i n mid-winter. Percent protein.  Percent protein increased i n  the spring, dropped o f f i n the late summer, climbed to a peak i n the autumn and then dropped slowly to a stable winter l e v e l . Percent lactose.  Percent lactose, which was less  v a r i a b l e than either percent milk f a t or protein, was lowest i n the summer and autumn. Milk shipment weight.  The amount of milk shipped  per herd was highest i n the spring and early summer and dropped to i t s lowest l e v e l s i n late summer and autumn. The e f f e c t of season on the composition and l e v e l of production of herd milk can mainly be a t t r i b u t e d to two f a c t o r s .  F i r s t l y , to the stage of l a c t a t i o n of the  O  Milk weight  •  O  % lactose  A % protein  % milk f a t 2400 . 2200 - 2000  cu  Di cd •P C  iC  5.0  n  -o  CU V U CO  cu  U  o-  .•  4.0  -P  , A-  (A  c o u  3.0 -  •H  f  -A•  P-  - 1200  ,A -  • A.  n  § co  - 1400  •P  2  1800  -1600  p<  •p c  £ o  S  J Apr | May  [ June [ July [Aug  1  3  2  4  5  6  [Sept [Oct ^ [Nov ^ j Dec ^ | Jan ^ | Feb ^ | Mar ^ [ 7  8  9  10  11  12  13  P E R I O D S Figure 4  Period average milk constituent percentages and milk shipment weight f o r thirteen periods  VO  96  cows  i n a herd i n a p a r t i c u l a r season ( i . e . the calving  d i s t r i b u t i o n ) and secondly to the e f f e c t of season of the year on milk production and composition on cows at a l l stages of l a c t a t i o n .  These factors can fluctuate from  year to year and therefore the seasonal e f f e c t s may. vary. However, the seasonal trends reported i n the current study agree with those reported by Waite and  Robertson  [ i g ] , Johnson et a l . [12] and Boswell et a l . [3 ].  Transformations Table 17 shows the r e s u l t s of the tests for skewness and kurtosis i n the d i s t r i b u t i o n s of the transformed untransformed  and  within herd-period variances of percent milk  f a t , protein and lactose.  In a l l cases the  untransformed  data showed s i g n i f i c a n t skewness and kurtosis, however, a f t e r transformation both skewness and kurtosis were nonsignificant.  Regression Analyses The l o g  e  transformed within herd-period variances  of percent milk f a t , protein and lactose, calculated with no strata and with four s t r a t a per period, were f i t t e d as dependent variables to regression models 30 to 35. A l l r e s u l t s are presented i n the transformed scale, so that regression c o e f f i c i e n t s measure the change i n the  TABLE 17 TESTS OF NORMALITY OF THE DISTRIBUTION OF WITHIN HERD-PERIOD VARIANCES BEFORE AND AFTER LOGARITHMIC TRANSFORMATION  ,Skewness  Kurtosis  Milk Constituent  Untransformed  Transformed  % Milk f a t  4.85*  0.20  0.139  40.91*  0.05  0.277  % Protein  2.01  .12  .139  5.10*  .10  .277  %  2.37*  .03  .139  8.61*  .01  .277  Lactose  significant  skewness o r k u r t o s i s .  Stan.Dev. U n t r a n s f o r m e d  Transformed  Stan.Dev.  98 log  e  of within herd-period variance of a given milk  constituent percentage associated with a u n i t change i n an independent v a r i a b l e .  Within Herd-Period Variance of Percent Milk Fat The estimates of the regression c o e f f i c i e n t s , t tests of the c o e f f i c i e n t s and the proportion of the sums 2  of squares (R ) accounted f o r by the regression equations are shown i n Table 18A f o r simple  linear regressions and  multiple l i n e a r regression; o v e r a l l , within period and within herd f o r the log of the within herd-period e of percent milk f a t .  variance  F-tests of the differences among  herds and among periods are also shown i n Table 18A. r e s u l t s f o r the l o g  e  The  of the within herd-period variance of  percent milk f a t with four s t r a t a per period are shown i n Table 18B. Milk shipment weight. regression of the l o g  e  The o v e r a l l simple  of the within herd-period  linear percent  milk f a t variance on the average weight Ckg.) of milk shipped was s i g n i f i c a n t and the regression c o e f f i c i e n t -4  was (-.2 64 ± ,083)xl0  ; the within period regression was -4  also s i g n i f i c a n t (-.249 ± 0.76)xl0 but the within herd regression was non-significant (-.295 ± .318)xl0"~^. These r e s u l t s  indicated that herds shipping larger amounts  TABLE 18A SIMPLE (SLR) AND MULTIPLE LINEAR (MLR) REGRESSION COEFFICIENTS FOR THE REGRESSION OF THE LOGARITHM OF THE WITHIN HERD-PERIOD VARIANCE OF PERCENT MILK FAT ON KILOGRAMS OF MILK, PERCENT MILK FAT, PROTEIN AND LACTOSE - NO STRATA Overall b±S.E. SLR  Within Period R  2  biS.E.  R  2  C  Within Herd F  b+S .E.  R  2  C  F  a  - .264±.083*  3.29  % Protein  .724+.186*  % Lactose  Milk wt. % Fat  b  - .2491.076*  0.32  2.14*  6.62*  - .2951.318 -.7531.238*  3.56  3.01*  6.67  6.73*  0 .3721.380  0.98  1.91*  .5291.378  0.69  5.61*  1.76  2.46*  - .1891.080*  1.81 2.13  -.5781.327  1 .3311.327* 0 .4051.414  5.34  0.6941.385 1 .1011.444*  3.65  6.2 8*  2.31  4.91  0 .1971.076* .7921.177*  .8891.318*  2.58  - .2491.086* - .4221.127*  2.54  1 .1841.282* 1 .0401.325*  5.36 3.13  0 .128±.082  0.83  .8971.406*  MLR  a  Milk wt.  b  % Fat % Protein % Lactose MLR equation  3.37  11.43  -.3771.147*  - .6581.240*  0.31 10.09  5.67*  1.09 2.62 1.13 2.14 6.56  degrees of freedom: SLR; 294,282 and 272; MLR; 291, 279 and 269 f o r o v e r a l l , within and within herds respectively. 3D regression c o e f f i c i e n t s x l 0 ~A . a  1.79*  periods  s i g n i f i c a n t : regression c o e f f i c i e n t s by t-tests and differences among l e v e l s (within subclass models) by F-tests. co ^° R^ calculated on the t o t a l within subclass sums of squares  TABLE 18B SIMPLE AND MULTIPLE LINEAR REGRESSION COEFFICIENTS PERCENT MILK FAT WITH FOUR STRATA PER PERIOD Overall b±S .E. SLR  Within R  2  b+S .E.  Within Herd  Period R  2c  F  b+S .E.  R  2  C  F  a  - .3081.080*  4.75 4.25  % Protein  0.282±.078* .774±.179*  % Lactose  Milk wt.  b  - .288±.074*  5.12  5.72*  - .610±.299* 1.51  2.89*  0.17 .04  2.77* 2.48*  .28  3.24*  6.24  6.22*  5.97  0 .317+. 073* .878±.171*  8.55  6.43*  - .154±.229 - .116±.359  .636±.311*  1.41  .547±.3.70  0.77  5.42*  0 .3391.387  - .238±.084* - . 035±.124  2.47 0.02  - .190±.078*  1.90  - .025±.143  0.01  % Protein  0 .615±.278*  1.53  0 .776±.320*  1.89  % Lactose  .5011.319  0.77  .040±.405  0.01  % Fat  MLR  a  Milk Wt. % Fat  MLR equation  b  9.00  10.49  - • 756±.315* 2.10 - .1021.232 0.07 0 .0401.372 0 .6091.428 6.03*  .01 .74 2.48  2.32*  degrees of freedom: SLR; 294, 282 and 272; MLR; 291, 279 and 269 f o r o v e r a l l , within periods and within herds respectively. a  regression c o e f f i c i e n t s xlO s i g n i f i c a n t : regression c o e f f i c i e n t s by t-tests and differences among l e v e l s (within subclass models) by F-tests. C o  R^ calculated on the t o t a l within subclass sums of squares  i-* o  101 of milk were associated with low within herd-period variance of milk f a t percent.  But, increased milk shipments  by a p a r t i c u l a r herd were not associated with a s i g n i f i c a n t reduction i n the within herd-period variance of milk f a t percent.  The range of milk shipment weights was much  greater (therefore the standard error of the regression c o e f f i c i e n t was much smaller) for both the o v e r a l l and the within period regressions than f o r the within herd regression. Milk f a t percent. gression of the l o g  e  The o v e r a l l simple l i n e a r r e -  of the within herd-period variance of  milk f a t percent on the average milk f a t percent was nons i g n i f i c a n t ; the regression c o e f f i c i e n t was 0.128±.082. Both the within period and the within herd regressions were s i g n i f i c a n t ; the regression c o e f f i c i e n t s were 0.197±.076 and-.753±.238 r e s p e c t i v e l y .  These r e s u l t s indicated that  high percent f a t herds tend to have large variances of percent milk f a t ; but that within herds, periods of low percent milk f a t (spring, see Figure 4) were associated with high variance of milk f a t percent.  The increase i n the  within herd-period variance of milk f a t percent that was associated with periods of low milk percent may be due to the r e l a t i v e l y rapid decline of milk f a t percent associated with the advent of spring grazing.  A consistent  d i r e c t i o n a l change i n a milk constituent percentage  102 across time would be expected to increase within herdperiod variance of the milk constituent percentage. Protein percent. period simple  The o v e r a l l and the within  linear regressions were s i g n i f i c a n t and  the regression c o e f f i c i e n t s were 0.724 ± .186 and,0.792 ± .177 r e s p e c t i v e l y .  However, the within herd regression  was non-significant; the c o e f f i c i e n t was 0.372 ± .380. Thus herds with high percent protein had higher than average within herd-period variance of milk f a t percent but changes i n protein content within a herd were not s i g n i f i c a n t l y associated with changes i n the variance of milk f a t percent. Lactose percent.  The simple l i n e a r regression  c o e f f i c i e n t s were s i g n i f i c a n t f o r o v e r a l l  CO.889 ± .318)  and within herds (0.897 ± .406) regression equations, but the within period regression c o e f f i c i e n t was non-significant.  (0.529 ± .378)  These r e s u l t s indicated that increases  in percent lactose within a herd were associated with an increase i n the within herd-period variance of milk f a t percentage, but that differences between herds i n percent lactose were not s i g n i f i c a n t l y associated with differences i n the variance of milk f a t percent. The F-tests of the differences among l e v e l s were s i g n i f i c a n t f o r both within subclass regression models and f o r a l l independent variables used.  These r e s u l t s  103 indicated that differences among both herd and period means were s i g n i f i c a n t when the independent variables were held constant  ( i . e . differences e x i s t among herd means  even a f t e r adjustment f o r the e f f e c t s of herd Overall multiple linear regression.  size). A l l coefficients  d i f f e r e d s i g n i f i c a n t l y from zero by a t - t e s t when a l l four independent variables were included i n the o v e r a l l multiple l i n e a r regression model.  The c o e f f i c i e n t s f o r  average milk weight and milk f a t percent were p o s i t i v e while those for percent protein and lactose were negative (Table 18A).  The model accounted f o r 11.43 percent of  the sums of squares of the dependent v a r i a b l e . Within period multiple l i n e a r regression.  Three of  the independent variables were s i g n i f i c a n t ; average milk shipment weight, percent milk f a t and percent protein when a l l four independent variables were included i n the within period multiple l i n e a r regression model.  These three  independent variables were also s i g n i f i c a n t when f i t t e d singly i n the simple  linear regression model.  However,  the sign of the c o e f f i c i e n t f o r percent milk f a t changed from p o s i t i v e i n simple  l i n e a r regression to negative when  the remaining three independent variables were held  constant.  The within period multiple regression model accounted f o r 10.09 percent of the t o t a l within period sums of squares of the dependent v a r i a b l e .  Differences among periods i n  104 the dependent v a r i a b l e were s i g n i f i c a n t by the F-test when the independent variables were held constant  (Table 18A).  Within herd multiple linear regression.  Two of  the independent v a r i a b l e s , percent milk f a t and percent lactose, were s i g n i f i c a n t sources of v a r i a t i o n when the within herd multiple l i n e a r regression model was f i t t e d . These two independent variables were also the only s i g n i f i c a n t sources of v a r i a t i o n when f i t t e d i n the simple l i n e a r regression models.  The within herd multiple  regression model accounted f o r 6.56 percent of the t o t a l within herd sums of squares of the dependent v a r i a b l e . Differences among herds i n the dependent v a r i a b l e were s i g n i f i c a n t by the F - t e s t when the independent variables were held  constant.  Within s t r a t a variance of milk f a t percent. of the within herd-period  The l o g  g  variances of percent milk f a t ,  calculated on a pooled within four s t r a t a basis, were f i t t e d as dependent v a r i a b l e s to the same regression models. The regression c o e f f i c i e n t s estimated when the variance was c a l c u l a t e d without s t r a t i f i c a t i o n  (Table 18A)  were not s i g n i f i c a n t l y d i f f e r e n t from the regression c o e f f i c i e n t s estimated with four strata per period (Table 18B1. However, f o r the independent variables of percent milk f a t and percent lactose the within herd  105 regression c o e f f i c i e n t s  (both multiple and simple) were  not s i g n i f i c a n t when the variance was calculated with s t r a t i f i c a t i o n but the regression c o e f f i c i e n t s were s i g n i f i c a n t l y d i f f e r e n t from zero when the variance was calculated without s t r a t i f i c a t i o n .  For milk shipment  weight the within herd regression c o e f f i c i e n t s were not s i g n i f i c a n t when the variance was c a l c u l a t e d without s t r a t i f i c a t i o n but were s i g n i f i c a n t when the variance was computed with four s t r a t a per period. Differences among herd and period means, tested by the F-test of the difference i n l e v e l s of the within herd and within period regressions, were s i g n i f i c a n t when the variance was c a l c u l a t e d with four s t r a t a per period. Within herd-period variance of a milk constituent percentage can be mainly a t t r i b u t e d to two factors (Materials and Methods);  (1) random day-to-day v a r i a t i o n s  i n the milk constituent percent and (2) d i r e c t i o n a l changes i n the milk constituent percent across time.  The  second factor (time trends) would be expected to account for more of the within herd-period v a r i a t i o n i n long periods than i n short periods  (strata).  The magnitude of  the random component would not be expected to change with length of periods.  The r e s u l t s of the regression  analyses  (strata vs. no strata) indicated that the r e l a t i o n s h i p s between the within herd-period  variance of milk f a t  percent and the independent variables can be mainly  106 attributed  t o t h e m a g n i t u d e o f t h e random p a r t o f t h e w i t h i n  herd-period  Within  variance  of milk  Herd-Period Variance  The protein,  log  f a t percent.  of Percent  of the w i t h i n  e  c a l c u l a t e d with  stratification,  four  Protein  herd-period  variance  s t r a t a per period  of  percent  and w i t h o u t  were u s e d a s d e p e n d e n t v a r i a b l e s i n  r e g r e s s i o n m o d e l s 30 t o 35. The of  estimates  the c o e f f i c i e n t s  of the regression  coefficients,  and t h e p r o p o r t i o n  t-tests  o f t h e sums o f  2 squares shown  i n Table  linear for  (R ) a c c o u n t e d  protein. periods log  e  19A f o r s i m p l e  regression;  the l o g  f o r by t h e r e g r e s s i o n e q u a t i o n s a r e  overall, within  of the within  e  linear  regressions  period  herd-period  and m u l t i p l e  and w i t h i n  variance  of  herd  percent  F - t e s t s o f t h e d i f f e r e n c e s among h e r d s a n d are also  shown  of the w i t h i n  i n Table  herd-period  c a l c u l a t e d with' f o u r  19A.  The r e s u l t s f o r t h e  variance  s t r a t a per period  of percent  protein  a r e shown i n T a b l e  19B. Milk significant within  shipment weight.  Milk  s h i p m e n t w e i g h t was  source of v a r i a t i o n f o r the o v e r a l l  period  simple  linear  regression  and the  equations.  The  -4 r e g r e s s i o n c o e f f i c i e n t s were (-.293 ±  .064)xl0  4  respectively.  was n o t a s i g n i f i c a n t herd  regression.  (-.306 ±  .071)xlO  Milk  shipment  and weight  source of v a r i a t i o n f o r the w i t h i n  These r e s u l t s i n d i c a t e d t h a t  herds  a  TABLE 19A SIMPLE (SLR) AND MULTIPLE LINEAR (MLR) REGRESSION COEFFICIENTS FOR THE REGRESSION OF THE LOGARITHM OF THE WITHIN HERD-PERIOD VARIANCE OF PERCENT PROTEIN ON KILOGRAMS'MILK PERCENT MILK FAT, PROTEIN AND LACTOSE - NO STRATA Overall b±S .E. SLR  Within R  2  b±S.E.  Within Herd  Period R  2  -  F  b± S . E .  - .524±.278  R  2  F  c c  a  - •306±.071*  5.92  - .293± .060*  7.07  7 .6 0*  % Fat  0 .131±.070  1.16  2.41  % Protein  0 .572±.161*  4.11  0 .171±.065* 0 .5121.152*  3.87  7 .61* 7 .14*  % Lactose  - .362±.278  0.57  - .075±.322  0.02  7 .05*  MLR Milk wt.  - .257±.075*  3.64  - .244+.068* - • 051± .125  4.18  Milk wt.  b  1.30  1.50  - ,296±.211  0.72  2.11*  0 .697±.330*  1.62  2.05*  - .636±.357  1.16  2.26*  - .440±.289 - .374±.213  0.83  0 .620±.342  1.10 1.18  - .410±.394  0.39  a  b  % Fat % Protein  - .153±.111  0 •710±.247*  0.60 2.58  % Lactose  - .383± .284  0.57  MLR equation  9 .33  0 .479± .280  0 .05 0.96  - ,304± .354  0.24 8.86  6 .99*  4 .18  1.38  degrees of freedom: SLR; 294, 282 and 272; MLR; 291, 279 and 269 f o r o v e r a l l , within periods and within herds respectively. a  regression c o e f f i c i e n t s xlO *  s i g n i f i c a n t : regression c o e f f i c i e n t s by t-tests and differences among l e v e l s (within subclass models) by F-tests.  C  R  2  calculated on the t o t a l within subclass sums of squares  i -  1  o ** J  TABLE 19B SIMPLE AND MULTIPLE LINEAR REGRESSION COEFFICIENTS PERCENT PROTEIN WITH FOUR STRATA PER PERIOD Overall  Within Period b±S.E. R  b±S.E.  R  Milk Wt. % Fat  - .188±.075*  2.06 2.81  - .167±.067*  % Protein % Lactose  0 •477±.169* 0 .057± .290  2.63  - .119±.080  0.73  % Fat  0 .123±.119  0.35  % Protein  0 .192±.264 - .143±.304  0.17  SLR  2  2  C  F  Within Herd R b±S .E. 2  C  F  a  b  0 .212± .073*  0.01  7.99*  - .729±.292*  2 .25  1.74*  0 .194±.066*  2.16 2.93  7.99*  0 .1211.224  0.11  1.33  0 .566±.156* 0 .058±.331  4.48 0 .01  8.56* 7.95*  - .065±.352 - .1161.380  0.01 0.03  1.34 • 1.72*  - .090± .072 - .008±.131  0 .54  - .776±.309*  2.29  <0.01  0 .129±.228  0 .12  1.29  .0241.365 0 .2201.421  MLR  a  Milk wt.  b  % Lactose MLR equation  0.07 3.96  0 .571±.293* - .341±.370  0.29 5.50  8.38*  <0.01 0.10 2.44  1.47  degrees of freedom: SLR; 294, 282 and 272; MLR; 291, 279 and 269 for o v e r a l l , within periods and within herds respectively. b -4 regression c o e f f i c i e n t s xlO a  s i g n i f i c a n t : regression c o e f f i c i e n t s by t-tests and differences among l e v e l s (within subclass models) by F-tests. c 2 R calculated on the t o t a l within subclass sums of squares  £ 0 0  109 shipping large amounts of milk were associated with low within herd-period variance of protein percent, but that increased milk shipments by a herd were not s i g n i f i c a n t l y associated with changes i n the within herd-period variance of protein percent. Milk f a t percent.  Milk f a t percent was a s i g n i f i c a n t  source of v a r i a t i o n f o r the within period simple l i n e a r regression model only.  The regression c o e f f i c i e n t was  0.171 ± .065. This r e s u l t indicated that herds shipping milk high i n milk f a t percent were associated with high within herd-period variance of p r o t e i n percent. Protein percent.  Percent p r o t e i n was a s i g n i f i c a n t  source of v a r i a t i o n f o r the o v e r a l l , within period and within herd simple l i n e a r regression models.  The regression  c o e f f i c i e n t s were 0.572 ± .161, 0.512 ± .152 and 0.697 ± .330 respectively.  These r e s u l t s indicated that herds shipping  milk high i n protein percent were associated with high within herd-period variance of protein percent.  The r e s u l t s  from the analyses of the within herd regression model indicated that an increase i n the l e v e l of p r o t e i n i n milk shipped by an i n d i v i d u a l herd was associated with an increase i n the within herd-period variance of p r o t e i n percent.  110 Lactose percent.  Lactose percent was not a s i g n i f i -  cant source of v a r i a t i o n for any of the three simple l i n e a r regression models. Overall multiple l i n e a r regression.  The p a r t i a l  regression c o e f f i c i e n t s associated with the independent variables of milk shipment weight and percent  protein  were s i g n i f i c a n t l y d i f f e r e n t from zero by a t - t e s t .  The  model accounted f o r 9.33 percent of the t o t a l sums of squares of the dependent v a r i a b l e . Within period multiple l i n e a r regression.  Only the  independent v a r i a b l e of milk shipment weight was a s i g n i f i c a n t source of v a r i a t i o n when the within period multiple l i n e a r regression model was f i t t e d .  The -4  regression c o e f f i c i e n t was (-.244 ± .068)xl0  . The  model accounted for 8.86 percent of the t o t a l sums of squares of the dependent v a r i a b l e .  Differences i n l e v e l s  were s i g n i f i c a n t by the F - t e s t . Within herd multiple l i n e a r regression.  When the  multiple regression was computed on a within herd basis none of the independent variables was a s i g n i f i c a n t source of v a r i a t i o n .  This model accounted f o r 4.18 percent of  the t o t a l sums of squares of the dependent v a r i a b l e . Differences between herd means were not s i g n i f i c a n t by the F-test of differences of l e v e l s .  Ill Within strata variance of protein percent.  The  regression c o e f f i c i e n t s estimated when the within herdperiod variance was calculated without s t r a t i f i c a t i o n were not s i g n i f i c a n t l y d i f f e r e n t from the c o e f f i c i e n t s estimated with four strata per period  Within Herd-Period Variance of Percent The within herd-period  (Table 19B).  Lactose  variances of percent lactose,  calculated without s t r a t i f i c a t i o n and with four s t r a t a per period were used as dependent v a r i a b l e s , a f t e r l o g  e  transformation, i n the regression models. The estimates  of the regression c o e f f i c i e n t s , t - t e s t s  of the c o e f f i c i e n t s and the proportion of the sums of 2  squares (R ) accounted for by the regression equations are shown i n Table 2OA for simple l i n e a r regressions and multiple l i n e a r regression, both o v e r a l l , within period and within herd for the l o g  e  variance of percent lactose.  of the within  herd-period  F-tests of the differences  among herds and among periods are also shown i n Table 2OA. The r e s u l t s f o r the l o g  e  of the within  herd-period  variance of percent lactose calculated with four s t r a t a per period are shown i n Table 2OB. Simple l i n e a r regression.  Milk shipment weight was  not a s i g n i f i c a n t source of v a r i a t i o n for any of the three simple l i n e a r regression models.  TABLE 2OA SIMPLE (SLR) AND MULTIPLE LINEAR (MLR) REGRESSION COEFFICIENTS FOR THE REGRESSTION OF THE LOGARITHM OF THE WITHIN HERD-PERIOD VARIANCE OF PERCENT LACTOSE ON KILOGRAMS MILK, PERCENT MILK FAT, PROTEIN AND LACTOSE - NO STRATA Overall  Within R  b±S.E. SLR  2  Within Herd  Period  biS .E.  R  2 C  F  biS.E.  R  2 C  F  a  Milk wt.  -.1001.074  0.61  -.1041.059  1.11  16 .49  % Fat % Protein  -.2331.071*  3.53  0.0401.167  0.02  -.1111.058 -.2591.138  1.27 1.24  % Lactose  -.3311.283  0.46  -.7331.285*  2.30  15.38* -l.481i.205* 16.19 16.78* 0 . 8 5 4 1 . 3 4 7 * 2 .18 17.04* -.0561.378 0.01  b  -.1701.076*  1.54  -.1451.063*  1.83  -.2971.279  % Fat % Protein  -.6431.112*  10.07 5.43  -.0341.115 -.2191.256  0.03 0.25  -1.5381.206*  % Lactose  0.1241.286  0.06  -.4931.325  0.79  -.1911.380  b  0.2821.294  0.34  1.22 3.32* 1.56 1.20  MLR Milk wt.  MLR equation  1.0511.249*  11.12  4.51  0.34  16.77 0.9201.330* 2.33  13.39*  0 .08 19.56  2.66*  degrees of freedom: SLR; 294, 282 and 272; MLR; 291, 279 and 269 f o r o v e r a l l , within periods and within herds respectively.  a  regression c o e f f i c i e n t s xlO *  .  s i g n i f i c a n t : regression c o e f f i c i e n t s by t - t e s t s and differences among l e v e l s (within subclass models) by F-tests. c 2 R calculated on the t o t a l within subclass sums of squares  M  y-> M  TABLE 2OB SIMPLE AND MULTIPLE LINEAR REGRESSION COEFFICIENTS PERCENT LACTOSE WITH FOUR STRATA PER PERIOD Overall  Within Period  Within Herd  b±S.E.  R  -.080±.077  0.37  -.0911.065  0.68  10.21*  -.0981.299  0.04  1.46  1.99  0.69 0.41  -1.0721.218*  8.23  0 .16  -.0911.065 -.1671.154  9.67*  % Protein  -.1791.073* 0.1201.172  0 .8981.352*  2.34  % Lactose  -.4441.290  0.79  -.4501.320  0.70  10.19 10.08*  2.46* 1.82*  .4871.383  0.59  1.48  MLR Milk wt.  -.1211.079  0 .73  -.1281.071  1.16  0 .0081.288  0.01  % Fat  -.5441.117*  6.87  -.1131.130  0 .27  -1.1731.219*  9.44  % Protein  1.0151.260*  4.83  -.0081.290  0 .01  0.8251.351*  1.82  % Lactose  -.1011.298  0.04  -.1861.367  0.09  -.6441.404  0.83  SLR  2  biS.E.  F  biS .E.  R  2  C  F  a  Milk wt.  b  % Fat  -  a  MLR equation  b  8 .18  2 .12  7.94*  11.93  2.07*  degrees of freedom: SLR; 294, 282 and 272; MLR; 291, 279 and 269 f o r o v e r a l l , within periods and within herds respectively. a  b regression c o e f f i c i e n t s xlO-4 *  s i g n i f i c a n t : regression c o e f f i c i e n t s by t-tests and differences among l e v e l s (within subclass models) by F-tests. R calculated on the t o t a l within subclass sums of squares C  2  114 Percent milk f a t was a s i g n i f i c a n t source of v a r i a t i o n when the regression was computed o v e r a l l and within herds.  The regression c o e f f i c i e n t s were -.233 ±  .071 and -1.481 ± .205 r e s p e c t i v e l y .  These r e s u l t s  indicated that the within herd-period variance of percent lactose increased, for a herd, when the f a t content of the milk dropped (spring, see Figure 4). Percent protein was a s i g n i f i c a n t source of v a r i a tion f o r the within herd regression only.  The regression  c o e f f i c i e n t was 0.854 ± .347. This r e s u l t indicated that the within herd-period variance of percent lactose increased when protein content o f herd milk increased (Figure 4). Percent lactose was a s i g n i f i c a n t source of v a r i a t i o n for the within period model only.  The regression c o e f f i c i e n t  was -.733 ± .285 i n d i c a t i n g that herds with low lactose l e v e l s were s i g n i f i c a n t l y higher i n the variance of percent lactose. The simple regression c o e f f i c i e n t s estimated when the variances were calculated without s t r a t i f i c a t i o n . (Table 2OA) were not s i g n i f i c a n t l y d i f f e r e n t from the regression c o e f f i c i e n t s estimated with four s t r a t a per period. Multiple l i n e a r regression. For the o v e r a l l regression milk shipment weight, percent milk f a t and percent protein were a l l s i g n i f i c a n t sources of v a r i a t i o n .  115 The p a r t i a l regression c o e f f i c i e n t s were; (-.170 ± .076) x l 0 ~ , -.643 ± .112 and 1.051 ± .249 r e s p e c t i v e l y . 4  The  model accounted f o r 11.12 percent of the t o t a l sums of squares of the dependent v a r i a b l e . On a within period basis only milk shipment weight was a s i g n i f i c a n t source of v a r i a t i o n ; the p a r t i a l -4 regression c o e f f i c i e n t was (-.145 ± .063)xl0  .  The model  accounted for 4.51 percent of the t o t a l within period sums of squares.  Period l e v e l s were s i g n i f i c a n t l y  d i f f e r e n t by the F - t e s t . On a within herd basis two of the independent v a r i a b l e s , percent milk f a t and percent p r o t e i n , were s i g n i f i c a n t sources of v a r i a t i o n . The p a r t i a l regression c o e f f i c i e n t s were -1.538 ± .206 and 0.920 ± .330 respectively.  The model accounted for 19.56 percent of  the t o t a l within herd sums of squares.  The F-test of  differences i n herd l e v e l s was s i g n i f i c a n t . The regression c o e f f i c i e n t s estimated when the variances were calculated without s t r a t i f i c a t i o n were not s i g n i f i c a n t l y d i f f e r e n t from the regression c o e f f i c ients estimated with four s t r a t a per period.  The F-test  of l e v e l s of both periods and herds were s i g n i f i c a n t i n both cases.  116  Conclusion of Regression  Analyses  Although the regression analyses  showed that i n  many cases the variances of milk constituent percentages were s i g n i f i c a n t l y associated with the independent variables used, the proportion of the t o t a l sums of squares accounted for by the regression equations was r e l a t i v e l y low and therefore the regression equations have l i t t l e value f o r predicting the within herd-period an i n d i v i d u a l  herd-period  subclass.  variance of  The regression  analyses also showed that differences among herds and among periods i n within herd-period variances of milk constituent percentages were s i g n i f i c a n t .  117  Herd and Period V a r i a t i o n The c r i t e r i o n of p r e c i s i o n used i n the current study was that random sample estimates of herd-period milk constituent percentages  should be a t least as precise  as composite estimates ( i . e . that the l e v e l of p r e c i s i o n of current sampling methods was acceptable to the industry).  The variance of estimates that w i l l meet t h i s  c r i t e r i o n were presented i n Table 14. By rearrangement of equation 24 to y i e l d :  a w 2  =  n(Na- - a )/(N - n) x a'  (37)  2  The maximum value of the within herd period variance of milk constituent percentages that w i l l s a t i s f y t h i s can be calculated f o r a given sample s i z e .  criterion  The values  presented i n Table 14 were substituted i n equation 37 f o r 2 2 a-X , values f o r acl (defined i n equation 2) were taken from Table 3, to calculate maximum values of within herd-period variance f o r two, three, four and f i v e random samples per f i f t e e n shipment period for a l l three milk constituents (Table 21). The values i n Table 21 were used to c a l c u l a t e the proportion of herds, periods or i n d i v i d u a l  herd-period  subclasses that would meet this c r i t e r i o n f o r various sampling schemes.  118 TABLE 21 MAXIMUM VALUE OF O FOR THE PRECISION OF A RANDOM SAMPLE TO MEET THE SPECIFIED CRITERION 2  Variance CxlO" ) 2  Milk Constituent  Sample Size Two  Three  Four  Five  % Milk f a t  0.737  1.198  1.742  2 .395  % Protein  0.280  0.373  0.542  0.745  % Lactose  0.473  0.676  0.983  1 .351  119 Season  Variation The  regression analyses  showed t h a t  differences  among p e r i o d s i n t h e w i t h i n h e r d - p e r i o d v a r i a n c e o f m i l k c o n s t i t u e n t p e r c e n t were s i g n i f i c a n t s t i t u e n t s when t h e i n d e p e n d e n t constant. over  As the data  o n l y one y e a r  years  f o r a l l milk  variables  were h e l d  i n t h e c u r r e n t s t u d y were  no c o m p a r i s o n s o f s e a s o n  are possible.  I f seasons  con-  collected  effects  are different  estimates  o f h e r d - p e r i o d means w o u l d be more p r e c i s e i n some than  i n o t h e r s under  therefore, in  t h e same random s a m p l i n g  i t c o u l d be w o r t h w h i l e  some s e a s o n s  than  i n others.  across  seasons  scheme;  t o t a k e more Alternatively  samples the sampling  frequency  s h o u l d be g r e a t e n o u g h t h a t t h e c r i t e r i o n o f  precision  i s satisfied  f o r t h e most v a r i a b l e  w o u l d mean t h a t t h e s a m p l i n g some s e a s o n s  frequency  than necessary.  This course  be w a s t e f u l o f r e s o u r c e s a n d w o u l d with  sampling  and t e s t i n g  will  bulk  variance of milk  p e r i o d by d i v i d i n g by  the pooled  with  milk.  results  showed  f a t percent.  f a t p e r c e n t was c a l c u l a t e d  o f freedom  two, t h r e e a n d f o u r s t r a t a  would  increase costs associated  the pooled w i t h i n herd  degrees  this  be g r e a t e r i n  of action  Within herd-period variance of milk period  seasons,  with  sums o f no  per period  The  f o r each squares  strata  and  ( T a b l e 22).  that the w i t h i n herd-period variance of  The  TABLE 22 PERIOD AVERAGE WITHIN HERD-PERIOD VARIANCE ( a ) OF PERCENT MILK FAT WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA w  Variance No S t r a t a Period Number  a ±S.E. w 2  Two S t r a t a DF  a ± S .E. w 2  (xlO  2  )  Three S t r a t a DF  a ±S.E. . w 2  Four S t r a t a DF  c ±S.E. w 2  DF  1  1 .570±.126  309  1.449± .121  286  0.974±.085  263  1 .124±.102  240  2  2 .584±.205  315  1.474± .122  292  1.153±.099  269  1 .173±.105  246  3  2 .291±.182  314 •  1.518± .125  291  1.374±.118  268  1 .257±.113  245  4  0 •905±.072  317  0.713± .059  294  0.527±.045  271  0 .601±.054  248  5  1 .364±.112  293  0.920± .079  266  0.850±.076  247  0 •992±.093  224  6  1 .563+.126  307  1.380± .115  284  1.276±.lll  261  1 .052±.096  238  7  1 .277±.105  295  1.005± .087  268  0.922±.082  249  0 •875±.082  226  8  1 .634±.132  304  1.389± .117  281  1.344±.118  258  1 .362±.125  235  9  1 .159±.093  310  0.734± .061  287  0.608±.053  264  0 .619±.056  241  10  1 .279±.102  315  1.086± .090  288  1.115±.096  269  1 .242±.112  246  11  1 .055±.092  261  0.993± .098  204  0.842±.079  227  0 .760±.074  207  12  0 ,614±.049  317  0.505± .041  294  0.509±.044  271  0 .441±.039  248  13  0 .677±.055  303  0.598± .050  288  0.545±.047  267  0 .491±.044  244  to  O  percent milk f a t was summer (Figure 5).  the highest i n the spring and earlyThese r e s u l t s agree with those r e -  ported by O'Keeffe [16]; however, Herrmann and Anderson [11] and Boswell et a l . [3 ] found that the variance  was  the highest i n the period October to December, although the work of Boswell et a l . [3] showed a secondary peak i n May.  In the current study these values  (without  s t r a t i f i c a t i o n ) , ranged from 0.0258 ± .00205 i n the second period  (second h a l f of A p r i l and the f i r s t h a l f of  May)  to 0.00614 ± .00049 i n the twelveth period (end of February and beginning of March). ' S t r a t i f i c a t i o n into four s t r a t a resulted i n a reduction of the within herdperiod variance of milk f a t percentage i n a l l periods; however, the reduction was,  i n general, greater i n those  periods of high variance than i n those periods of low variance  (Figure 5).  The period variances estimated with  four s t r a t a per period were a l l lower than the maximum values shown i n Table 21 for four samples per period. Therefore with four samples per period (one from each of four strata) the c r i t e r i o n of p r e c i s i o n would be met i n a l l periods.  Three samples (possibly two i n some months)  would be adequate i n the winter i f the seasonal trends reported i n the current study are consistent across years. The differences among seasons may  be due to changes  i n b i o l o g i c a l variance or testing variance (Part 1 ) .  123 However, as w i t h i n herd-period v a r i a n c e was g e n e r a l l y large i n those seasons Cspring and autumn) associated with changes i n herd feeding and handling, the seasonal differences i n within herd-period variance can probably be a t t r i b u t e d mainly to differences i n b i o l o g i c a l variance. Within herd-period variance of protein percent. Period means within herd-period variance of percent protein were also calculated with no strata and with two, three and four s t r a t a  (Table 23). The r e s u l t s  (graphed Figure 6) showed  two peaks; one i n the spring (period two) and one i n the autumn (period e i g h t ) .  S t r a t i f i c a t i o n resulted i n a reduction  i n the estimates of the within herd-period variance of percent p r o t e i n i n a l l periods.  With four samples per  period the variance was higher than the maximum allowable for four of the periods (period two, f i v e , eight and ten). However, as estimates of within herd-period variance of percent p r o t e i n were lower than that of percent milk f a t the standard error of the estimate of percent p r o t e i n would be lower than the standard error of the estimate of percent milk f a t . Period means within herd-period variance of percent lactose  were  calculated  (Table 24).  The  results  (graphed i n Figure 7) showed that with four samples per period the c r i t e r i o n of p r e c i s i o n was met i n a l l periods.  TABLE 23 PERIOD AVERAGE WITHIN HERD-PERIOD VARIANCE ( a ) OF PERCENT PROTEIN WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA 2  Variance (x!0~^) No Strata  Two Strata  Three Strata  Four Strata  Period Number  aw±S.E.  DF  a ±S.E. w  . DF  1  0 .536±.043  309  0 .509±.042  286  0 .504±.044  263  0 .433±.039  240  2  1 .238±.098  315  1 .120±.092  292  1 .037±.089  269  0 •656±.059  246  3  0 .650±.052  314  0 .585±.048  291  0 .543±.047  268  0 •563±.051  245  4  0 .714±.057  317  0 .453±.037  294  0 •388±.033  271  0 .310±.028  248  5  0 .753±.062  293  0 .757±.065  266  0 .741±.066  247  0 .638±.060  224  6  0 .941±.076  307  0 .641±.054  284  0 .727±.063  261  0 .529±.048  238  7  0 •723±.059  295  0 .596±.051  268  0 •482±.043  249  0 .373±.035  226  8  1 .490±.121  304  1 .035±.087  281  0 .887±.078  258  1 .007±.093  235  9  0 •777±.062  310  0 .598±.050  287  0 .603±.052  264  0 .535±.049  241  10  0 •887±.070  315  0 .790±.066  288  0 .740±.064  269  0 .8081.073  246  11  0 .494±.043  261  0 .369±.036  204  0 •409±.038  227  0 .421±.041  207  12  0 .611±.048  317  0 .551±.045  294  0 •563±.048  271  0 •486±.043  248  13  0 •565±.046  303  0 .450±.037  288  0 .325±.028  267  0 •263±.024  244  2  2  o ±S.E. 2  DF  0 ±S.E. w 2  DF  to •ti  2.0  O No  A Four s t r a t a per p e r i o d  1.5 A CN I  strata  O  CU CJ  1.0 H  c  cd  •H  u  >  0.5  H  [ A p r i l J May i  2  | June | J u l y | Aug 3  4  5 P  Figure 6  E  | Sept | Oct  6 R  I  7 O  D  8  | Nov 9  | Dec  | Jan  1 Feb  10  11  12  [March 13  S  W i t h i n h e r d - p e r i o d v a r i a n c e of l a c t o s e p e r c e n t f o r t h i r t e e n p e r i o d s  r—1  to  TABLE 24 PERIOD AVERAGE WITHIN HERD-PERIOD VARIANCE ( a j ) OF PERCENT LACTOSE WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA Variance No S t r a t a Period Number  2 a ±S.E. w  (xlO~ )  Two S t r a t a DF  2 a ±S.E. w  Three S t r a t a DF  o  O 1S.E.  w  Four S t r a t a DF  2  a iS.E.  w  DF  1  0.392±.031  309  0.333±.028  286  0 .3381.029  263  0 .2601.024  240  2  0.822±.065  315  •568±.047  292  .4991.043  269  .4141.037  246  3  .511+.041  314  .519±.043  291'  .537+.046  268  .5561.050  245  4  • 972± .077  317  •737±.061  294  .6381.055  271  .5271.047  248  5  .525±.043  293  .442±.038  266  .4261.038  247  .4521.043  224  6  .675+.054  307  .5351.045  284  .6051.053  261  .6021.055  238  7  .8881.073  295  .833±.072  268  .8211.073  249  .7301.068  226  8  .541±.044  304  .437±.037  281  .345+.030  258  .4071.037  235  9  .223±.018  310  .2151.018  287  .1821.016  264  .1931.018  241  10  .4041.032  315  .374±.031  288  .27110 .23  269  .3661.033  246  11  .364+.032  261  .2781.027  204  .3651.034  227  .3451.034  207  12  .544±.043  317  .5271.043  294  .5131.044  271  .4581.041  248  13  .253± .020  303  .2651.022  288  .2551 .022  267  .2481.022  244  Figure 7  Within herd-period variance of lactose percent for t h i r t e e n periods  128 Herd V a r i a t i o n The within period regression analyses  (Tables 18A to  2OB) showed that large herds were lower i n within herdperiod variance of milk constituent percent than smaller herds and a l s o , i n general, that higher variance was associated with high herd l e v e l s of milk f a t and p r o t e i n . Herd means of within herd period variance of milk constituent percent were calculated f o r the twenty-three  herds used i n  the regression analyses with no strata and with two, three and four s t r a t a f o r the within herd-period variance of milk constituent percent. The within herd-period variance of percent milk f a t (Table 25) herd means ranged from 0.02955 ± .00313 for the most v a r i a b l e herd to 0.00590 ± .00063 f o r the l e a s t v a r i a b l e without s t r a t i f i c a t i o n .  With four s t r a t a  per period the range was from 0.01695 ± .00205 to 0.00380 ± .00046.  Therefore with four samples and four s t r a t a per  period the c r i t e r i o n  (Table 21) was met f o r a l l herds.  Herd means of within herd-period variance of percent protein (Table 26) ranged from 0.01227 ± .00132 to 0.00458 ± .00049 without s t r a t i f i c a t i o n and from 0.00968 ± .00117 to 0.00322 ± .00039 with four s t r a t a per period. For nearly half the herds the c r i t e r i o n  (Table 21) of  p r e c i s i o n w i l l not be met with four samples (one from each of four s t r a t a ) .  TABLE 25 HERD AVERAGE WITHIN HERD-PERIOD VARIANCE (a ) OF PERCENT MILK FAT WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA w  Variance (x!0~ ) No Strata  Two Strata  Three Strata  a ±S.E. w 2 .955±.313  176  1.402±.154  163  1 .215±.139  2 .081± .221  176  1.8581.207  159  2 .020±.215  17 4  1.624±.182  1 .889± .202  173  1 .675±.184  DF  a ±S.E. w  150  a iS.E. w 1 .453±.174  137  1 .714±.197  150  1 .5401.185  137  157  1 .728±.200  148  1 .6951.205  135  1.741±.193  160  1 .4841.172  147  1 .5791.192  134  163  1.137±.130  151  1 .053±.124  142  1 .1821.146  130  1 .667±.185  161  0.9041.102  156  0 .774±.091  143  0 .8011.098  131  1 .580±.168  175  1.370±.151  162  1 .039±.120  149  1 .1731.141  136  1 .526±.164  172  0.962±.109  155  0 .818± .095  146  0 .6321.077  133  1 .4431.156  170  0.8541.097  153  0 .6691.078  144  0 .6021.074  131  1 .423±.156  164  1.103±.123  159  0 .964±.lll  148  0 .9961.120  135  1 .418±.152  173  1.312±.147  156  0 .8471.098  147  0 .8881.108  134  1 .313±.139  177  1.024±.112  164  0 .9011.103  151  0 .9091.109  138  1 .244±.133  174  1.128±.127  157  0 .8671.100  148  0 .9881.119  135  2  2  DF  a ±S.E. w  Four Strata  2  DF  2  DF  TABLE 25 (continued) Variance (xlO" ) No Strata  Two Strata  Three Strata  a ±S.E. w  DF  1 .201± .130  170  1 .138± .129  154  0.997±.117  1 .165± .125  172  0 .983± .111  155  1 .152± .123  174  0 .792± .088  1 .117± .119  175  1 .005± .109  o- +S.E. w  a ±S.E. w  DF  143  0 .907±.111  131  1.003+.117  146  0 .939±.114  133  161  0.750± .087  148  0 .688±.083  135  0 .912± .102  158  0.759± .087  149  0 •757±.091  136  169  0 .977± .111  152  0.837±.098  143  0 .726±.089  130  0 .979± .103  179  0 .811± .088  166  0 .725± .082  153  0 .732±.087  140  0 • 937±.100  173  0 .599+ .067  156  0 .549± .064  147  0 .508± .062  134  0 .889± .095  17 2  0 .705± .080  155  0.649± .075  146  0 .536± .065  133  0 .664± .071  17 5  0 .581± .065  158  0.552±.063  149  0 • 562± .068  136  0 .590± .063  173  0 • 420± .047  156  0.371± .043  147  0 .380±.046  134  2  2  DF  a ±S.E. w  Four Strata  2  DF  2  OJ O  TABLE 26 HERD AVERAGE WITHIN HERD-PERIOD VARIANCE (a~) OF PERCENT PROTEIN WITHOUT. STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA Variance No S t r a t a a ±S.E. 2  w  Two DF  Strata  a iS.E. w 2  (x!0~ ) Three S t r a t a  DF  a i S .E. w 2  Four DF  Strata  a +S.E. w 2  DF  1 .227±.132  172  1 .0671.120  155  1.0081.117  146  0 .8531.104  137  1 .199±.127  176  0 .8561.095  159  0.8461.097  150  0 .6721.081  137  1 .175± .128  170  0 .8781.099  154  0.9641 .113  143  0 .7931.097  135  1 .075±.118  164  0 .8261.092  159  0.8451.097  148  0 .7251.088  134  1 .060±.113  173  1 .0641.118  160  1.0831.125  147  0 .9691.117  130  0 .8821 .094  174  0 .6081.067  161  0.4911.057  148~  0 .5111.062  131  0 .876±.096  163  0 .6741.077  151  0.7271.086  142  0 .5611.069  136  0 .872±.093  174  0 .7781.087  157  0 .6951.080  148  0 .6681.081  133  0 .864±.091  179  0 .7641.083  166  0.7201.082  153  0 .6661.079  131  0 .7911.086  169  0 .5201.059  152  0.4361.051  143  .0 .3431.042  135  0 .791±.084  175  0 .6581.073  162  0.5291.061  149  0 .4761.057  134  0 .776±.082  175  0 .5571.062  158  0.4651.062  149  0 .4261.051  138  0 .7521.083  161  0 .5811.065  156  0.5421.064  143  0 .4971.061  135  TABLE 26 (continued) Variance (xlO~ ) No Strata c ±S.E. w 2  Two Strata DF  Three Strata  Four Strata  a ±S.E. w  DF  a ±S.E. w  DF  a ±S.E.  DF  2  2  2  W  0 ,748±.080  172  0 .604±.068  155  0 .538±.063  146  0.4361.053  131  0 .732±.078  176  0 .664±.073  163  0 .579±.066  150  0.5611.067 '  133  0 .712±.076  174  0 .637±.071  157  0 .605+.070  148  0.572 + .069  135  0 .702±.075  173  0 •427±.048  156  0 .455+.053  147  0.3781.045  136  0 .693±.074  175  0 .597±.067  158  0 .5631.065  149  0.5341.064  130  0 .599*.064  17 2  0 .499*.056  155  0 .4051.047  146  0.3291.040  140  0 .537±.057  173  0 .503±.057  156  0 .4771.055  147  0.3761.046  134  0 .478±.051  173  0 .427±.048  156  0 .3391.039  147  0.3221.039  133  0 .464±.049  177  0 .426±.047  164  0 .3611.041  151  0.3651.044  136  0 .458±.049  170  0 .459±.052  153  0 .4291.050  144  0 .3951.048  134  133 The within herd-period variance of percent lactose (Table 27) herd means ranged from 0.00836 ± .00089 to 0.00352 ± .00038 without s t r a t i f i c a t i o n and from 0.00714 ± .00086 to 0.00276 ± .00034 with four s t r a t a per period. The herd means are a l l below the maximum value allowed i f the c r i t e r i o n of precision  (Table 21) i s to be met and  four samples are taken each period. Laboratory determinations were done f o r a l l herds at approximately the same time; therefore differences among herds can mainly be a t t r i b u t e d to differences i n b i o l o g i c a l variance.  D i s t r i b u t i o n of Within Herd-Period Variances Table 28 shows the frequency d i s t r i b u t i o n of the within herd-period variances of milk f a t percent calculated with no s t r a t a and with two, three and four s t r a t a per period.  A histogram of the d i s t r i b u t i o n i s shown i n  Figure 8 f o r no- s t r a t a and f o r four s t r a t a .  With four  samples taken at random i n a period (no strata) 77.57 percent (Table 31) of the i n d i v i d u a l herd-periods were predicted to meet the s p e c i f i e d c r i t e r i o n of p r e c i s i o n (Table 14). With three s t r a t i f i e d random samples (one sample from each of three strata) 77.14 percent of the herd-periods w i l l also meet the same standard; therefore, s t r a t i f i c a t i o n w i l l r e s u l t i n the saving of one sample  TABLE 27 HERD AVERAGE WITHIN HERD-PERIOD VARIANCE OF PERCENT LACTOSE WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA Variance (x!0~ ) No Strata  Two Strata  Three Strata  a ±s.E. w  DF  a ±s.E. w  DF  0.836±.089  175  0.786±.087  .806±.087  170  .726±.077  2  2  a  Four Strata  .E.  DF  c ±S.E. w  DF  162  0.744± .086  149  0.714±.086  137  .763±.086  154  .615±.072  143  .618± .076  137  175  .564±.063  158  .532± .061  149  .522± .063  135  .681±.072  176  .622± .069  159  • 630± .072  150  .574±.069  134  •616±.066  172  .482±.054  155  .438±.051  146  .4161.051  130  .609±.065  175  .590± .065  163  .574± .066  150  •582±.070  131  .599±.064  175  .525±.059  158  .4701 .054  149  .459+.055  136  •581±.061  177  .521±.057  164  .443± .051  151  .4471.053  133  .580± .062  172  .446+.050  155  .460± .054  146  .4201.051  131  .540±.060  161  .460±.052  156  .440±.052  143  .4171..051  135  .537±.057  173  .449±.050  160  .449±.052  147  .4371 .053  134  .510±.054  174  .413±.046  161  .398±.046  148  .3721 .045  138  .507±.056  164  .450±.050  159  .423±.049  148.  .3771.046  135  2 ± S  2  TABLE 27 (continued) Variance (xlO*" ) No Strata a ±S.E. w 2  Two Strata DF  Four Strata  Three Strata  C iS.E. w  DF  a is.E.  DF  cr iS.E.  DF  2  2  w  2  W  .503±.054  17 3  .4121.046  156  .4411.051  147  .3891.047  131  .501±.054  173  .3311.037  156  .3161.037  147  .2761.034  133  .486+.054  163  .4331.049  151  .3621.043  142  .4321.053  135  .473±.050  174  .3791.042  157  .3651.042  148  .3411.041  136  .467±.051  169  .3841.044  152  .3661.043  143  .3451.042  130  .4461.047  179  .4111.045  166  .3841.044  153  .3961.047  140  .4341.046  174  .3711.042  157  .4011.046  148  .3681.044  134  .4201.045  172  .3491.039  155  .3461.040  146  .3421.042  133  .4201.045  173  .3301.037  156  .3331.039  147  .2801.034  136  .3521.038  170  .3121.035  153  .3001.035  144  .2821.035  134  on  TABLE 28 FREQUENCY DISTRIBUTION OF THE VARIANCE OF PERCENT MILK FAT CALCULATED WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA PER PERIOD Relative and Cumulative Frequencies Number of Strata Class Limits  None Cum.  %  0.0 .005 .010 .0150 .0200 .0250 .0300 .0350 .0400 .0450 .0500 .0550 .0600 .0650  -  —  0.0049  17.95  .0099 .0149  31.73  Two a  %  Three Cum.  a  24.52  %  Cum.  a  32.06  40.13 15.92  64.65  38.41  21.47  49.68 71.15  80.57  .0249  10.90 5.13  82.05 87.18  8 .60 4.46  .0299 .0349  5.13 2.56  92.31 94.87  .0399  1.28  .0449  0.96  .0499 .0549  0.96 0.32  .0599 .0649  .0199  Four %  Cum.  a  33.33 35.56 15.87  68.89  14.60  70.47 85.07  89.17 93 .63  6.03 3.81  91.10 94 .91  6.98  91.74  3.17  94.91  2.87 1.59  96.50 98.09  2.22 1.90  97 .13 99 .03  2.54 0.32  97.45 97.77  96.15 97.11  0.0 0.0  98 .09  0.0 0 .32  99.03  1.59  99.35  99.36 99.36  98.07  1.27  99.36  0.0  0.32  98.39 98.71  0.32  99.36 99.68  0.32  99.03  0.0  99.68  0.96  99.99  0.32  r  98 .09  100.00  84 .76  0.0 0.32  99.35  0.0 0.32  99.67  0.0  99.68  0.0  99.67  0.0  99.68  0.0  99.67  0.3.2  0.32  99.99  99.68  100.00  LO CA  TABLE 28  R e l a t i v e and C u m u l a t i v e  (continued)  Frequencies  Number o f S t r a t a None Mean Stan.  Smallest  0 .00928  0 .00926  .01373  .00897  .00785  .00796  .1592  .0685  .06503  .06358  .00146  .00065  .00133  .00089  Value  Cumulative  312  frequencies.  Four  0.01052  Value  Number  Three  0.01371 Dev.  Largest  Two  314  315  315  138  50  40  -  1  30  -  20  -  10  -  Four s t r a t a No  1.0 Figure 8  2 0  _2  strata  4.0  >6.0  Variance (xlO ) D i s t r i b u t i o n of the within herd-period variance of milk f a t percent (no strata and four strata)  139 per period i n order to meet the same c r i t e r i o n f o r the same proportion of herd-periods.  With four s t r a t i f i e d random  samples per period 89.53 percent of the subclasses w i l l meet the c r i t e r i o n . Table 29 shows the frequency d i s t r i b u t i o n of the i n d i v i d u a l herd-period variances of percent protein without s t r a t a and with two, three and four strata per period. histogram i s presented i n Figure 9.  A  With four simple  random samples per period 41.03 percent of the herd-periods were below the l i m i t s s p e c i f i e d i n Table 21.  The  percentages f o r s t r a t i f i e d random sampling were; 33.97 and 68.89 f o r three and four strata respectively (Table 31). Table 30 shows the frequency d i s t r i b u t i o n of the within herd-period variances of percent lactose with no s t r a t a and with two, three and four s t r a t a per period. A histogram i s presented i n Figure 10.  With four simple  random samples per period 89.42 percent of the herdperiod were below the l i m i t s s p e c i f i e d i n Table 21.  With  s t r a t i f i e d random sampling the percentages were 83.81 and 95.87 f o r three and four s t r a t a per period r e s p e c t i v e l y .  A l l Possible Samples f o r Seven Sampling Schemes Experiment  I  A l l possible samples f o r seven random sampling schemes (Material and Methods) from the data of Experiment  were computer generated  I . The deviation of each  TABLE 29 FREQUENCY DISTRIBUTION OF THE VARIANCE OF PERCENT PROTEIN CALCULATED WITHOUT STRATA AND WITH TWO, THREE AND FOUR STRATA PER PERIOD R e l a t i v e and C u m u l a t i v e F r e q u e n c i e s Number o f S t r a t a None  Two  Three  Four  t  Class Limits  a  Cum.  %  a  Cum.  %  a  Cum.  %  a  0.0049  37.18  .0099  39.10  76.28  39.17  87.58  32.70  87.94  27 .30  91.43  .0100  .0149  13.14  89.42  6.37  93.95  94 .29  3.49  94 .92  .0150  .0199  . 6.41  95.83  3.50  97.45  6.35 3.49  97 .78  3 .17  98 .09  .0200  .0249  1.92  97.75  98 .72  1.28  99.03  1.27 0.32  0.63  .0299  98.72 99.04  99 .05  .0250  1.27 0.32  99.37  0.95  99.67  .0300  .0349  0.64  99.67  0.96  100.00  0.32  99.69  99.67  .0350  .0399  0.32  99 .99  0.32  100.01  0.0 0.32  0.0 .0050  -  Cum.  %  48.41  55.24  64 .13  99.99  Mean S t a n . Dev.  =00790 .00581  0 .00644 .00480  0 .00600 .00485  0. 00528 o 00452  Largest Value  .03954  .03261  • 03655  Smallest Value Number  .00061  .00053  .03700 .00054  312  Cumulative frequencies.  314  315  »  00059  315 £ o  141 70  T  60  50  -  40  -  Four s t r a t a No 30  -  20  -  10  _  1.0  2.0  Variance Figure 9  strata  3.0  CxlO"* ) 2  D i s t r i b u t i o n of the within herd-period variance of protein percent (no strata and four strata)  TABLE 30 FREQUENCY DISTRIBUTION OF THE VARIANCE OF PERCENT LACTOSE CALCULATED WITHOUT STRATA AND WITH TWO, THREE AND FOUR STRATA PER PERIOD Relative and Cumulative Frequencies Number of strata  Class Limits 0.0  % 0.0049  .0050 - . .0099  %  68.79 .25.16  93.95  67 .94 27.94  95.88  25.40  95. 88  1.59 1.90  97.47 99.37  3.17 0.0  99. 05 99. 05  0.0 0.32  99.37  0.32  99.37  99.69  0.67  100. 00  %  57.69 31.73  89.42  .0100 .0150 -  .0149 .0199  7.05 2.56  96.47 99.03  3.18  97.13  2.23  99.36  .0200 .0250 -  .0249 .0299  0.32  99.35 99.67  0.0 0.0  99.36  .0300 Mean  .0349  0.32  0.32  99.99 0.00552  Stan. Dev. Largest value Smallest value Number  99.36  0.64 100.00 0. 00470  % 70.48  0.32 100.01 0. 00447  0.00427  03195 00027  •  00026  .00028  «  00372  o  03061  e  •  00054  •  Cumulative frequencies.  a  .00331 .02788  00412  314  Cum.  00363 Q 03325  c  312  Cum .a  _, a Cum.  Cum.  a  Four  Three  Two  None  <*  315  315  143 70  -f  1  60 -  50 -  W O <  40 -  Four s t r a t a No  strata  W U P4  30 -  W  20 -  10 -  —I  1.0  2.0  3.0  4.0  -2 Figure 10  Variance CxlO ) D i s t r i b u t i o n of the within herd-period variance of lactose percent Cno strata and four strata)  144  TABLE 31 PERCENTAGE OF HERD-PERIOD SUBCLASSES PREDICTED TO MEET THE CRITERION OF PRECISION (TABLE 21) . Number of Strata Number of Samples  None  Three  Four  Percent milk f a t 3  60.26  4  77.57  77.14 89.53  Percent protein 3  19.87  4  41.03  33.97 68.89  Percent lactose 3  73.08  4  89.43  83.81 95 .87  145 sample mean  (percent milk  f a t and  protein)  from  the  estimate  o f t h e h e r d - p e r i o d mean were c a l c u l a t e d  possible  samples  periods. of  Frequency d i s t r i b u t i o n s  these  schemes.  expected  the herds  These  results  relative  and  in Tables  32A  33B  and  The the  those  confidence schemes  and  of the  limits  33B)  value  limits  sampling  standard  and  F o r example:  seven  for  sampling  protein.  limits  schemes a r e f a t and  shown  presented  of the d e v i a t i o n s .  f o r t h e mean p e r c e n t m i l k f a t  experiment  and  two  32B)  protein  than the  agree  (Table 9).  protein  for percent milk  reduced  the  i n Tables  Histograms are  ( T a b l e s 3 2A  estimated  the  t h e number o f a l l p o s s i b l e  are  are fat.  well  (Tables the  s m a l l e r f o r each Therefore, protein  t h e mean p e r c e n t frequency  the magnitude of the t h r e e s a m p l e s and  with  The  larger  however  scheme, t h e mean p e r c e n t  with  been used  e r r o r o f t h e mean,  for percent milk  for percent  Stratification  deviations  had  seven  indicated  frequencies of  than p r e d i c t e d (Table 9);  be more p r e c i s e l y fat.  cumulative  for percent  schemes t h a n  any  distributions  of the d i s t r i b u t i o n  p r e d i c t e d from  confidence  for  t o 14  seven  confidence 33A  32B  f o r percent  i n F i g u r e s 11  for  frequency  a b s o l u t e d e v i a t i o n and  samples f o r each o f the  and  o f the a b s o l u t e  I.  absolute d e v i a t i o n s , the  33A  subclass f o r a l l herd-  i f random s a m p l i n g  i n Experiment  The  largest  for a l l  d e v i a t i o n s were c o n s t r u c t e d f o r e a c h o f t h e  sampling the  i n each h e r d - p e r i o d  fresh  would milk  of large  largest deviation. three strata  the  TABLE 32A FREQUENCY DISTRIBUTION OF THE ABSOLUTE DEVIATIONS OF ALL POSSIBLE SIMPLE RANDOM SAMPLES, WITH ONE TO FOUR OBSERVATIONS PER SAMPLE, FROM THE PERCENT MILK FAT FRESH MEAN Relative and Cumulative Frequencies One Class Limits 0.00 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160 0.180 0.200 0.250  —  %  Number of Observations per Sample Two Three Cum.  %  a  Cum.  a  %  0.019 0.039  16 .43 15 .45  31.88  24 .02 21.19  0.059 0.079  14 .08 12 .21  45.96 58.17  17.31 12.49  0.099  10 .00  68.17  8.79  83.80  0.119  7 .47  75.64  5.60  89 .40  0.139  81.65  3.68 2.42  93.08  0.179  6 .01 4 .72 3 .44  3.93 2 .18  95.50 97.01  0.199  2 .86  0.249 0.299  0.159  86.37 89.81  1.51  Four Cum.  a  30 .22  %  Cum.  a  45.21  25 .03  55.25  35 .78 27 .48  62.52  17.73 11.20 6.74  17 .26 9.47  80.52  75.01  72 .98 84 .18 90.92  4 .91  94.85  2 .49  94.90 97.39  1.19 0.72  97.03 98 .22 98 .94  1 .21 0.64  98 .60 99.24  0.32  99.56  0.40 0.43  99.34 99.77  0.18 0 .18  99.74 99.92  0.15  99 .92  0.05  99.97  1.04  3 .95  92.67 96.62  1.23  98.05 99.28  1 .80  98.42  0.42-  99.70  63.26 89.99  TABLE 32A (continued) Relative and Cumulative Frequencies Number of Observations per Sample  % .300 ~ Largest Value . Mean Deviation Stan. Dev.* 99% C.L. No. of Samples 3  C  Cum.  a  1.59  100.01  %  Cum.  a  0.30  Four  Three  Two  One  100.00  %  Cum.  a  0.08  100.00  %  Cum.  a  0.03  100.00  0.742  0.623  0 .605  0.538  0.085  0.058  0.038  0.045  0.113  0.077  0.060  0.050  ±.292  ±.199  ± .156  ± .129  4,511  30,154  124,803  357,659  Cumulative r e l a t i v e frequencies. ^Standard deviation of the d i s t r i b u t i o n of deviations. °99% confidence l i m i t s .  TABLE 32B FREQUENCY DISTRIBUTION OF THE ABSOLUTE DEVIATIONS OF A L L POSSIBLE STRATIFIED RANDOM SAMPLES, WITH ONE OBSERVATION PER STRATA AND TWO TO FOUR STRATA, FROM THE PERCENT MILK FAT FRESH MEAN R e l a t i v e and C u m u l a t i v e  Frequencies  Number o f S t r a t a Two Class Limits 0.00 .020 .040 .060 .080 .100 .120 .140 .160 .180 .200 .250 .300  -  —  %  Three Cum.  a  %  Four Cum.  a  %  Cum.  0.019  26.33  .039  22.78  49.11  27.45  62.37  29.72  70.33  .059  18.76  67.87  17.87  80.24  16.18  86.51  .079  11.79  79.66  9.92  90.16  7.57  94.08  .099  8.16  87.82  5.01  95.17  3.42  97.50  .119  4.85  92.67  2.49  97.66  1.39  98 .89  .139  2.89  95.56  1.21  98 .87  0.65  99.54  .159  1.86  97.42  0.52  99 .39  0.28  99.82  .179  0.89  98 .31  0.30  99.69  0.09  99.91  .199  0.73  99.04  0.14  99.83  0.05  99.96  .249  0.65  99.69  0.13  99.96  0.04  100.00  .299  0.20  99.89  0.04  100.00  0.11  100.00  34.92  40.61  CO  a  TABLE 32B  (continued)  Number of Strata  Largest Deviation Mean Deviation Stan. Dev.* 99% C L .  Three  Four  0.465 .051  0.295 .038  0.236 .032  .0675 .174  3  Number of Samples  Two  16,081  Cumulative r e l a t i v e frequencies. 'standard deviation of the d i s t r i b u t i o n of deviations. 99% confidence l i m i t s .  .0529 .136 34,240  .0416 .117 50,469  TABLE 33A FREQUENCY DISTRIBUTION OF THE ABSOLUTE DEVIATIONS OF A L L POSSIBLE SIMPLE RANDOM SAMPLES, WITH ONE TO FOUR OBSERVATIONS PER SAMPLE, FROM THE PERCENT PROTEIN FRESH MEAN  Relative  and C u m u l a t i v e  Relative  Frequencies  Number o f O b s e r v a t i o n s p e r Sample One Class Limits 0.0 .020 .040 .060 .080 .100 .120 .140 .160 .180 .200 .250 .300  —  %  Two Cum.  a  %  Three Cum.  a  %  Four Cum.  a  %  Cum.  a  0.019  21.41  .039  19.53  40.94  25.25  55.32  28.73  65.68  30.44  73.67  .059  16.47  57.41  18 .30  73.62  17.32  83 .00  15.59  89.26  .079  12.28  69 .69  11.40  85.02  9.05  92.05  6 .47  95.73  .099  9.47  79.16  6.71  91.73  4 .25  96.30  2.60  98.33  .119  7.27  8 6.43  3.75  95.48  1.93  98.23  1.04  99.37  .139  4.61  91.04  1.94  97.42  0.90  99.13  0.40  99.77  .159  3.08  94.12  1.08  98.50  0.49  99.62  0.15  99.92  .179  1.97  96.09  0.59  99.09  0.22  99.84  0.05  99.97  .199  1.42  97.51  0.34  99.43  0.10  99.94  0.02  99.99  .249  1.42  98 .93  0.41  99.84  0.06  100.00 •  0.01  100.00  .299  0.53  99 .46  0.13  99.97  0.01  100.01  0.53  99.99  0.03  100.00  30.07  36.95  43.23  TABLE 33A  (continued)  Number o f O b s e r v a t i o n s p e r Sample One  TWO  Three  Four  0.598  0.447  0.305  0.249  Mean D e v i a t i o n  .064  .044  .035  .029  Stan.  .0856  .0585  .0460  .0381  .221  .151  .119  .099  4,511  30,154  124,803  357,659  Largest Deviation  99%  Dev.*  C.L.  3  C  Number o f S a m p l e s  Cumulative r e l a t i v e frequencies. Standard  c  99%  d e v i a t i o n o f the d i s t r i b u t i o n  confidence l i m i t s .  of deviations.  TABLE 33B FREQUENCY DISTRIBUTION OF THE ABSOLUTE DEVIATIONS OF ALL POSSIBLE STRATIFIED RANDOM SAMPLES, WITH ONE OBSERVATION PER STRATA AND TWO TO FOUR STRATA, FROM THE PERCENT PROTEIN FRESH MEAN R e l a t i v e and C u m u l a t i v e R e l a t i v e F r e q u e n c i e s Number of Class Limits  Strata  Two  Four  Three  0.019  32.36  .020 .040  .039 .059  26.85 18.31  59 .21 77.52  30.47 16.07  71.86 87 .93  30.97 12.56  81.68 94.24  .060 .080  .079 .099  11.36 5.67  88 .88  7.05 2.82  94 .98 97.80  3.80 1.18  98 .04 99.22  .100 .120  .119 .139  2.64 1.20  97.19 98.39  1.12 0.57  98 .92 99.49  0.50 0.19  99.72 99.91  .140 .160  .159 .179  0.66 0 .30  99.05 99.35  0.30 0.12  99.79 99.91  0.07 0.01  99.98 99.99  .180  .199  0.25  0.01  100.00  .249  0.24  0.05 0.02  99.96  .200  99.60 99.84  .250  .299  0.12  99.96  0.01  99.99  0.03  99 .99  0.0  .300  41.39  94.55  50.71  99.98  TABLE 33B  (continued)  Number o f S t r a t a  Largest Mean  Dev.*  Four  0.377  0 .309  0.182  .040  .031  .024  .0531  .0404  .0317  C.L.  .137  .104  .082  16,081  34,240  50,469  5  C  Number o f S a m p l e s  a  Three  Deviation  Stan. 99%  Deviation  Two  Cumulative relative  frequencies.  ^Standard d e v i a t i o n of the d i s t r i b u t i o n c  99%  confidence  of deviations.  limits.  r-  1  Cn LO  50  cu  T  154  40 .  rH  Cu  CO rd 4->  30 -  %  Protein  %  Milk f a t  O  tri MH  o  4J  20 "  CD O  u CD PH  10 -  40 co <D  rH  §•  rd CO  30 -  •H rd 4-1  o  tH  20 -  m O  +J C (D O  u  10 -  CD PH  0.02  Q.06  0.10  0.14  0.18  0.22  >0.3  Absolute Deviation From Fresh Sample Mean Figure 11  D i s t r i b u t i o n of absolute deviations of a l l possible single samples (n=l> from the fresh sample estimatepercent milk f a t and p r o t e i n  50 -, 155 CO  cu  40 _  •H  0)  c •rH  c  cu ^  o u «  ft B rd CO rr dH -P O  30 -  % Protein  <4H  0  -P  20 _  CU  TJ  CD  -P rd  O  U  (U  o  10 -  •H TJ  C  CP •H rH  g rd co TJ  40  cu  •rH <4H  •rH  •P rd U -P co MH O  o  •H  •P  co cu  rH  ft  rd -P O EH  •rH  a  Q  % Milk f a t 20 -  UH  O -P  (0  30 -  CO  H •P  •H  n  cu O M  10 -  CU  PU  0.02  0.06  0.10  0.14  0.18  0.22  >0.3  A b s o l u t e D e v i a t i o n From F r e s h Sample Mean Figure  12  D i s t r i b u t i o n o f absolute d e v i a t i o n s of a l l p o s s i b l e samples o f s i z e two (n=2) from t h e f r e s h sample e s t i m a t e - p e r c e n t m i l k f a t and p r o t e i n  50 156 01  cu  H  B fd  40  .  1  LO CD  c  -H  r-H  rd  •P  fl CD  M O M CQ  >i  X) T3  CD  o  30 -  % Protein  E-«  MH O •P  c CD o  20 -  >H  0)  PH  4->  rd  10 .  o •rH  fl tn fl  -H rH  Ht rd  co  40 _  cu -rH MH -rH  -P  id  01  U  -P  co MH  rd  w  fl fd o •P  •H -P  0  Xi  •H H 4J 01 •H  Q  30  O EH  % Milk f a t  20 .  MH  O -P  fl  CD O  10  -  n  CD PH  0.02  0.06  Absolute F i g u r e 13  0.10  0.14  0.18  0.22  >0.3  D e v i a t i o n From F r e s h Sample Mean  D i s t r i b u t i o n o f a b s o l u t e d e v i a t i o n s of a l l p o s s i b l e samples of s i z e t h r e e (n=3) from the f r e s h sample e s t i m a t e - p e r c e n t m i l k f a t and p r o t e i n  50  T 157  40 "  %  30 -  Protein  20  10 -  40  30  . % Milk f a t  20  .  ia  ~1  0.02  0.06  0.10  Absolute Deviation Figure  14  0.14 From F r e s h  0.18  0.22  r >0.3  Sample Mean  D i s t r i b u t i o n of absolute deviations of a l l possible s a m p l e s o f s i z e f o u r (n=4) f r o m t h e f r e s h s a m p l e estimate-percent m i l k f a t and p r o t e i n  158 largest deviation was 0.295 i n absolute value but the largest deviation with three simple random samples was 0.605 for percent milk f a t ; with four samples the largest values were 0.23 6 and 0.538 percent milk f a t for s t r a t i f i e d and simple random sampling r e s p e c t i v e l y .  In both cases  the largest deviation from the fresh mean with  stratified  sampling was less than one-half as large as the l a r g e s t deviation with simple random sampling.  Monitoring Random Sampling A milk sampling scheme should contain provisions f o r resolving a disputed r e s u l t  ( i . e . the producer considers  that a p a r t i c u l a r estimate i s too low).  As producers  receive the r e s u l t s of the analyses a f t e r the period to which i t applies i s over, any additional samples taken i n order to s e t t l e a disputed r e s u l t are from milk shipped i n the next period and consequently are an unsatisfactory check of the estimate of the previous period mean.  There-  fore i t would be worthwhile  test  to monitor the observed  r e s u l t s as they are accumulated  so that the decision to  eliminate or replace observations which show large deviations from p r i o r tests could be made before the period i s over. With s t r a t i f i e d random sampling  (one observation  per strata) differences between consecutive milk samples can be attributed to three sources:  1.  technical errors such as; sample m i s i d e n t i f i c a t i o n , equipment malfunction, e t c .  2.  e r r o r s , i n the s t a t i s t i c a l  sense, due  to sampling  from adjacent strata with the same means and variances. 3.  to true but unknown differences between adjacent s t r a t a means.  Large deviations between consecutive milk samples due  to  points 2 and 3 above are expected to occur but are v a l i d unbiased estimates of the true mean and i n general observations  the  should not be replaced or eliminated.  deviations due to point one above however should  Large  be  detected and the offending observation should be replaced or eliminated i f the error cannot be corrected. i t may  However,  not be possible to determine the cause of large  deviations; therefore, under p r a c t i c a l conditions an a d d i t i o n a l sample would have to be taken i f large deviations  unexplained  occurred.  The expected d i s t r i b u t i o n of the deviations under the conditions of point two w i l l have a mean equal to zero and a variance equal to twice the within s t r a t a variance. On t h i s basis 99 percent of the deviations are expected to 2 l i e within the i n t e r v a l , ±2.575 2 , where a i s the 2 a  within s t r a t a variance.  w  Values for  various sampling 2 schemes were calculated using estimates of a from w Experiments I and II (Table 34). The d i s t r i b u t i o n of  160  TABLE 34 99 PERCENT CONFIDENCE LIMITS OF THE DIFFERENCE BETWEEN TWO RANDOM MILK SAMPLES  Milk Constituent  % Fat  % Protein  % Lactose  Experiment Number  Number of Strata None  Two  .426  .374  .351  .351  II  .420  .394  ,393  .381  I  .323  .283  .266  II  .259  .225  .195  .201  I  .270  .249  .243  .237  II  .253  .246  .237  .232  I .  , .293  Three  Four  161 differences as  above  under p o i n t three w i l l  ( p o i n t two)  but  will  difference  between  two  The  general  principle  monitor value  (of m i l k  sample  to  so  that technical  so  t h a t d e v i a t i o n s due  critical percent values  values  t o be  limits  between  exceeds the  critical  determine  error milk  sample  strata  and  should  i s detected  be  detected are  the  sample  the  o r when c h a n g e s  i n the  of  the  particular  of  the  evaluation of  however, the  on  herd  proposed  are  be  and  or  whether  checked  management o r  technical  replace the  when t h e  The  i f  i f an  I f no  (across herds)  considered.  be  additional  strata.  deviation i s reasonable  considered  these  alone  should  occurred  based  strata  of  If a deviation  d e c i s i o n to r e t a i n  two  99  the  sampling  c o r r e c t e d an  between  enough  Reasonable  Use  zero.  e r r o r has  difference  enough  large  34.  observation  h a v e t o be  the  critical  schemes a r e  means w e r e the  but  to  i n a hundred would  drawn f o r t h i s  then  a  ignored.  to chance of  c a n n o t be  be  observation w i l l  magnitude of  value  technical  i s detected  error the  i f a  select  shown i n T a b l e  due  the  system designed  f o r various sampling  differences  to  i n any  to chance  replaced  to  sample d i f f e r e n c e ) s m a l l  w o u l d mean t h a t one  expected  variance  strata.  scheme i s t o  e r r o r s can  confidence  same  h a v e a mean e q u a l  adjacent  a random sampling  have the  average are composition  latter  part  d e v i a t i o n s i s somewhat s u b j e c t i v e ; monitoring  scheme  should  reduce  the  number of observations with r e a l errors and therefore increase the producers' confidence i n the random sampling scheme.  The monitoring system could be made more objective  i f estimates of the expected difference between s t r a t a means could be associated with the month or season i n which the s t r a t a  fell.  Data c o l l e c t e d i n a random sampling program should be analysed regularly so that the program can be evaluated and modified i f necessary.  Computer handling  of milk t e s t r e s u l t s make regular analyses r e l a t i v e l y simple.  Factors that should be considered i n such  analyses include: 1.  the e f f e c t of seasonal changes i n milk cons t i t u e n t percentages on the differences between s t r a t a means;  2.  the e f f e c t of season on within herd-period variance of milk constituent percentages;  3.  the i d e n t i f i c a t i o n of herds with large shipment to shipment v a r i a t i o n i n milk c o n s t i t uent percentages; and  4.  the estimation of t e s t i n g variance by regular r e p l i c a t e t e s t i n g of milk samples from randomly selected herds.  The r e s u l t s of these analyses could be used to modify the sampling program f o r c e r t a i n herds or seasons.  The  r e s u l t s could also be used to adjust the c r i t i c a l values (Table 341 i f needed.  164  CONCLUSIONS  Variances associated with the procedures of sampling bulk milk and of forming composites were concluded to be small r e l a t i v e to the t o t a l within herd-period variances of milk constituent percentages.  Sampling and laboratory  (including testing) procedures used i n t h i s study—except for the formation of composites i n Experiment I I I — w e r e those usually followed i n B r i t i s h Columbia and the work was done by the people who are r e g u l a r l y employed to do t h i s work. Therefore, estimates of variances associated with sampling and laboratory procedures were estimates of v a r i a b i l i t y under normal f i e l d conditions. Variances associated with the laboratory analyses of milk samples were concluded to be r e l a t i v e l y large. Milk testing procedures were found to be the main source of v a r i a t i o n of estimates of percent lactose.  Therefore  the variance of estimates of percent lactose depends mainly on the number of samples analysed to obtain these estimates.  Testing variances for percent milk f a t and  percent protein were concluded to vary from time to time.  165 If  testing  dictions and  variances vary  then  statistically  of the variance o f estimates  protein  cannot  be made.  However p r a c t i c a l c o n s i d e r a t i o n of the  variance of these estimates; a reasonable sufficient.  current Green  study,  Estimates the study  [10] i n d i c a t e d  milk  f a t and p r o t e i n  than  0.007. Biological  the  total  of testing by Dunn  expected  approximation  v a r i a n c e s from; t h e  [6] a n d t h e r e v i e w  t h a t the t e s t i n g  t o be  v a r i a n c e i s sampling  variance  the s t a t i s t i c a l  of  t h e v a r i a n c e o f e s t i m a t e s o f h e r d - p e r i o d mean m i l k percentages  a n d i s , t h e r e f o r e , n o t a component  obtained  which a l l shipments a r e sampled. estimated  However,  sampling the  i t was  was w o r t h w h i l e  frequency  herd-period  variance  that s t r a t i f i e d  r e l a t i o n s h i p s were  and m u l t i p l e l i n e a r  than i n  to  random reduce  t h e t r u e mean. found  between w i t h i n  shipment weight,  and p e r c e n t  was  relatively  a s i t w o u l d be e x p e c t e d  v a r i a n c e s and m i l k  schemes i n  (strata)  r e d u c t i o n was  concluded  f a t , percent protein  simple  sampling  Biological  o f l a r g e d e v i a t i o n s from  Significant  milk  from  t o be s m a l l e r i n s h o r t p e r i o d s  long p e r i o d s , but the average small.  half  milk  in  constituent  sense  less  f o r approximately  within herd-period variance of percent Biological  by  variances of percent  c o u l d u s u a l l y be e x p e c t e d  v a r i a n c e accounted  f a t and p r o t e i n .  pre-  of percent milk f a t  do n o t r e q u i r e p r e c i s e d e t e r m i n a t i o n s  is  valid  lactose  percent by  regression techniques.  However,  1 6 6  the proportion of the sums of squares accounted f o r by the regression equations was r e l a t i v e l y small f o r a l l equations.  Therefore, the r e l a t i o n s h i p are not useful f o r  predicting herd-period variances. Two-week composite  samples were concluded to y i e l d  biased estimates of true means.  Random samples are  expected to y i e l d unbiased estimates- Deviations of random sample estimates from the true mean should cancel out and, therefore, the mean deviation over a period of time should be close to zero. The variance of the estimates of the mean herd-period milk constituent percentages obtained from milk samples from four randomly selected shipment was predicted to approximate the variance of estimates obtained by the compositing method currently i n use.  The costs associated  with the c o l l e c t i o n and analyses of four randomly chosen milk samples are expected to be lower than the costs associated with the composite method now used. Therefore on the basis of cost comparisons, precision and unbiasedness preferred to composite  expected  random sampling i s to be  sampling.  The p r e c i s i o n of  estimates obtained by s t r a t i f i e d random sampling—four strata and one sample per strata—was concluded to be acceptable,on the average, to the industry.  However, f o r  c e r t a i n herds or periods the sampling frequency may need to be greater to achieve an acceptable l e v e l of p r e c i s i o n  167  A l t e r n a t i v e l y costs could be reduced by taking fewer than four samples f o r c e r t a i n herds or periods and s t i l l an acceptable l e v e l of p r e c i s i o n .  achieve  The r e s u l t s i n t h i s study  indicated that i n the i n i t i a l stages of a random sampling program four samples should be taken f o r each herd-period. The program could be assessed and modified, i f necessary, by using the  r e s u l t s obtained i n the i n i t i a l period.  Starting a random sampling program during periods when within herd-period variance of milk constituent percentages is  expected to be low (winter i n t h i s study) would reduce  the p r o b a b i l i t y of obtaining samples with large deviations from the true mean and allow time to accumulate data to assess the program p r i o r to the advent of more variable seasons.  168  LITERATURE CITED 1.  Anderson and Bancroft. 1952, S t a t i s t i c a l Theory i n Research. McGraw-Hill Book Co., New York.  2.  Biggs, D.A. 1967. Analyzer.  3.  Boswell, R.C, E. Green and D.I. Jenkins. 1967. D a i l y v a r i a t i o n i n the compositional q u a l i t y of ex-farm milk. B r i t . Milk Marketing Board. Tech. Div. Report #58.  4.  Cochran, W.G. 1946. Relative accuracy of systematic and s t r a t i f i e d random samples for a c e r t a i n class of populations. Ann. Math. Stat. 17: 164-177.  5.  Dimick, P.S. and H.V. Atherton. 1962. Factors influencing b u t t e r f a t sampling accuracy i n bulk cooled milk. Vermont Agr. Expt. Sta. B u l l . 626.  6.  Dunn, L.K.  7.  Edwards, R.A. and E. Donaldson. 1966. A study of the v a r i a b i l i t y of the composition of mixed herd milks. J . Soc. Dairy Technol. 19: 110-113.  8.  F i s h e r , R.A. and F. Yates. 1957. S t a t i s t i c a l Tables. O l i v e r and Boyd, Edinburgh. 5th ed.  9.  Freese, F. 1964. Linear Regression Methods for Forest Research. U.S. Forest Serv. Research Pub. FBL17.  1973.  Milk Analysis with the Infrared Milk J . Dairy S c i . , 50: 799-803.  (unpublished data).  10.  Green, E. 1970. Automatic measurement of the f a t and protein contents of milk. Jour. Soc. Dairy Technol. 23: 190193.  11.  Herrmann, L.F. and E.D. Anderson. 1965. Butterfat sampling and t e s t i n g problems. U.S.D.A. Tech. B u l l . # 1336.  12.  Johnson, K.R., D.L. Fourt, R.A. Hibbs and R.H. Ross. 1961. E f f e c t of some environmental factors on the milk f a t and s o l i d s - n o t - f a t content of cows milk. J . Dairy S c i . 44: 658.  169 13.  L i s k a , B.J. and H.E. of a g i t a t i o n samples from Technol. 17:  Calbert. 1954. Study of the influence time on the Babcock t e s t of milk farm bulk holding tanks. J . Milk Food 14-17.  14.  Morris, H.A., S.T. Coulter and C.E. Gates. 1968. within herds i n composition of herd milk. Dairy S c i . 51: 1207-1209.  15.  O'Keefe, M.G. 1967. Factors a f f e c t i n g the design of milk t o t a l s o l i d s testing schemes. J . Dairy Res. 34: 207210.  16.  • 1968. The use of single or composite milk samples for the determination of f a t . J . Dairy Res. 35: 291294.  17.  Preston, H.J. 1954. Developing b u t t e r f a t sampling and t e s t ing programs. U.S. Dept. Agr. Farmer Co-op. Serv. B u l l . 5 (52pp).  18.  Snedecor, G.W. and W.G. Cochran. 1967. S t a t i s t i c a l Methods. 6th ed. Iowa State University Press. Ames, Iowa.  19.  Waite, R., J.C.D. White and A. Robertson. 1956. Variation i n the chemical composition of milk with p a r t i c u l a r reference to s o l i d s - n o t - f a t . I. The e f f e c t of stage of l a c t a t i o n , season of year and age of cow. J . Dairy Res. 23: 65-81.  20.  Welch, B . L . 1956. variances.  Variation J.  On l i n e a r combinations of several J . Amer. Stat. Assoc. 51: 132-148.  

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