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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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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Title | Study of within herd variability in milk fat, protein and lactose content of bulk milks in British Colunbia and factors affecting the design of herd milk sampling programs |
Creator |
Williams, Christopher John |
Publisher | University of British Columbia |
Date Issued | 1973 |
Description | Three sets of data were used to estimate variation, from all 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 April 24, 1971 (Experiment I). The set of three samples was handled as follows: (1) one sample was used in the formation of a two-week composite; (2) one sample was used in the formation of a one-week composite; and (3) one sample was analysed fresh. Four milk samples were taken from each shipment of 22 different herds from November 17 to December 16, 1971. Three of the four samples were analysed fresh in duplicate (Experiment II). The fourth sample was divided into three parts and each part was used in the formation of a composite. Each composite was analysed in duplicate after a two-week collection period (Experiment III). Herd milk was shipped on alternate days. All milk samples (8,894) were analysed for milk fat, 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 fat, protein and lactose were; 0.01371 ± .00030, 0.00787 ± .00017 and 0.00548 ± .00012 respectively. Estimates were obtained from Experiment II of within herd-period variance and its components by the analyses of variance of a hierarchial model. The estimates of these variances for percent milk fat, protein and lactose respectively were: (1) within herd-period variance — 0.01329 ± .00064, 0.00507 ± .00031 and 0.00483 ± .00017; (2) biological (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 significantly different from estimates from Experiment II. Orthogonal polynomials were used to estimate the relationship between the serial correlations (calculated from Experiment I) of milk constituent percentage and the number of shipments separating two shipments for which the correlations were calculated. Only the linear term was significant for percent protein and lactose and accounted for 99.7 and 98.4 percent of the total sums bf squares for these two milk constituents respectively. Linear and quadratic after linear were significant for percent milk fat serial correlations and accounted for 98.4 and 1.3 percent of the total sums of squares respectively. Strata within periods was fitted as an effect (Experiments I and II) in a hierarchal model and was a significant source of variation. The variances of estimates of herd-period mean milk constituent percentages obtained from various simple and stratified random sampling schemes were calculated. Stratification resulted in a relatively small reduction in 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 I and II were near zero. 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 for percent milk fat, protein and lactose respectively. It was calculated that four random samples would estimate herd-period mean milk constituent percentages at least as precisely as two two-week composite samples. Two-week composite samples underestimated percent milk fat by 0.045 percent milk fat 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. Simple and multiple regression techniques were used in an attempt to predict herd differences in within herd-period variance from the average amount of milk shipped and percent milk fat, protein and lactose. In general, large within herd-period variances of milk constituent percentages were significantly associated with small herd milk shipments and high levels of milk fat and protein. However, the proportion of the total sums of squares accounted for by the various regression equations was relatively low; therefore the equations were not useful for predicting herd-period variances. Within herd-period variance of percent milk fat was highest in the spring and autumn; therefore sampling frequency may need to be greater at some seasons than at others. Differences among herds in within herd-period variance of milk constituent percentages were significant; therefore random sampling schemes may have to be modified to suit individual herds. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-03-10 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0093070 |
URI | http://hdl.handle.net/2429/32316 |
Degree |
Doctor of Philosophy - PhD |
Program |
Animal Science |
Affiliation |
Land and Food Systems, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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