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UBC Theses and Dissertations

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

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A STUDY OF WITHIN HERD VARIABILITY IN MILK FAT, PROTEIN AND LACTOSE CONTENT OF BULK MILKS IN BRITISH COLUMBIA AND FACTORS AFFECTING THE DESIGN OF HERD MILK SAMPLING PROGRAMS by CHRISTOPHER JOHN WILLIAMS B.S.A., University of Brit i s h Columbia, 1967 M.Sc, University of Brit i s h Columbia, 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Animal Science We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1973 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission fo r extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of vV The University of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT Three sets of data were used to estimate variation, 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 in the formation of a two-week composite; (2) one sample was used in the formation of a one-week composite; and C3) 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. A l l 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 i t s components by the analyses of variance of a hierarchial model. The estimates of these variances for percent, milk. fat, protein and lactose respectively were: Cl) 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 se r i a l 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 s e r i a l correlations and accounted for 98.4 and 1.3 iv percent of the total sums of squares respectively. Strata within periods was f i t t e d 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 s t r a t i f i e d 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. V 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. v i TABLE OF CONTENTS PAGE ABSTRACT . . i i TABLE OF CONTENTS v i LIST OF TABLES ix LIST OF FIGURES xiv ACKNOWLEDGEMENTS xv i INTRODUCTION . 1 LITERATURE REVIEW 4 PART 1 - ESTIMATION OF POPULATION PARAMETERS 9 INTRODUCTION 9 MATERIALS AND METHODS . . . . . . . . 10 Collection and Analyses of Milk Samples 10 The Problem and Definition of Terms Used 12 S t a t i s t i c a l Methods . . . . . . . . . . . 17 RESULTS AND DISCUSSION 29 Estimates of Within Herd-Period Variance and Components . 29 Effects of Strata „ 35 Within Strata Variance . . . 44 Vari a b i l i t y of Estimates from Various Sampling Schemes . . . . . . . 55 Composite Sampling . . . . . 59 Calculation of the Criterion of Precision 67 v i i PAGE Composite Sampling v e r s u s Random Sampling . . . . 69 CONCLUSIONS . . . . . 79 PART 2 82 INTRODUCTION 82 MATERIALS AND METHODS 85 Source of Data 85 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 on M i l k Shipment Weight! and M i l k C o n s t i t u e n t Percentages . . . . . . . . . . -94 Tr a n s f o r m a t i o n s 96 R e g r e s s i o n A n a l y s e s . . . . . 96 W i t h i n Herd-Period V a r i a n c e o f P e r c e n t M i l k F a t 98 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 P e r c e n t P r o t e i n . . . . . 106 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 P e r c e n t L a c t o s e . . . . . I l l C o n c l u s i o n o f R e g r e s s i o n A n a l y s e s . . . 116 Herd and P e r i o d V a r i a t i o n . 117 Season V a r i a t i o n . . . . . . . . . . 119 Herd V a r i a t i o n 128 D i s t r i b u t i o n o f W i t h i n Herd-Period V a r i a n c e s 133 A l l P o s s i b l e Samples f o r Seven Sampling Schemes - Experiment I . . . . . 139 M o n i t o r i n g Random Sampling . . . . . . . . . . . 158 v i i i PAGE CONCLUSIONS 164 LITERATURE CITED . . . . . 168 i x LIST OF TABLES TABLE PAGE 1. ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS EXPERIMENT I 30 2. ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS EXPERIMENT II . . . . . . . . 31 3. COMPONENTS OF WITHIN HERD-PERIOD VARIANCE (±S.E.) ESTIMATED FROM EXPERIMENT II AND WITHIN HERD-PERIOD VARIANCE (±S.E.) ESTIMATED FROM EXPERIMENT I PERIODS ARE FIFTEEN CONSECUTIVE SHIPMENTS . 32 4. WITHIN HERD SERIAL CORRELATIONS FOR PERCENT MILK FAT, PROTEIN AND LACTOSE . . . . 37 5A. THE REDUCTION IN SUMS OF SQUARES DUE TO SUCCESSIVE TERMS IN THE POLYNOMIAL OF EQUATION 19. PERCENT MILK FAT SERIAL CORRELATIONS 38 5B. THE REDUCTION IN SUMS OF SQUARES DUE TO SUCCESSIVE TERMS IN THE POLYNOMIAL OF EQUATION 19. PERCENT PROTEIN SERIAL CORRELATIONS . 39 5C. THE REDUCTION IN SUMS OF SQUARES DUE TO SUCCESSIVE TERMS IN THE POLYNOMIAL OF EQUATION 19. PERCENT LACTOSE SERIAL CORRELATIONS . . . . . . . . . . . . 40 6A. ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS EXPERIMENT I — TWO STRATA PER PERIOD . . . . . . . . . . . . 45 6B. ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS. EXPERIMENT I — THREE STRATA PER PERIOD 46 6C. ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS. EXPERIMENT I — FOUR STRATA PER PERIOD 47 X TABLE PAGE 7A. ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS.EXPERIMENT II — TWO STRATA PER PERIOD 48 7B. ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS.EXPERIMENT II — THREE STRATA PER PERIOD 49 7C. ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS. EXPERIMENT II — FOUR STRATA PER PERIOD 50 8. 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 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 . . . . . . o . . . . . 57 10. ANALYSIS OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS EXPERIMENT III — ESTIMATE OF COMPOSITING VARIANCE . . . 60 11. ESTIMATES OF COMPOSITING AND TESTING VARIANCE EXPERIMENT III . . . . . . . . . . . 62 12A. ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES FITTING HERDS AND PERIODS (MODEL 8) EXPERIMENT I FRESH SAMPLE ESTIMATES . . . . . . 64 12B. ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES FITTING HERDS AND PERIODS (MODEL 8) EXPERIMENT I TWO-WEEK COMPOSITE ESTIMATES 65 12C. ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES FITTING HERDS AND PERIODS (MODEL 8) EXPERIMENT I TWO ONE-WEEK CONPOSITE ESTIMATES . . . . . 66 x i TABLE PAGE 13. VARIANCE OF COMPOSITES (xlO 2) 68 14. VARIANCES OF HERD-PERIOD MEAN MILK CONSTITUENT PERCENT ESTIMATED BY TWO TWO-WEEK COMPOSITES PER PERIOD . . . . . . . 70 15. PAIRED t-TEST OF DIFFERENCES BETWEEN THE FRESH ESTIMATE OF A TWO WEEK PERIOD MEAN AND BOTH KINDS OF COMPOSITE ESTIMATES . . . 71 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 76 17. TESTS OF NORMALITY OF THE DISTRIBUTION OF WITHIN HERD-PERIOD VARIANCES BEFORE AND AFTER LOGARITHMIC TRANSFORMATION 97 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 . 99 18B. SIMPLE AND MULTIPLE LINEAR REGRESSION COEFFICIENTS PERCENT MILK FAT WITH FOUR STRATA PER PERIOD 100 1 9 A . S I M P L E ( S L R ) A N D M U L T I P L E L I N E A R ( M L R ) R E G R E S S I O N C O E F F I C I E N T S F O R T H E R E G R E S S I O N O F T H E L O G A R I T H M O F 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 P E R C E N T P R O T E I N O N K I L O G R A M S M I L K P E R C E N T M I L K F A T , P R O T E I N A N D L A C T O S E -N O S T R A T A 107 1 9 B . S I M P L E A N D M U L T I P L E L I N E A R R E G R E S S I O N C O E F F I C I E N T S P E R C E N T P R O T E I N W I T H F O U R S T R A T A P E R P E R I O D 1 0 8 2OA. 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 112 x i i TABLE PAGE 20B. SIMPLE AND MULTIPLE LINEAR REGRESSION COEFFICIENTS PERCENT LACTOSE WITH FOUR STRATA PER PERIOD 113 21. MAXIMUM VALUE OF a w FOR THE PRECISION OF A RANDOM SAMPLE TO MEET THE SPECIFIED CRITERION 118 22. PERIOD AVERAGE WITHIN HERD-PERIOD VARIANCE Caw) OF PERCENT MILK FAT WITHOUT STRATIFI-CATION AND WITH TWO, THREE AND FOUR STRATA 120 23. PERIOD AVERAGE WITHIN HERD-PERIOD VARIANCE (oh OF PERCENT PROTEIN WITHOUT STRATIFI-CATION AND WITH TWO, THREE AND FOUR STRATA 124 24. PERIOD AVERAGE WITHIN HERD-PERIOD VARIANCE (a w) OF PERCENT LACTOSE WITHOUT STRATIFI-CATION AND WITH TWO, THREE AND FOUR STRATA . . . . . . . . . 126 25. HERD AVERAGE WITHIN HERD-PERIOD VARIANCE (a w) OF PERCENT MILK FAT WITHOUT STRATIFI-CATION AND WITH TWO, THREE AND FOUR STRATA o 129 26. HERD AVERAGE WITHIN HERD-PERIOD VARIANCE (a w) OF PERCENT PROTEIN WITHOUT STRATIFI-CATION AND WITH TWO, THREE AND FOUR STRATA 131 27. HERD AVERAGE WITHIN HERD-PERIOD VARIANCE OF PERCENT LACTOSE WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA . . . . 134 28. FREQUENCY DISTRIBUTION OF THE VARIANCE OF PERCENT MILK FAT CALCULATED WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA PER PERIOD 136 29. FREQUENCY DISTRIBUTION OF THE VARIANCE OF PERCENT PROTEIN CALCULATED WITHOUT STRATA AND WITH TWO, THREE AND FOUR STRATA PER PERIOD 140 X l l l TABLE PAGE 30. 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 31. PERCENTAGE OF HERD-PERIOD SUBCLASSES PRE-DICTED TO MEET THE CRITERION OF PRECISION (TABLE 211 . . . . . . . . 144 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 146 32B. 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 33A. 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 33B. FREQUENCY DISTRIBUTION OF THE ABSOLUTE DE-VIATIONS OF ALL POSSIBLE STRATIFIED RANDOM SAMPLES, WITH ONE OBSERVATION PER STRATA AND TWO TO FOUR STRATA, FROM THE PERCENT PROTEIN FRESH MEAN . . . . . . . . 152 34. 99 PERCENT CONFIDENCE LIMITS OF THE DIFFERENCE BETWEEN TWO RANDOM MILK SAMPLES . . . . . . . . . 160 xiv LIST OF FIGURES FIGURE PAGE • r 1. Serial correlations of percent milk fat,protein and lactose . . . . . . . . . . . . . 41 2. Within herd-period biological and total variance estimated with no strata and with two, three and four strata per period for percent milk fat protein and lactose . . . . 52 3. The number of samples required (n) for random sampling to equal the precision of composite sampling for various ratios of biological to testing variance (r) calculated from equation 29 . . . . . . . . . 78 4. Period average milk constituent percentages and milk shipment weight for thirteen periods . . . . . . . . . . 95 5. Within herd-period variance of milk fat percent for thirteen periods . . . . . . . . 122 6. Within herd-period variance of protein percent for thirteen periods . . . . . . . . . . . . 125 7. Within herd-period variance of lactose percent for thirteen periods . . . . . . . . . . . . 127 8. Distribution of the within herd-period v a r i -ance of milk fat percent (no strata and four strata) 138 9. Distribution of the within herd-period variance of protein percent (no strata and four strata) 141 10. Distribution of the within herd-period variance of lactose percent (no strata and four strata) 143 11. Distribution of absolute deviations of a l l possible single samples (n=l) from the fresh sample estimate-percent milk fat 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 Distribution of absolute deviations of a l l possible samples of size two (n=2) from the fresh sample estimate-percent milk fat and protein 155 Distribution of absolute deviations of a l l possible samples of size three (n=3) from the fresh sample estimate-percent 155 milk fat and protein Distribution of absolute deviations of a l l possible samples of size four (n=4) from the fresh sample estimate-percent milk fat and protein 157 xv i ACKNOWLEDGEMENTS The author wishes to thank Dr. R.G. Peterson, under whose supervision this study was conducted, for his assistance in planning the project and in analysing the results. The author also thanks Dr. CW. Roberts and Dr. J. Hodges for their suggestions and criticisms. Thanks are extended to the personnel of the Br i t i s h Columbia Department of Agriculture, Dairy Branch: Mr. T.C.T. Chao, Technical Director; Mr. G.D. Johnson, Officer-in-Charge, Dairy Laboratory; and Mr. E.N. Jenstad, Dairy Specialist for supervision of data collection 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 for whole milk i s usually estab-lished per hundred pounds of milk of a given milk fat (and/ or other milk constituent) percentage. This basic price is adjusted for deviations of the milk shipped by individual dairy farmers from the given percentage. Therefore, determining the percent composition of herd milk i s important in paying producers accurately. At the present time, Br i t i s h Columbia producer milk prices are established each month by a pricing formula which includes a d i f f e r e n t i a l for deviations from the given percentage for milk fat only. The accounting period, in B r i t i s h Columbia, i s a calendar month and i t is necessary to sample herd milk in order to determine the monthly average percent milk fat. 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 collection 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 selection 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 in both composite and random sampling. The f i r s t scheme i s currently used in Bri t i s h Columbia and the usual compositing period i s two weeks. The main disadvantages of composite sampling l i e in the labor required to sample each shipment and to transfer the sample to a composite bottle. In addition the compositing procedures and storage of the composites could introduce bias and/or variation in the test results. The second scheme removes the need for forming and maintaining a composite bottle for each herd but i t requires the same number of samples as the compositing method and more laboratory analyses. However, i t i s the most precise of the three schemes. The third 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 test results 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 collection and handling of milk samples. Estimates of herd-period means from random samples contain variation due to true differences between shipments; this is not a source of variation in 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 yield unbiased estimates of the true herd-period percent milk composition. The purpose of this 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 fat, protein and lactose in bulk tank milk shipments and to consider ways of assessing the percent milk fat, protein and lactose in herd milk without composite samples. Only a small amount of research has been done on sampling and testing bulk tank milk and there i s a need for a thorough analysis of a l l sources of variation based on more comprehensive data and longer time periods than has been done in most reported studies. Estimates of the variances associated with bulk tank milk sampling and testing are needed i f the precision 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 fat and total solids content of bulk herd milk in Scotland, O'Keeffe . [15,16] sampled each daily bulk tank shipment of ten herds for twelve months. The milk samples were analysed for percent milk fat by the Gerber method and for percent total solids by the Claesson milk testing machine. From these data he estimated the between daily shipment within herd-month variances as 0.0246 and 0.039 for percent milk fat and percent total solids respectively; estimates of the within herd-year variances were 0.043 and 0.085 for the same two milk components respectively. 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 for per-cent milk fat, 0.181 for percent protein and 0.147 for percent solids-not-fat. Edwards and Donaldson [7] sampled daily bulk tank milk shipments of thirty-two B r i t i s h herds for thirteen days. The milk samples were analysed for percent milk fat by the Gerber method and for total solids by a gravimeter method. The solids-not-fat percentage was 5 calculated by difference. 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 fat, solids-not-fat and total solids respectively. 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 fat, but the largest difference was 0.63 percent milk fat. In O'Keefee's [16] study the largest day-to-day difference was 1.0 percent milk fat. Edwards and Donaldson [7] reported that the 95 percent confidence interval of the difference between two randomly selected single milk samples was ±.39 percent milk fat. These workers found that while there was a tendency for small herds to have greater between shipment variation than large herds the differences among herds were not significant (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] in a comprehensive study of milk fat testing in the U.S.A., sampled 49,117 milk shipments (two days production in each shipment) from herds shipping to eleven different milk plants over a period of four months for most herds and over a one year period for the remaining herds. The milk samples were tested for 6 percent fat by the Babcock method. These workers estimated that the within herd-month standard deviation of percent milk fat was 0.146. Boswell et a l . [3] sampled daily milk shipments from 86 herds throughout England and Wales for a period of one year. The milk samples were analysed for percent milk fat, solids-not-fat and total solids; 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 fat was highest in November; these two studies reported values of 0.177 percent and 0.19 percent respectively for this month. The lowest standard deviations occurred in the late winter and early spring in 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%). The study of Boswell et a l . [3] showed a secondary peak in May CO.17%). O'Keeffe's [16] study showed the highest within herd-month variance of milk fat percent in May CO.0458) with a second peak in October (0.0381); the lowest values occurred in the winter months with January (0.0085) being the lowest. Boswell et a l . [3] reported that high within herd-month variance of percent milk fat was associated with small herds. Herrmann and Anderson [11] used multiple 7 regression techniques to estimate the effects of: (1) level of milk fat, (2) amount of milk shipped, (3) the coeffic-ient of variation of the amount of milk shipped and (4) the variance of environmental temperature on the within herd-month variance of percent milk fat. The regression model accounted for a significant (p<_.05) reduction in the total sums of squares. Of the four independent variables used only the variance of environmental temperature was not a significant (p£.05) source of variation. The remain-ing independent variables were negatively associated with the herd-month variance of percent milk fat. Herrmann and Anderson [11] found that composite milk samples underestimated percent milk fat as compared to the percentage calculated from fresh milk samples. The average amount of bias was -.011 percent milk fat but varied from -.095 percent to 0.031 percent by milk plant; thus indicating that the amount of bias in composite samples depended on the handling of the samples. Preston [17] also reported that the percent milk fat estimated from composite samples was lower that the corresponding percentage 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 daily bulk shipments of eight herds for eight days. The milk samples were analysed in duplicate for percent 8 milk fat by the Milko-Tester Mark II and for percent total solids by the Claesson milk testing machine. He estimated the variances associated with sampling the bulk tank were 0 . 0 0 1 7 and 0 . 0 0 3 2 for percent milk fat and total solids respectively. The variances associated with testing were 0 . 0 0 1 0 and 0 . 0 0 2 4 for percent of the same two milk components respectively; however, O'Keeffe [16 ] suggested that the testing variance of percent milk fat estimated in this study was much lower than i s usually encountered under practical conditions. In a study of bulk tank sampling methods, Dimick and Atherton [5] reported that bulk cooled milk was thoroughly mixed after three minutes of agitation and therefore that the variance associated with sampling bulk tanks i s generally low i f sampling procedures are carefully followed; these results are supported by Liska et a l . [ 1 3 ] . in a review of automatic testing of milk for fat and protein Green [10] reported estimates of the standard deviations associated with testing milk samples on Infrared milk analysers (IRMA) under practical laboratory conditions ranging from 0 . 0 6 to 0 . 0 9 for both percent milk fat 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 specified precision requires knowledge of the appropriate population variances. The purpose of Part 1 of this thesis was to estimate the variances associated with sampling bulk tank milk shipments under various sampling schemes. Two main sources of variation are assumed; (1) variation between the true percent composition of shipments (i.e. sampling variance in the s t a t i s t i c a l sense) and (2) variation associated with the various procedures of estimation. These estimates are used to predict standard errors of herd-period mean milk constituent percentages estimated under different sampling schemes and to determine the number of samples needed i f estimates obtained by random sampling are to equal the precision of estimates obtained by composite sampling. 10 1. MATERIALS AND METHODS Collection and Analyses of Milk Samples Three sets of data were collected for this study-by 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 pro-duction Ofour milkings). A l l milk samples were analysed by the British Columbia Department of Agriculture (BCDA), Dairy Branch Laboratory for percent milk fat, 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 five 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 in the shipment was recorded at the time of sampling. The samples were maintained on ice 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 in the formation of a two-week composite of seven fresh samples; 2. the second sample was used in the formation of a one-week composite of either three or four fresh samples; 3. the third sample was analysed fresh. Mercuric chloride and potassium dichromate were used as perservatives for the composite samples. The composites were formed in the plant receiving the milk. The total numbers of samples analysed were; 4,701 fresh, 697 two-week and 1,334 one-week. Experiment II. Four milk samples were taken from each bulk shipment of twenty-two different herds, a l l shipping to the same milk plant Ca different plant than the herds in Experiment I), for a period of one month, from November 17 to December 16, 1971. Fifteen shipments were sampled per herd. Three of the four samples were 12 analysed fresh in duplicate, with duplicates randomly assigned to analysers (1,910 analyses). The fourth sample was used in Experiment III. Experiment III. The fourth sample collected i n Experiment II 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 (252 analyses). Potassium dichromate was used as a preservative for these composites. This set of composites was accumulated in the BCDA, Dairy Branch Laboratory. The total number of observations for a l l three experiments was 8,894. The Problem and Definition of Terms Used The purpose of this study was to design a random sampling scheme to estimate, with a level of precision acceptable to the dairy industry, the percent milk fat, protein and lactose in milk shipped by farmers during an accounting period. Accounting periods in Br i t i s h Columbia are currently one month long and milk i s usually shipped on alternate days therefore a period in this study (.unless otherwise specified) was defined as fifteen consecutive shipments. Each herd-period of fifteen 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 in i t s s t a t i s t i c a l sense and also to refer to a small quantity of milk removed from a shipment of milk for analysis. The meaning intended should be clear from the context in which the word was used. The term "milk constituent" was used to refer to the three main milk constituents (milk fat, protein and lactose) only. The precision of an estimate may be considered acceptable i f differences between estimates of herd-period means can be mainly attributed to true differences associated with herd, period or herd-period effects and only to a small degree be attributed to the vagaries of sampling. It was assumed in this study that the precision of the compositing sampling method most commonly used (two composites of seven or eight shipments in each period) was acceptable to the industry and that interest in a random sampling scheme was motivated by a desire to reduce the cost involved in sampling every shipment and in building and storing composite samples. Therefore the criterion of precision in this study was 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 size in a sampling scheme in which three t r a i t s are measured and the same precision i s required for 14 each t r a i t i s determined by the most v a r i a b l e t r a i t . How-ever, the c r i t e r i o n of p r e c i s i o n used i n t h i s study c o u l d be d i f f e r e n t f o r each m i l k c o n s t i t u e n t and t h e r e f o r e the c o n s t i t u e n t which i s now e s t i m a t e d most p r e c i s e l y c o u l d determine the sampling scheme i f the scheme i s to s a t i s f y the c r i t e r i o n 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 . However, m i l k f a t i s the o n l y m i l k c o n s t i t u e n t c u r r e n t l y used i n e s t a b l i s h i n g m i l k p r i c e s ; t h e r e f o r e the p r e c i s i o n o f p e r c e n t m i l k f a t e s t i m a t e s was the o n l y e s t i m a t e whose p r e c i s i o n c o u l d be assumed to have been accepted by the i n d u s t r y . For t h i s r e a s o n the main emphasis i n t h i s study was on the p r e c i s i o n o f e s t i m a t e s o f p e r c e n t m i l k f a t and a sampling scheme was deemed to be adequate i f the c r i t e r i o n was met f o r t h i s m i l k c o n s t i t u e n t . P r o t e i n or s o l i d s - n o t -f a t c o n t e n t may be i n c l u d e d i n f u t u r e m i l k p r i c i n g formulae i n which case e s t i m a t e s o f sampling v a r i a b i l i t y would be u s e f u l to the i n d u s t r y , t h e r e f o r e , p e r c e n t p r o t e i n and l a c t o s e , the two main components of s o l i d s - n o t - f a t , were i n c l u d e d i n t h i s study. The d e s i g n of a sampling scheme t o meet the s p e c i f i e d c r i t e r i o n r e q u i r e s the e s t i m a t i o n o f the v a r i a n c e of c u r r e n t composite e s t i m a t e s and the 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 m i l k c o n s t i t u e n t p e r c e n t . The w i t h i n h e r d -2 p e r i o d v a r i a n c e (a ) measures the v a r i a b i l i t y of e s t i m a t e s w •* of the m i l k c o n s t i t u e n t percentages o f each shipment o f m i l k from a herd f o r a g i v e n p e r i o d and can be w r i t t e n , 15 after O'Keeffe [15]: 2 2 2 2 <T = a' + a + a* (1 w a s t where 2 biological variance—measures the va r i a b i l i t y due to true differences among shipments in milk constituent percentages; 2 a sampling variance—measures the v a r i a b i l i t y among s 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 results of analyses (done at different times on different analysers) on the same sample. Sampling and testing variances are due to procedures of estimation and may be combined: a 2 = a 2 + a 2 C2) a s t 2 where a i s the within shipment variance—due to both a sampling and testing. Biological variance can be attributed mainly to: 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 individual cows-this factor would give rise to random shipment to shipment fluctuations; 2. removals from or additions to the milking herd; 16 3 . changes i n r o u t i n e s and/or p e r s o n n e l a s s o c i a t e d w i t h m i l k i n g and h a n d l i n g the h e r d — t h e s e changes may occur r e g u l a r l y and g i v e r i s e to c y c l i c f l u c t u a t i o n s ; 4 . c o n s i s t e n t d i r e c t i o n a l trends a c r o s s time i n the percentage of a m i l k c o n s t i t u e n t , which may be due t o the i n f l u e n c e o f changing s e a s o n s — t h i s f a c t o r would be expected to g i v e r i s e t o a p o s i t i v e c o r r e l a t i o n between the p e r c e n t o f any m i l k c o n s t i t u e n t i n one shipment w i t h the p e r c e n t o f the same c o n s t i t u e n t i n another shipment c l o s e to i t i n time; t h e i r c o r r e l a t i o n being a f u n c t i o n o f t h e i r d i s t a n c e a p a r t and d i m i n i s h i n g as the d i s t a n c e i n c r e a s e s ( s e r i a l c o r r e l a t i o n s ) . I f t h i s f a c t o r i s r e l a t i v e l y important the b i o l o g i c a l v a r i a n c e would be expected t o be lower i n a s h o r t p e r i o d than i n a long p e r i o d . I n which case d i v i s i o n o f p e r i o d s i n t o s u b-periods o r s t r a t a and randomly s e l e c t i n g shipments f o r sampling from each s t r a t a ( s t r a t i -f i e d random sampling) would be expected to reduce the standard e r r o r o f the e s t i m a t e d h e r d - p e r i o d mean as compared t o simple random sampling. Each m i l k shipment i n a p e r i o d i s sampled to form a composite; t h e r e f o r e the v a r i a n c e of the e s t i m a t e d mean, between composite v a r i a b i l i t y , i s due to w i t h i n shipment v a r i a b i l i t y and to va r i a b i l i t y introduced by the procedures associated with the formation of a composite (compositing variance). Biological variance—which measures the va 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 effects 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 fat, protein" and lactose using the analysis of variance. The linear mathematical model assumed was: y + h . + p . , . v i 3 U ) + w. k(ij) (3) where the observed milk consistuent percent of the kth shipment in the j * * 1 period of the i herd ; the general mean ; the effect associated with the i * - * 1 herd, 2 2 N(0,ov) , a, i s the variance among herd means; 18 >;j ( i ) t * 1 S e f f e c t °f t h e J*"* 1 p e r i o d i n t h e i ^ 1 2 2 h e r d , N t O , a ) , a i s t h e v a r i a n c e among p p p e r i o d means w i t h i n h e r d s ; 4* V* w k ( i j ) t * i e e f f e c t o f t h e k s h i p m e n t w i t h i n t h e Ti 2 2 j t h p e r i o d a n d i t h h e r d , UtO,o^)f a i s 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 . The w i t h i n s t r a t a v a r i a n c e s f o r t w o , t h r e e a n d f o u r s t r a t a p e r p e r i o d w e r e e s t i m a t e d f r o m E x p e r i m e n t I f o r p e r c e n t m i l k f a t , p r o t e i n a n d l a c t o s e u s i n g t h e a n a l y s i s o f v a r i a n c e . T h i s a n a l y s i s p a r t i t i o n e d 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 i n t o ; t h e v a r i a n c e among s t r a t a means a n d t h e r e s i d u a l w i t h i n s t r a t a v a r i a n c e . I f s t r a t a w e r e a s i g n i f i c a n t s o u r c e o f v a r i a t i o n t h e n s t r a t i f i e d r a n d o m s a m p l i n g w o u l d be e x p e c t e d t o b e w o r t h w h i l e . The l i n e a r m a t h e m a t i c a l m o d e l a s s u m e d w a s ; v i j k l = * + h i + P j ( i ) + s t k ( i j ) + w s l ( i j k ) ( 4 ) w h e r e y ^ j k l t h e o b s e r v e d m i l k c o n s t i t u e n t p e r c e n t o f t h e i t * 1 s h i p m e n t i n t h e k t n s t r a t a i n t h e j t h p e r i o d o f t h e i t n h e r d ; s t k ( i j ) t h e e f f e c t o f t h e k t h s t r a t a i n t h e j t * 1 p e r i o d o f t h e i 1 " h e r d , N ( 0 , a ) , a i s t h e S t S L v a r i a n c e among s t r a t a means w i t h i n p e r i o d s 2 2 a n d h e r d s (a . - two s t r a t a , a ., - t h r e e s t s t 2 s t r a t a a n d a ,, - f o u r s t r a t a ) ; S t 19 ws l( i j k ) the effect of the l f c ^ shipment in the -th strata, j*-* 1 period and i 1 : 1 : 1 herd, th 2 2 N(0,o" ), o is the within strata ' ws ' ws 2 2 variance (a - two strata, a , - three ws ws 2 strata and a , , - four strata) ; ws 1 1 and the remaining symbols have been defined in equation 3. Estimation of the components of within herd-period  variance. The replicated sampling and testing data collected for one period in Experiment II was used to estimate the components, given in equation 1, of the within herd-period variance. The linear mathematical model assumed was: y. i = u + h. + d . / . i + s, , • • \ +t,,.., x - ^ i j k l i u Ci) k ( i j ) l C i j k ) (5) where i j k l u h. l j Ci) *kCij) the observed milk constituent percent of the 1 ^ test on the k t n sample from the jt* 1 shipment of the i * - * 1 herd; the general mean; the effect of the i t h herd, N(0,a 2); the effect of the j*-* 1 shipment of the 2 2 i t h herd, N(0,a^), is the biological variance; the effect of the k t n sample from the j t n 2 2 shipment of the i ^ * 1 herd, N(0,a ) , a i s s s the sampling variance; t. ,. . the effect of the 1 t h test on the k t h sample from the jt* 1 shipment of the i t * 1 2 2 herd, N(0,a tl, a is the testing variance. Mean squares were set equal to their expectations and the resulting equations were solved to obtain estimates of the components of the within herd-period variance. The within strata biological variances for two, three and four strata per period were estimated from Experiment II for percent milk fat, protein and lactose. The linear mathematical model assumed was: v.... =u + h. + st. ,., + ds, ,. .. + t ,, ., , x (6) •^ljklm ^ I D d) k(ij) mCijkl) v ' where ^ijklm t* i e observed milk constituent percent of the mt*1 test on the i t * 1 sample from the kth shipment in the jt* 1 strata and the i t * 1 herd; stj ^  the effect of the jt* 1 strata of the i t h 2 2 herd, N(.0,a . ) , a i s the variance among 2 strata means within herds (a . - two strata, st ' 2 2 a - three strata and a .,, - four strata) st. st d s k ( i j ) t h e e f f e c t o f the kth shipment in the 2 *rh 2 2 strata and i * - 1 1 herd, N(0,a d s), o^s i s the within strata biological variance 2 2 (a, - two strata, a, , - three strata ds ' ds' 2 and a . ,, - four strata); 5 1 and the remaining symbols have been defined in equation 5. The difference between estimates of within herd-period 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 III (.triplicate composites and duplicate tests) were used to estimate compositing variances for percent milk fat, protein and lactose using the analysis of variance. The linear mathematical model assumed was: y. ., , = \x + h. + g. ,. . + c, ,. .. + t, ,. ., » ( 7 ) • ' i j k l H i ^3 Ci) k C i } ) l U;jk) ' where i j k l i s the observed milk constituent percent of the l ^ h t e s t on the k*-*1 composite i n the compositing period of the i * - * 1 herd; H the general mean; h ± the e f f e c t of the i t h herd, N (0,a 2); g j Ci) t* 1 S e f f e c t of the j*** 1 compositing period i n the i t h herd, N ( 0,a 2), a 2 i s the variance among compositing period means within herds; °kCij) t* 1 S e ^ e c t °f t^ i e k***1 composite in the j*-* 1 • i _ compositing period of the i herd, N(0,a ), a i s the compositing variance; the effect of the l r n test on the k composite in the jt* 1 compositing period and the i ^ herd, NC0,a 2), a2, i s the testing variance. The number of degrees of freedom associated with estimates of compositing variances from Experiment III were relatively small. Also composites were formed by the staff of the BCDA, Dairy Branch Laboratory; usually com-posites are formed in the laboratories of the milk plants to which the herd milk is shipped (as was the case in Experiment I ) . For these reasons estimates of compositing variances of percent milk fat, protein and lactose were obtained from Experiment I by an indirect method using estimates of sampling and testing variances from Experiment II. The linear mathematical model assumed was: y + h. + p! + r.. 1 j ID (8) where the mean milk constituent percent of the i t h h e r c j f o r the jth period, periods in this analysis were seven consecutive shipments (two weeks); h. x the effect of the i th the effect of the j seven shipment 2 2 period, N(0,a ,), a , i s the variance among p p period means; r. the joint effect of the i , th herd and the i j j r period which includes the interaction between the ±*-h herd and the j*"* 1 period (hp . .) and the random error (ei- k) . Model 8 does not yield direct 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 variables in model ( 8 ) . These were: (a) the mean of seven fresh samples weighted by the weight of milk in 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 residual variation of the fresh sample mean and the residuals of the two kinds of composites were equated to their expectations in 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 written: 24 where xf n. s n. t o2 = a 2 + i CT2 + ol (10) xc c n. s t 2 2 .j 0 n,+n0 n ^ a 2 . = o% + -^v- 2 - a 2 + • (11) x2c 2c 2 s 2 t n. n. 2 a ^ the within herd-period variance of the mean of seven shipments each sampled and tested once; 2 a the within herd-period variance of a two-c week composite; 2 a 9 the within herd-period variance of the weighted mean of two one-week composites; 2 2 o*c and o"2C the variances associated with the formation of a two-week composite and two one-week composites respectively; n^ and n2 the number of shipments in each of the two one-week composites (n^ = 3 and n2 = 4); n. the number of shipment in a two-week composite, (n. = n^ + xi^ = 7); 2 2 a and a. defined in equation 1. Equations 9, 10 and 11 represent the expectations of the random error of the residual mean square arising from r ^ j in model 8; therefore, the expectations of the residual 2 5 mean square for each kind of sampling may be written as follows: .2 _2 . 1 2 . 1 2 a . rf = a , ,... + -=<s + —^2 (12) ph(f) n. s n. t v ' where a 2 = a 2 , . + a 2 + -^a 2 + a 2 (13) rc phCc) c n. s t 2 2 2 2 2 n 1 + n 2 2 n l + n 2 2 n • n • 2 o"r£ is the residual mean square of the mean of seven shipments; 2 a is the residual mean square of two-week composites; 2 a"r2C is the residual mean square of the mean of two one-week composites; 2 aph(f) is the variance associated with the herd-period interaction effect for fresh sampling; 2 °* is the variance associated with the herd-phCc) period interaction effect for two-week composite sampling; 2 a , »„ . is the variance associated with the herd-ph(2c) period interaction effect for two one-week composite sampling; 2 2 o"s and o"t defined in equation 1. The remaining symbols and the coefficients associ-ated with variances have been defined in equations 9, 10 and 11. If the interaction variance is assumed to be equal for a l l three kinds of sampling then the following equations hold: For two-week composites: a 2 - a 2 = a 2 + % a 2 (15) rc r f c 7 t by rearrangement and substitution in equation 10; 2 2 2 1 2 1 2 ,, v^ °xc = a r c - a r f + 1 as + T °t <16> For two one-week composites: a 2 - a 2 = a 2 + i | a 2 (17) r2c r f . 2c 49 t and i t follows that: a 2 = cr2 - a 2 + % a 2 + \ a 2 (18) x2c r2c r f 7 s 7 t 2 2 The estimates of sampling (a ) and testing (a ) variances obtained from the analysis of the data of Experiment II were used to solve equations 15 to 18 for the variance associated with the formation of composites 2 2 (a or a 9 ) and for the variance of a composite estimate 2 2 Ca or o , ) of the period mean. Serial correlations. The se r i a l correlations, r u , of y.. with y.., were calculated on a within herd basis; Jxj J X j + U 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 se r i a l correlation coefficients were plotted on a correlogram versus u. The relationship between the se 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, after Snedecor and Cochran [18]. The mathematical model assumed was: 5 r = b + L b. u 1 C19) u o . = 1 x where r^ is the within herd s e r i a l correlation coefficient; b Q the population mean when u equals zero; u i s the number of shipments separating the two shipments for which r u was calculated Cu = 1,14} ; b^ i s the regression coefficient of r u on u 1. The graph of the equation which included only those powers in u which produced a significant reduction in the sums 28 of squares was plotted on the correlogram. Other s t a t i s t i c a l methods. Paired t-tests were used on the data of Experiment I to test for bias in 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 correspond-ing 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 level of significance 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 linear combinations of variances were computed after Welch [ 2 0 ] . 1. RESULTS AND DISCUSSION Estimates of Within Herd-period Variance and Components The analysis of variance tables, showing the expec-tations of mean squares, of hierarchal models 3 and 5 used for the analysis of Experiments I and II are presented in Tables 1 and 2 respectively. The estimates from Experiment II of biological, sampling and testing variances for a l l three milk constituents are reported in Table 3. The within herd-period variances estimated from Experiment I. are shown in column 5 of Table 3. Sampling variance. The estimate of the sampling variance for percent milk fat was 0.00094 ± .00027 which was 7.1 percent of the total within herd-period variance. The estimates of sampling variance for percent protein and lactose were small and negative. These results indicated that drawing a milk sample, by the method used in the current study, was not an important source of variation for 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 prior to taking a sample. TABLE 1 ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS EXPERIMENT I Source DF SS a MS a F a EMS Herds (h) 25 1127.81 180.47 37.70 45.11253 7.21873 1.50787 99.09* 38.85* 9.50* 2 2 2 w 2 p 3 h P e r i o d s (p)/h 289 131.58 53.70 45.88 0.45528 .18581 .15877 33.20* 23.60* 29.00* 2 2 w 1 p Shipments/p&h 4188 57.43 32.98 22.93 .01371 .00787 .00548 a 2 w T o t a l 4502 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 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 of v a r i a t i o n . TABLE 2 ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS EXPERIMENT I I Source DF ss a MS a „a F EMS Herds (h) 21 158.53 51.53 10.35 7.54901 2.45382 0.49296 171.00* 113.80* 50.06* 2 2 2 2 c~+k,a +k-,a*+k.at t I s 3 d 4 h Shipments (d)/h 300 13.24 6.47 2.95 0.04416 .02156 .00985 5.40* 15.15* 2.95* 2 2 2 t I s 2 d Samples (s)/h&d 633 5.17 0.90 2.11 .00817 .00142 .00333 1.30* .77 .84 oj+k..a2 t I s T e s t s / h , d&s 955 6.00 1.76 3.82 .00628 .00184 .00400 T o t a l 1909 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 of v a r i a t i o n . k x = 2 k 2 = 5.93 k 3 = 5.95 k 4 = 86.81 L O 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 FIFTEEN CONSECUTIVE SHIPMENTS V a r i a n c e (xlO ) M i l k B i o l o g i c a l Sampling T e s t i n g T o t a l a T o t a l b C o n s t i t u e n t (1) (2) (3) (4) (5) % M i l k f a t 0.607±.061 0.094±.027 0.628±.029 1.329±.064 1.371+.030 % P r o t e i n .340±.029 -.021±.006 c •167±.006 0.507±.031 0.787±.017 % L a c t o s e .110±.014 -.033±.013 c .373±.013 .483±.017 .5481.012 di 2 2 2 2 t o t a l o f columns one to three; i . e . a = a, + a + a. w d s t b 2 w i t h i n h e r d - p e r i o d v a r i a n c e (a ) estimated from experiment one. when e s t i m a t e s of sampling v a r i a n c e were n e g a t i v e the t e s t i n g v a r i a n c e was estimated w i t h i n c r e a s e d degrees of freedom by combining the sums of squares f o r sampling and t e s t i n g . t o 33 T e s t i n g v a r i a n c e . E s t i m a t e s o f t e s t i n g v a r i a n c e s and t h e i r s t a n d a r d e r r o r s 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 were; 0.00628 ± .00029, 0.00167 ± .00006 and 0.00373 ± .00013 r e s p e c t i v e l y (Table 3 ) . E s t i m a t e s of t e s t i n g v a r i a n c e by Dunn [6] f o r the p e r c e n t of the same thr e e m i l k c o n s t i t u e n t s (the a n a l y s e s were performed i n the same l a b o r a t o r y as the a n a l y s e s i n the c u r r e n t study) were; 0.00612, 0.00631 and 0.00505 r e s p e c t i v e l y . In a review of automatic m i l k a n a l y s e r s , Green [10] r e p o r t e d t h a t t e s t i n g v a r i a n c e s w i t h IRMA. under p r a c t i c a l l a b o r a t o r y c o n d i t i o n s were i n the range 0.0036 to 0.0081 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 . E s t i m a t e s of t e s t i n g v a r i a n c e i n the c u r r e n t study f o r p e r c e n t m i l k f a t and l a c t o s e (Table 3) f e l l i n the range g i v e n by Green [10] and the e s t i m a t e f o r p e r c e n t m i l k f a t c l o s e l y agreed w i t h the e s t i m a t e by Dunn [ 6 ] . The e s t i m a t e of 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 p r o t e i n i n the c u r r e n t study was s m a l l e r than the e s t i m a t e by Dunn [6] and below the range r e p o r t e d by Green [10]. The d i f f e r e n c e between 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 t e s t i n g v a r i a n c e by Dunn [6] and the e s t i m a t e i n the c u r r e n t study may i n d i c a t e t h a t t e s t i n g v a r i a n c e v a r i e s from time to time under p r a c t i c a l l a b o r a t o r y c o n d i t i o n s . T e s t i n g v a r i a n c e as d e f i n e d i n the c u r r e n t study i n c l u d e d ; sample p r e p a r a t i o n , machine to machine v a r i a t i o n and machine p r e c i s i o n and thus r e p r e s e n t e d the t o t a l v a r i a n c e a s s o c i a t e d w i t h t e s t i n g 34 and sample handling procedures. The difference between estimates of testing variance in 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 listed above other than machine precision. The testing variances were 47.2, 32.9 and 77.2 per-cent (calculated from Table 3) of the total within herd-period variance for percent milk fat, protein and lactose respectively. Therefore testing was an important source of variation 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 relatively independent of the sampling scheme. Biological variance. Estimates of biological variances and their standard errors were 0.00607 ±.00061, 0.00340 ± .00029 and 0.00110 + .00014 for percent milk fat, protein and lactose respectively. The biological variances were 45.7, 67.1 and 22.8 percent of the total within herd-period variance for percent milk fat, protein and lactose respectively. Within herd-period variance. Estimates of the total within herd-period variance are shown in columns four (Experiment III and five (Experiment I) of Table 3. These estimates from the two experiments were not s i g n i f i -cantly different, by two-tailed F-tests, for percent milk fat or percent lactose. However, differences between the estimates were significant for percent protein. The components of within herd-period variance were defined to be; biological, 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 attributed to differences in biological variance and/or in testing variance in the two experiments. If biological variances d i f f e r in the population, then random sampling schemes may have to be modified for different herds and/or different 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. Effects of Strata Two shipments of milk which are close together in time can be expected to be more similar i n milk constituent percent than two shipments which are more widely separated. (Materials and Methods}. 36 Serial correlation. The sets of product moment correlations, r u for pairs of shipments u shipments apart were calculated for values of u from one to fourteen on a within herd basis for twenty-three herds of Experiment I across thirteen periods for percent milk fat, protein and lactose. The results are presented in Table 4 and a correlogram shown in Figure 1. The sets of product moment correlations were f i t t e d to equation 19. The values of u 1 (i = 1,5) in equation 19 were replaced by orthogonal polynomial coefficients from Fisher and Yates [8]. The reduction in sums of squares was tested as each successive term was added. As the objective was to find the polynomial of lowest degree that was a good f i t , calculations were stopped when two successive additions were both non-significant (Tables 5A, 5B and 5C for percent milk fat, protein and lactose respectively). The coefficients in the resulting polynonomial equations were transformed to yield equations expressed in terms of u. These equations and graphs of these equations are shown in Figure 1. Cochran [4] has shown that when a s e r i a l correlation exists in 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 is a straight line the variance of systematic sampling was equal to the variance of a st r a t i f i e d random sample, provided that there was no 37 TABLE 4 WITHIN HERD SERIAL CORRELATIONS3, FOR PERCENT MILK FAT, PROTEIN AND LACTOSE Number of Shipments Apart (u) Serial Correlations Number of Paired Values Milk Fat Protein Lactose 1 0.826 0.736 0.749 4227 2 .777 .693 .724 4196 3 .742 .656 .684 4168 4 .707 .604 .648 4145 5 .685 ,590 .633 4120 6 .663 .548 .631 4099 7 .632 .518 .597 4084 8 .603 .479 .590 4057 9 .572 .432 .564 4037 10 .560 .396 .531 4020 11 .541 .366 .528 3988 12 .519 .346 .505 3969 13 .502 .302 .465 3955 14 .478 .263 .438 3933 the se r i a l correlation, r u of with y ^ + u computed on a within herd basis. 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 DF SS MS F Total 13 0 .1513275 Reduction to Linear 1 .1490156 Deviations from Linear 12 .0023119 0.000193 773.5* Reduction to Quadratic 1 .0019158 Deviations from Quadratic 11 .0003961 .0000360 53.2* Reduction to Cubic 1 .0001011 Deviations from Cubic 10 .0002950 .0000295 3.4 Reduction to Quartic 1 .0000218 Deviations from Quartic 9 .0002732 .0000304 0.7 significant reduction of sums of squares. 39 TABLE 5B THE REDUCTION IN SUMS OF SQUARES DUE TO SUCCESSIVE TERMS IN THE POLYNOMIAL OF EQUATION 19. PERCENT PROTEIN SERIAL CORRELATIONS Source . DF SS MS F Total 13 0.2913929 Reducation to Linear 1 .2906398 Deviations from Linear 12 .0007531 0.0000628 4631.7* Reduction to Quadratic 1 .0000859 Deviations from Quadratic 11 .0006672 .0000607 1.4 Reduction to Cubic 1 .0000107 Deviations from Cubic 10 .0006565 .0000657 0.2 * significant reduction in sums of squares. 40 TABLE 5C THE REDUCTION IN SUMS OF SQUARES DUE TO SUCCESSIVE TERMS IN THE POLYNOMIAL OF EQUATION 19. PERCENT LACTOSE SERIAL CORRELATIONS Source DF SS MS F Total 13 0.1130053 Reduction to Linear 1 .1112329 Deviations from Linear 12 .0017724 0.0001477 753.1* Reduction to Quadratic 1 .0000135 Deviations from Quadratic 11 .0017589 .0001599 0.1 Reduction to Cubic 1 .0005543 Deviations from Cubic 10 .0012046 .0001205 4.6 significant reduction i n sums of squares. Figure 1 Serial correlations of percent milk fat, protein and lactose periodic fluctuation in the population. However, when the correlogram i s concave upward he reported that the. variance of systematic sampling was less than the variance of st r a t i f i e d random sampling. When periodic variation exist in a population then the variance of systematic samples and the amount of bias in estimates provided by systematic samples depend on the relationship between the sampling frequency and the period of the fluctuations. Therefore, when fluctuations of unknown or variable period may exist in a population, s t r a t i f i e d random sampling i s to be pre-ferred to systematic sampling. Cyclic fluctuations in milk constituent percentages may be present in the population currently under study (Material and Methods). The period of these fluctuations may dif 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 rejected in the current study. The relationships between the ser i a l correlations of percent protein and percent lactose and u were estimated as linear. The reduction in sums of squares due to the linear f i t was 99.7 percent and 98.4 percent (calculated from Tables 5B and 5C respectively) of the total sums of squares of the ser i a l correlations of percent protein 43 and percent lactose respectively. The relationship between the se r i a l correlations of percent milk fat and u contained a significant contribution due to the quadratic term in the equation. The reduction in sums of squares due f i t t i n g both linear and quadratic terms was 99.7 percent of the total sums of squares of the se r i a l correlations of percent milk fat; the reduction due to f i t t i n g the linear term only was 98.4 percent of the total sums of squares (calculated from Table 5A). Therefore, although the graph describing the relationship between the ser i a l correlations of percent milk fat and u was concave upwards, the departures from a linear relationship were relatively small. These results indicated that, for a l l milk constituents, the seria l correlations decreased regularly 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, for 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 size, 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 biological variance were obtained from Experiment II using s t a t i s t i c a l model 6. Three levels of st r a t i f i c a t i o n of fifteen shipment periods (one month) were used: (a) two strata, one of seven and one of eight ship-ments; (b) three strata of five shipments each; and (c) four strata with four shipments in three strata and three ship-ments in the fourth stratum! The analysis of variance table showing expectations of mean squares of Experiments I with two, three and four strata are presented in Tables 6A, 6B and 6C respectively. The results for Experiment II are presented in Tables 7A to 7C. Table 8 shows the biological variance for Experiment II and the within herd-period variance for both Experiments for a l l three milk constituents and for two, three and four strata. The effect of strata was a significant source of variation in a l l analyses. Therefore f i t t i n g strata reduced the magnitude of the within herd-period variance (within strata variance), The results were plotted in Figure 2 for percent milk fat, protein and lactose for both experiments. The values plotted in Figure 2 for no strata were from Table 3. TABLE 6A ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS EXPERIMENT I — TWO STRATA PER PERIOD Source DF SS a MS a F a EMS Herds (h) 25 1127.81 180.47 37.70 45.11253 7.21873 1.50787 99.09* 38.85* 9.50* 2 2 2 2 a +k„a 4.+kca +k,a' ws 4 s t 5 p 6 h Peri o d s (p)/h 289 131.58 53.70 45.88 0.45528 .18581 .15877 8.60* 7.39* 10.. 05* 2 , 2 , 2 ws 2 s t 3 p S t r a t a ( s t ) / h & p 315 16.68 7.92 4.98 .05296 .02515 .01580 5.03* 3.89* 3.41* 2 2 aws +Vst Shipments/h,p & s t 3873 40.75 25.06 17.95 .01052 .00647 .00464 a 2 ws T o t a l 4502 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 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 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 , k, = 7.08 k 0 = 7.22 k- = 14.29 k„ = 7.24 k c = 14.33 k. = 172.88 1 2 3 4 5 6 TABLE 6B ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS. EXPERIMENT I — THREE STRATA PER PERIOD Source DF s s a MS a F a EMS Herds (h) 25 1127.81 180.47 ' 37.70 45.11253 7.21873 1.50787 99.09* 38.85* 9.50* 2 2 2 2 aws' +Vst' + k5VVh P e r i o d s (p)/h 289 131.58 53.70 45.88 0.45528 .18581 .15877 11.74* 10.21* 14.02* a w s ' + k 2 a s t ' + k 3 a p S t r a t a ( s t ) / h & p 629 24.38 11.45 7.21 .03876 .01821 .01132 4.18* 3.01* 2.55* 2 , 2 a w s ' + k l a s t ' Shipments/h/p & s t 3559 33.04 21.53 15.81 .00928 .00605 .00444 a 2 , ws' T o t a l 4502 a t h e 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 re 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 of v a r i a t i o n . k, = 4.75 k 0 = 4.81 k- = 14.29 k. = 4.82 k c = 14.33 kc 172.88 1 2 3 4 5 6 TABLE 6C ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS. EXPERIMENT I ~ FOUR STRATA PER PERIOD Source DF S S a MS 3 F a EMS Herds (h) 25 1127.81 180.47 37.70 45.11253 7.21873 1.50787 99.09* 38.85* 9.50* a 2 , ws 1 • + k 4 a s f ,+kt,a2+kca? 1 5 p 6 h P e r i o d s (p)/h 289 131.58 53.70 45.88 0.45528 .18581 .15877 15.7 4* 11.15* 16.37* a 2 , ws 1 ' + k 2 a s f ' + k 3 0 p S t r a t a ( s t ) / h & P 944 27.31 15.73 9.15 .02893 .01666 .00970 3.12* 3.13* 2.28* 2 ws 1 • + k l a s f i Shipments/h,p & s t 3244 30.12 17.25 13.78 .00928 .00532 .00425 aws' i T o t a l 4502 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 a re f o r p e r c e n t m i l k f a t , p r o t e i n nd 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 of v a r i a t i o n . k x = 3.55 k 2 = 3.66 k 3 = 14.29 k 4 = 3.66 k c ='14.33 kc = 172.88 O TABLE 7A ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS EXPERIMENT II — TWO STRATA PER PERIOD Source DF s s a MS3 _a F EMS Herds (h) 21 158.53 51.53 10.35 7.54901 2.45382 0.49296 46.30* 21.33* 16.82* 2 , 2 , 2 , 2 , 2 t I s 3 ds 5 s t 6 h Strata (st)/h 22 3.59 2.53 0.64 .16304 .11505 .02931 4.69* 8.12* 3.53* 2 , 2 , 2 , 2 t I s 3 ds 4 s t Shipments (ds)/h & s t 278 9.66 3.94 2.31 .03474 .01416 .00831 4.25* 9.95* 2.49* 2 2 2 t I s 2 ds Samples (s)/h,st & ds 633 5.17 0.90 2.11 .00817 .00142 .00334 1.30* .77 .84 2 2 c:+k,cr t I s Tests/h,st,ds & s 955 6.00 1.76 3.82 .00628 .00184 .00400 To t a l 1909 a t h e three values l i s t e d for each source of v a r i a t i o n are for 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 . k x = 2 k 2 = 5.93 k 3 = 5 . 9 5 k 4 = 43.16 k 5 = 4 3 . 6 6 k g = 86.81 TABLE 7B ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS EXPERIMENT II — THREE STRATA PER PERIOD Source DF ss a MSa F a EMS Herds (h) 21 158.53 51.53 10.35 7.54901 2.45382 0.49296 75.54* 25.17* 4.97* 2 2 2 2 2 V k l V k 3 a d s ' + k 5 a s t ' + k 6 a h Strata (st)/h 44 4.40 4.29 1.36 .09993 .09748 .03094 2.89* 11.45* 4.97* a t + k l a s + k 3 a d s ' + k 4 a s t ' Shipments (ds/h & s t 256 8.85 2.18 1.59 .03456 .00851 .00622 4.23* 5.98* 1.86* a t + k l a s + k 2 a d s ' Samples (s)/h,st & ds 633 5.17 0.90 2.11 .00817 .00142 .00334 1.30* 0.77 0.84 aj+k-.a2 t I s Tests/h,st,ds & s 955 6.00 1.76 3.82 .00628 .00184 .00400 < T o t a l 1909 a t h e three values l i s t e d for each source of v a r i a t i o n are for 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 . k, = 2 k 0 = 5.93 k, = 5.95 k. = 28.88 k c = 29.06 k, = 86.81 1 2 3 4 5 6 ^ TABLE 7C ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES OF HERD BULK MILKS. EXPERIMENT I I — FOUR STRATA PER PERIOD Source DF SS' MS EMS Herds (h) S t r a t a ( s t ) / h Shipment (ds)/h & s t Samples ( s ) / h , s t & ds T e s t s / h , s t , d s & s T o t a l 21 66 234 633 955 1909 158.53 51.53 10.35 6.18 4.21 1.71 7.07 2.25 1,24 5.17 0.90 2.11 6.00 1.76 3.82 7.54901 2.45382 0.49296 .09357 .06386 .02592 .03020 .00963 .00531 .00817 .00142 .00334 .00628 .00184 .00400 80.68 38.43* 19.02* 3.10* 6.63* 4.88* 3.70* 6.77* 1.57* 1.30* 0.77 0.84 o\+k,a 2+k_c 2 , ,+k,a2J., ,+k,a 2 t I s 3 d s " 5 s t " 6 h a t + k l a s + k 3 a d s " + k 4 a s f a t + k l a s + k 2 a d s " 2 2 < + k l a s a t h @ t h r e e v a l u e s l i l t e d fea? © a e h §©u£@© ©f v a r i a t i e n a r © f©E p t r e t n t m i l k f a t , g ^ e t t 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 of V a r i a t i o n . k x = 2 k 2 = 5.93 k 3 = 5,95 k 4 = 21.50 k 5 = 22.31 k g = 86.81 o 51 TABLE 8 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 . Number of Strata _2 (Components of Variance (xlO ) Experiment II Experiment I Biological Total Total A. Milk fat None Two Three Four : percent 0.607+.061 .448+.051 .445+.052 .372±.048 1.329± .064 1.170±.054 1.167±.056 1.094± .049 1.371±.030 1.052±.024 0.928±.022" .928±.023 B. Protein None Two Three Four percent 0.340+.029 .215±.020 •119±.013 .139+.015 0.507±.031 .382± .021 .286±.014 •306±.016 0.787±.017 •647±.015 .605±.014 .532±.013 C. Lactose None Two Three Four percent 0.110+.014 •084±.012 .049±.010 .033±.009 0.483±.017 .457±.016 .422±.014 •406±.014 0 .548±.012 .464±.011 .444±.011 •425±.011 52 CN I -P cd 4-1 o r* rH rH •H s CU cn td -P CO o H (U •p a CD d -p •H CD -P •P U) 0 c: U o Pi o M rH -r( e MH 0 CD O c: rd CD •rl 01 0 rd •P > O rd a 1.5 n 1.0 . 0.5 F i g u r e 2 0.0 1.0 0.5 0.0 1.0 0.5 0.0 o A • 2 w 2 w 2 d from Experiment I . from Experiment I I from Experiment I I T " • 1 0 2 3 4 Number of s t r a t a 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, three and f o u r s t r a t a per p e r i o d 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 53 Within strata variance of percent milk fat. The estimates of within herd-period variance—no s t r a t a — o f percent milk fat obtained from the two experiments were not significantly different by an F-test. However the estimates of within herd-period variance of percent milk fat calculated within strata (within strata variance) from Experiment I were significantly lower than the estimates from Experiment II. Stratification is expected to reduce the effect 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 attributed to either differences between biological variances or to differences between testing variances in the two sets of data. Stratification would be expected to reduce biological variance only: testing variance (within shipment variation) would not be altered 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) in Experiment I may have been a more important source of variation than time trends in the single period in Experiment II. Thus directional changes in milk constituent percentages may be greater in some periods (seasons) than in others. In which case, unless s t r a t i f i c a t i o n can effectively stabilize within herd-period variances, i t may be necessary to take more 54 milk samples in some seasons than in other seasons i f the same level of precision is to be achieved for a l l seasons. Within strata 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 , for percent protein obtained from Experiment I were a l l significantly lower than the corresponding estimates obtained from Experiment II (Table 8 and Figure 2} by an F-test. The difference between the estimates was relatively constant for a l l levels of str a t i f i c a t i o n . The results indicated, by use of the same reasoning that was applied to the results for the within herd-period variance of milk fat percentage, that testing variance was different 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, va r i a b i l i t y of testing variance would affect the precision of a l l milk sampling schemes; although, the magnitude of the change in precision may not be the same for a l l schemes. The difference between biological variance and within herd-period variance was assumed to be equal to testing (and sampling!variance, equation 1 . If biological variances were approximately the same in the two data sets, then testing variance in Experiment I could be approximated by the difference between biological variance estimated 55 from Experiment II and within herd-period variance from Experiment I. The estimate of testing variance for percent protein in Experiment I obtained by the average of these differences was 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 , for percent lactose obtained from Experiments I and II were non-significant. Variability 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 written: where the symbols have been defined in 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: a2- = i ta2(l-£)+ a 2 + a 2] x n d N s t (20) (21) where VL . 1 I the weight attached to the i str a t a and i s equal to the number of shipments i n the i stra t a divided by the t o t a l number of shipments i n the period; m the number of str a t a i n the period; n^ the number of observations drawn from the i * " * 1 s t r a t a ; the number of shipments i n the i * - * 1 s t r a t a ; 2 the within s t r a t a b i o l o g i c a l variance, which i s assumed equal for a l l s t r a t a . With equal s t r a t a s i z e and equal number of observations per stratum, equation 2 1 reduces tos a f = 4 - [ a 2 Cl " 2 l E ) + a 2 + a 2) ( 2 2 ) x n m ds N 3 t where n' i s the number of observations per stratum. The other symbols remain as previously defined. The estimates of the variances obtained from the analyses of Experiment II were substituted into the appropriate equations to cal c u l a t e the variance of the mean and the 99% confidence i n t e r v a l about the mean for various sampling schemes (Table 9). The r e s u l t s showed that the reduction i n the confidence 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 Number3 % Milk Fat % Protein % Lactose 2b O"- 99%CLC 2b o- 99%CLC 2b o- 99%CLC X X X A. Simple random sampling 1 1.289 ± .293 0.484 + .180 0 .475 ±.178 2 0.624 ±.203 .231 + .124 .234 ±.125 3 .403 + .164 .146 + .099 .154 ±.104 4 .292 ±.139 .104 + .083 .113 ±.087 5 .225 ±.122 .079 + .072 .089 ±.077 6 .181 + .111 .062 + .064 .073 ±.070 7 .149 ±.100 .050 + .058 .062 ±.064 8 .126 ±.092 .041 + .052 .053 ±.059 9 .107 + .084 .034 + .047 .046 ±.056 15 .048 ±.057 .011 + .027 .025 ±.041 B. Stratified random sampling Two Strata 2 0.558 + .193 0.178 + .109 0 .224 ±.122 4 .264 + .133 .082 + .074 .109 ± .085 6 .182 + .110 .05 0 + .057 .071 ±.069 8 .126 ± .092 .034 ± .047 .052 ±.059 Three Strata 3 0.359 ± .154 0.087 + .076 0 .137 ±.096 6 .165 ± .105 .036 + .049 .067 ±.067 9 .100 ± .082 .024 + .040 .044 ± .054 Four Strata 4 0.254 + .130 0 .068 ± .067 0 .101 ± .082 8 .114 ±.087 .030 ±.044 .049 ± .057 anumber of samples per period for both simple and s t r a t i f i e d random sampling variance of the mean xlO 99% confidence interval of the mean. 58 relatively small and diminished as n increased. The re-duction in the confidence intervals was greatest for percent protein and least for percent lactose. These results can be attributed mainly to two factors. F i r s t l y , biological variance (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 testing variances (within shipment variation) would not be altered. Therefore, s t r a t i f i c a t i o n would be expected to reduce the confidence intervals to a greater extent for those milk constituents for which biological variance was a major component of the within herd-period variance. Secondly, the f i n i t e population correction factor applied only to the biological variance therefore the contribution of biological variance to the standard error of the mean would be reduced more rapidly as sample size increased than the contribution of sampling and testing variances. Thus for relatively large n the contribution of biological variance to the standard error would be small and therefore the effect of any reduction in the magnitude of biological variance by s t r a t i f i c a t i o n on the standard error would diminish as n increased. Stratification could s t i l l be worthwhile i f i t resulted in a reduction in the frequency of large deviations from the true mean by eliminating the probability of drawing a l l observations from either the beginning or the end of a period. Although large deviations may occur with relatively low frequency their occurrance could be of concern to the individual milk producer as his payment for the period's milk shipments are based on the results of the estimate of the mean percent milk fat. Composite Sampling Variance of composites—Experiment III. The criterion of precision in the current study was that a random sampling scheme should estimate herd period means at least as pre-cisely as two-week composite sampling. Experiment III was designed to provide estimates of the standard error of herd-period 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 in the formation of composites; therefore, the variance of a composite estimate was entirely attributable to procedures of estimation; C l ) sampling, (2) testing and C3) formation of a composite sample. The data of Experiment III 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 is presented in Table 10. Compositing was not a significant source of variation for any of the three TABLE 10 ANALYSIS OF VARIANCE OF MILK CONSTITUENT PERCENTAGE OF HERD BULK MILKS EXPERIMENT III: ESTIMATE OF COMPOSITING VARIANCE Source DF s s a MSa F a EMS Herds Ch) 20 20.7105 7.2451 1.3535 1.03553 0.36225 .06768 48.40* 22.08* 6.93* 2 2 2 2 cf+2a^+6a^+12a^ t e g h Periods Cg)/h 21 0.4493 .3445 .2050 .02140 .01641 .00976 15.41* 3.08* 4.53* a2+2a2+6a2 t e g 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 the three values listed for each source of variation are for percent milk fat, protein and lactose respectively. significant source of variation. o 61 m i l k c o n s t i t u e n t s s t u d i e d . E s t i m a t e s of compositing and t e s t i n g v a r i a n c e (Table 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 equating the mean squares to t h e i r e x p e c t a t i o n s and s o l v i n g the r e s u l t i n g e q u a t i o n s . The e s t i m a t e of the v a r i a n c e a s s o c i a t e d w i t h the f o r m a t i o n of a composite sample was low (0.000087 ± .00013) f o r p e r c e n t m i l k f a t , and was low and n e g a t i v e (-.000251 ± .00054 6) f o r p e r c e n t p r o t e i n . The e s t i m a t e f o r p e r c e n t l a c t o s e w h i l e n e g a t i v e was r e l a t i v e l y l a r g e i n a b s o l u t e v a l u e (-.001226 ± .000332). Es t i m a t e s o f t e s t i n g v a r i a n c e o b t a i n e d from Experiment I I (Table 3) were compared w i t h e s t i m a t e s o b t a i n e d from Experiment I I I (Table 11). F - t e s t s showed t h a t the e s t i m a t e of 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 m i l k f a t was s i g n i f i c a n t l y lower i n Experiment I I I then i n Experiment I I ; 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 t e s t i n g v a r i a n c e was s i g n i f i c a n t l y h i g h e r . The e s t i m a t e s of 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 l a c t o s e were not s i g n i f i c a n t l y d i f f e r e n t between the two experiments. These r e s u l t s support the c o n c l u s i o n s , based on the comparison of the e s t i -mates of 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 Experiments I and I I , t h a t the 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 p r o t e i n may va r y from time to time. The r e s u l t s from Experiment I I I i n d i c a t e d t h a t 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 m i l k f a t a l s o may v a r y from time to time. The e s t i m a t e s of t e s t i n g v a r i a n c e s from Experiment I I were based on a n a l y s e s done 62 TABLE 11 ESTIMATES OF COMPOSITING AND TESTING VARIANCE EXPERIMENT III Variance (xlO Milk Constituent Compositing Testing % Milk fat 0.0087 ± .0130 0.1214 ± .0152 % Protein -.0251 ± .0546 .5627 ± .0547 % Lactose -.1226 ± .0332 .3629 ± .0352 63 over a period of one month. The analyses of the composite in Experiment III were done on two days (two weeks apart) in the same month as the analyses for Experiment II. Therefore, testing variances would appear to be subject to considerable short-term fluctuations. Variance of composites — Experiment I. The results from the analyses of Experiment III indicated that the formation of composites is not an important source of variation of composite sample estimates of the period mean percent composition. However, as the number of degrees of freedom associated with these estimates was relatively low the data from Experiment I were analysed by s t a t i s t i c a l model 8. The residuals from these analyses were equated to their expectations to yield estimates of the variance of composite formation and the variance of a composite estimate as shown in equations 9 to 18 (Statistical Methods) . The analysis of variance tables (model 8) for fresh sample, two-week composite and two one-week composite estimates are presented in Tables 12A, 12B and 12C respec-tively. The estimates of sampling and testing variance (Table 3) and the residual mean squares (Tables 12A, 12B, and 12C) were used to solve equations 15 through 18 for 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 DF s s a MS a F a Herds 25 1 5 1 . 9 3 8 2 3 . 9 8 2 4 . 9 5 5 6 . 0 7 7 5 1 0 . 9 5 9 3 0 . 1 9 8 2 1 3 9 3 . 3 * 1 1 0 . 1 * 5 5 . 3 * Periods 25 1 1 . 5 7 9 4 . 1 5 8 5 . 2 4 1 . 4 6 3 1 6 . 1 6 6 3 2 . 2 0 9 6 2 3 0 . 0 * 1 9 . 1 * 5 8 . 5 * Residual 5 6 4 8 . 7 1 4 4 . 9 1 5 2 . 0 2 3 . 0 1 5 4 5 . 0 0 8 7 1 . 0 0 3 5 9 T o t a l 614 the three values l i s t e d for each source of v a r i a t i o n are for percent milk f a t , p r o t e i n 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 . 65 TABLE 12B ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES FITTING HERDS AND PERIODS (MODEL 8) EXPERIMENT I TWO-WEEK COMPOSITE ESTIMATES Source DF ssa MSa F a 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 the three values l i s t e d for each source of variation are for percent milk fat, protein and lactose respectively. significant source of variation. 66 TABLE 12C ANALYSES OF VARIANCE OF MILK CONSTITUENT PERCENTAGES FITTING HERDS AND PERIODS (MODEL 8) EXPERIMENT I TWO ONE-WEEK COMPOSITE ESTIMATES Source DF s s a MSa F a 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 the three values l i s t e d for each source of variation are for percent milk fat, protein and lactose respectively. significant source of variation. associated with the formation of a composite sample for both types of composite samples used in Experiment I. The estimates of compositing variances for a l l three milk constituents (Table 13) were smaller, or only slightly larger, than their standard errors for both two-week composites and two one-week composites. These results support the conclusions, based on Experiment III, that the formation of a composite i s not an important source of variation. If sampling and compositing variances are both small then testing was the most important source of variation of composite estimates (equation 10). Calculation of the Criterion of Precision The estimates of the variance of a composite for a two-week compositing period (Table 13) were used to calculate 2 the variance of the mean of two two-week composites (a- ). The equation can be written: 0 m n. 2 „ 4 = 2 - i - o2 (23) x c i=l N x c i is the number of composite samples in a period (month) ; is the number of shipments in the i * * n composit-ing period; where m n. l 68 TABLE 13 VARIANCE OF COMPOSITES C x l O " 2 ) Two-week Two One-week Milk Constituent xc c 2 C x2c °2c % Milk fat 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 in a period (month) ; a 2 c_ variance of the i^h composite, (which is defined in equation 10 for two-week composites). The variance of the mean of two two-week composite are presented in Table 14. These values were the maximum variances of the means of random samples allowable i f the criterion of precision was to be met. Composite Sampling versus Random Sampling . Accuracy of composites. Experiment I was used to test the accuracy of composite sampling. Three estimates of herd mean milk constituent percentages for each two-week 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 in each shipment (fresh sample estimates). Paired t-tests were used to test differences between fresh sample estimates and means estimated by; (a) the observed value of a two-week composite and (b) the mean of two one-week composites weighted by the amount of milk represented by each composite. The results (Table 15) indicated that percent milk fat was significantly 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 fat 0.3675 % Protein 0.1105 % Lactose 0.2050 TABLE 15 PAIRED t-TEST OF DIFFERENCES BETWEEN THE FRESH 3 ESTIMATE OF A TWO WEEK PERIOD MEAN AND BOTH KINDS OF COMPOSITE ESTIMATES M i l k C o n s t i t u e n t F r e s h v s . Two-week F r e s h v s . Two One Week D i f f . b S.D. t D i f f . b S.D. t % M i l k f a t % P r o t e i n % L a c t o s e DF -.045 0.023 .010 69 0.113 .085 .064 4 -10.6* 7.0* 4.2* -.045 0.003 .010 6 0.080 .060 .057 56 -14.5* 1.3 4.3* mean o f f r e s h samples from seven c o n s e c u t i v e shipments. bmean d i f f e r e n c e : composites minus f r e s h , s i g n i f i c a n t d i f f e r e n c e . of composite estimates was 0.045 percent milk fat. Percent protein was overestimated by two-week composites (the difference was 0.025 percent protein), but two one-week composite estimates were not significantly different from fresh estimates. Both types of composites overestimated percent lactose by 0.010 percent lactose. Herrmann and Anderson [11] and Preston [17] also reported that percent milk fat was lower in composite samples than fresh samples. Estimation of sample size. The criterion used in the current study was that the standard error of an estimate from a random sample should be at least as low as the standard error of the estimate from composite samples currently in use. The number of random samples (n) required to give a predetermined variance of the mean can be found by rearranging equation 20 to yield; n = a 2 / (a- + a 2) (24) w ' x d N where 2 a- i s the predetermined variance of the mean; N i s the number of shipments in the period; and the remaining symbols have been defined in equation 1. This equation also holds for s t r a t i f i e d random sampling i f the strata size are equal, the sampling fraction i s the same for a l l strata and a, is defined as the within strata d biological variance. 2 The appropriate predetermined variances (a-) in the current study for each milk constituent were the variances of the mean milk constituent percentages, for a period of fifteen (N) shipments, estimated by two two-week composite samples (Table 14). The numbers of random samples required per month for each milk constituent were calculated from 2 equation 24 by using these variances as a- and the X 2 estimates of biological variances (o^) and within herd-2 period variances (a^) from Experiment II (Table 3). The results of the calculations are presented in Table 16. The criterion specified that random sampling should be at least 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 calculation 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 entirely to procedures of estimation; i.e. compositing, sampling and testing. The proceeding analyses indicate that testing variance is the most important (Tables 3 and 11) 7 4 of the three. The variance of estimates based on random sampling are due to both procedures of estimation (sampling and testing) and to biological 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 precision equal to that of composite samples depends on the relationship between biological variance and pro-cedural variance. An expression defining this relationship 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 represent-ing the same number of shipments (N/m) can be written: a2- = i (a 2 + a 2) + (25) xc m c t N Equating this equation to equation 20 and rearranging yields: n = m(r + 1)/{1 + ^ [a 2 + a 2 (| - 1)]+ (26) a a where n i s the number of random samples required to give an estimate of the mean with precision equal to that from a composite scheme; N is the number of shipments in the period for which an estimate of the mean is desired; m is the number of compositing periods in the current composite scheme; 2 a is the variance associated with the formation c of a composite; 2 2 r is the ratio a, / a ; d a and the remaining symbols have been defined in equations 1 and 2. The term: g2 C S N a in the denominator of equation 26 reduces to zero i f : a 2 = (1 - | ) a\ (28) 2 2 and i s near zero i f both a and a are small relative to c s 2 a t, that i s , i f the main procedural source of variation is testing. If the term shown in formula 27 can be assumed to be very close to or equal to zero then equation 26 reduces to: 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 Method3 Percent Percent Percent Milk fat Protein Lactose A 3.26 3.81 2.27 B 3.31 4.78 2.49 aMethod. A 2 2 Calculated from formula 24, with and o ^ estimated from experiment two. calculated from formula 29. 7 7 2 I f a c i s large the use of the s i m p l i f i e d equation w i l l 2 tend to overestimate n . I f a g i s large then n w i l l be underestimated. The advantage of using equation 29 rather than equation 24 i s that only the r a t i o of b i o l o g i c a l to t e s t ing plus sampling variance need to be known or estimated i n order to ca l cu l a te the number of random samples needed to replace a compositing scheme with a random sampling scheme of equal p r e c i s i o n . Estimates of sample s i ze ca l cu la ted by equation 2 9 are presented i n Table 16 and agree w e l l with those ca l cu la ted by equation 24. The number of samples required (n i n formula 29) were graphed (Figure 3) versus the r a t i o of b i o l o g i c a l to t e s t ing variance (r i n equation 29) for two, three and four composites per p e r i o d . The graph can be used to f i n d the number of samples required (n) for var ious values of r for three compositing schemes. 2 T " 1 T " 2 T 3 T 4 T 5 I 6 8 l 9 10 11 12 13 — r 14 R a t i o 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 s (r) F i g u r e 3 The number of samples r e q u i r e d (n) f o r random sampling t o e q u a l the 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 o f b i o l o g i c a l t o 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 from e q u a t i o n 29 — I 15 C O 79 1. CONCLUSIONS Estimates of sampling variance for a l l three milk constituents were very small relative to the total within herd-period variance of milk constituent percentages. From these results i t can be concluded that the method of sampling bulk milk used in this study introduced l i t t l e variation 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 i t follows that testing i s the main source of variation in composite sample estimates of herd-period mean milk constituent percentages. Estimates of testing variances for percent milk fat and percent protein obtained from Experiment II were significantly different from the corresponding estimates obtained from Experiment III. Therefore i t can be con-cluded that testing variances vary from time to time in the laboratory. If this conclusion is true then s t a t i s -t i c a l l y valid predictions of the variances of estimates, obtained from any sampling scheme, of herd-period mean milk constituent percentages cannot be made. However, 80 practical considerations demand that reasonable limits 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 least approximated. More data would be required to estimate the amount of variation in testing variances. The analyses indicated that biological variances may vary from time to time or from herd to herd. Varia-tion in biological variances would mean that a random sampling scheme may have to be modified for different seasons or herds. However, i t may be possible to associate differences in biological 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 suit different herds or seasons. Two-week composite samples were concluded to yield biased estimates of a l l three milk constituent percentages. One-week composites were biased estimates of percent milk fat and lactose. Random sampling would be expected to yield 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 accur-ate (unbiased) than, estimates obtained from two two-week composites. It 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. As both testing and biological variances may vary, these con-clusion apply to the average condition and may not be valid for a l l herds or periods. 82 PART 2 2. INTRODUCTION Estimates of population variances associated with bulk milk sampling and testing were obtained in Part 1. These estimates were used to predict 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 this thesis investigated some of the practical problems associated with random sampling schemes. The results presented in 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 attributed to different sources. The variance of estimates obtained by composite sampling was attributed to procedures of estimation (sampling, testing and compositing). The variance of estimates obtained by random sampling was attributed to true differences among shipments (bio-logical variance) and to procedures of estimation (sampling and testing). 83 The magnitude of variances associated with pro-cedures of estimation may vary from time to time (Part 1) but would be expected to be essentially the same for a l l herds at a given time. The magnitude of biological variance, however, i s not necessarily the same for a l l herds. Therefore, the variance of estimates obtained by composite sampling would be similar for a l l herds; but the variance of estimates obtained from random sampling could di f f e r among herds. Thus, although a particular random sampling scheme may, on the average, meet a specified acceptable level of precision, estimates of herd-period mean milk constituent percentages obtained by this scheme could be much more variable for some herds than others. Each estimate i s economically important to the individual producer; therefore, ideally, the variance of estimates should be the same for a l l herds. If the va r i a b i l i t y of estimates obtained by random sampling is to be approximately equal for a l l herds then different sampling schemes may be necessary for some herds. For the above reasons the data of Experiment I were used to estimate i f herds differed in within herd-period variances of milk constituent percentages and i f these differences (if any) were large enough to warrant different sampling schemes for certain herds. The data of Experiment 2 were also used to estimate i f herd-period variances can be predicted from easily measured variables associated with herds. Within herd-period variances of milk constituent percentages may di f f e r among seasons. These differences may be attributed to changes in either biological variance or testing variance (Part 1). If biological variance is higher at certain seasons, sampling frequency should be increased in these seasons. On this basis sampling schemes may need to be modified not only for certain herds but also for certain periods. Mistakes in sample identification, analyses etc., can occasionally be made; therefore, test results should be systematically checked for gross errors. Factors i n -volved in checking the results from a random sampling scheme are discussed in Part 2 . 2 . MATERIALS AND METHODS 85 Source of Data The fresh sample data collected in Experiment I (defined in Part 1) from the twenty-three herds that shipped milk throughout the thirteen periods of Experiment I were used for the analyses in Part 2 . A l l periods in Part 2 were of fifteen consecutive milk ship-ments (approximately one month). S t a t i s t i c a l Methods The average amount of milk in each shipment in each of the thirteen periods was calculated (the arithmetic average of a l l shipments). The mean milk constituent percent for each period was also calculated (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 fat, protein and lactose were calculated for each herd-period subclass for twenty-three herds and thirteen periods. The within herd-period variances of percent milk fat, protein and lactose were also calculated with two, three and four strata per period by dividing the sums of squares pooled over the strata by the pooled 86 degrees of freedom. Frequency distributions of these variances were constructed. Herd and period mean within herd-period variances of percent milk fat, protein and lactose were calculated for each herd and each period by dividing the pooled sums of squares (pooled over periods for herd means and pooled over herds for period means) by the pooled degrees of freedom. Regression analyses. Simple and multiple regression techniques were used to estimate the effects 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 to yield a normal distribution, Snedecor and Cochran [18] . Therefore the distributions of the within herd-period variances and the natural logarithm (log e) of these variances were both tested for skewness and kurtosis by the method of Snedecor and Cochran [18]. The regressions were f i t t e d overall, within herds and within periods. Simple linear regressions of the l o g e of the within herd-period variances of percent milk fat, protein and lactose were f i t t e d on each of four independent variables which are defined as follows: ID ID ID w a s : 87 M^j t h e mean w e i g h t (kg.) o f m i l k i n e a c h s h i p m e n t o f t h e i * " * 1 h e r d a n d j t n p e r i o d ; t h F.. t h e mean p e r c e n t m i l k f a t a s s o c i a t e d w i t h t h e i h e r d a n d j t n p e r i o d ; t h P.. t h e mean p e r c e n t p r o t e i n a s s o c i a t e d w i t h t h e i h e r d j t n p e r i o d ; t h L . . 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 w i t h t h e i t h h e r d a n d j p e r i o d . The o v e r a l l s i m p l e l i n e a r r e g r e s s i o n m o d e l a s s u m e d w h e r e y. . = b A + b,X. . + e. . (30) • ' l ] 0 1 i j i j y ^ j t h e n a t u r a l l o g a r i t h m o f 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 e a c h m i l k c o n s t i t u e n t t h t h p e r c e n t o f t h e i h e r d a n d j p e r i o d ; bp t h e p o p u l a t i o n mean when X^.. e q u a l s z e r o ; b ^ t h e s i m p l e 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 X. . ; ID X.. was s e t e q u a l t o M.., F.., P.. a n d L . . i n ID 1D 1D ID ID t u r n ; 2 e.. t h e r a n d o m e r r o r , N(0,a ) . xj e 88 The within herd simple linear regression model assumed was: y. . = b n + h. + b,X. . + e. . (31) 2 13 0 l 1 i ] i j where bp the population mean when equal frequencies exist in a l l subclasses and when X^ .. equals zero; th h^ the effect associated with the i herd; b^ the within subclass simple regression co-eff i c i e n t of y.. on X..; e. . the random error N(0,a2) ; and the remaining symbols were defined in equation 30. The within period simple linear regression model assumed was: where y. . = brt + p. + b,X. . + e. . (32) J i ] 0 1 13 13 th Pj the effect associated with the j period; 2 e. . the random error, N(0,o* ) ; 13 e y^j defined in equation 30; and the remaining symbols were defined in equation 31. Multiple regressions of the l o g e of within herd period variances of percent milk fat, protein and lactose 89 were f i t t e d on the four independent va r i a b l e s . The o v e r a l l multiple regression model assumed was: y.. = b n + b,M.. + b~F.. + b,P.. + b.L.. + e.. (33) * xi 0 1 in 2 i i 3 in 4 i i IT where b n the population mean when M.., F.. P.. and L^j a l l equal zero; b^ the p a r t i a l regression c o e f f i c i e n t of y^j on M. . ; ID >_ the p a r t i a l regression c o e f f i c i e n t of y.. z i ] o n F. .; ID >2 the p a r t i a l regression c o e f f i c i e n t of y ^ on P. . ; 3-D >4 the p a r t i a l regression c o e f f i c i e n t of y^.. o n L. . ; ID 2 e. . the random error N (0, a ); i ] e y.. was defined i n equation 30 and M.., F.., P.. ^ i ] ^ ID ID ID and L.. were defined on page 87. ID The within herd multiple regression model assumed was: y. . = b n + h. + b,M.. + b 0F.. + b 0P.. + b.L.. + e.. (34) •*13 0 1 1 ID 2 13 3 i j 4 13 13 where bg the population mean when equal frequencies e x i s t i n a l l subclasses and when M.., F... ID ID P.. and L.. a l l equal zero; ID ID M t h h. was the effect associated with the i herd; 1 was the within subclass par t i a l regression coefficient of y^j on ..; b 2 was the within subclass par t i a l regression coefficient of y.. on F..; ID 3 was the within subclass par t i a l regression coefficient of y. . on P. .; JiD ID b^ was the within subclass par t i a l regression coefficient of y. . on L. .; a i D ID 2 e.. was the random error, N(0,a ); ID ® y^j was defined in equation 30; 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. + bnM. . + b 0F. . + b-,P. . + b.L. . + e. . (35) •'lD 0 r ] 1 I D 2 !D 3 I D 4 I D I D where th p^ was the effect associated with the j period; 2 e.. was the random error, N(0,a ); xj ' v ' e ' y^j was defined in equation 30; and the remaining symbols were defined in equation 34. Simple and part i a l regression coefficients were tested for significance by t-tests. Differences among adjusted herd and period means were tested by F-tests of the reduction in the residual sums of squares of the overall Cboth simple and multiple) regressions obtained by f i t t i n g the within subclass regression models. The F-value i s calculated, after Freese [9] as follows: F , = S S E " S S E ' / MSE' (36) S _ i ' V s - 1 where SSE the residual sums of squares from the overall regression models, (equation 30 for simple and equation 33 for multiple regressions); SSE1 the residual sums of squares from the within subclass regression models (equations 31 and 32 for simple and equations 34 and 35 for multiple regressions); MSE* residual mean square from within subclass regression models; s the number of subclasses in the within sub-class regression models; v the number of degrees of freedom associated with the error mean squares in the within subclass regression models; The above F-test i s identical to the F-test of the main effects in the analysis of covariance in the one-way cla s s i f i c a t i o n . 92 Within period regressions measured the extent to which herd differences in the within herd-period variance of milk constituent percent can be attributed to herd differences in the independent variables. The within herd regressions measured the extent to which changes in the value of the independent variables in a herd were associated with changes in the within herd-period variance of milk constituent percent. A l l possible samples for seven sampling schemes were computer generated from the data of Experiment I. Frequency distributions of the absolute value of the deviation of each sample from the fresh mean were construc-ted for percent milk fat and percent protein. The fresh mean was the mean of a l l fresh samples, weighted by the amount of milk in the shipment, in 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 for 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 strata Cb) Two strata Cone of seven and one of eight shipments1. 3 . Three shipments sampled per period: 93 (a) No strata Cb) Three strata (of five shipments each). 4. Four shipments sampled per period; (a) No strata (b) Four strata (three of four shipments and one of three shipments). 94 2- RESULTS AND DISCUSSION Period Effects on Milk Shipment Weight and Milk Constituent  Percentages Figure 4 shows the mean milk shipment weight and mean milk constituent percent for each of the thirteen periods used in this study. Percent milk fat. Percent milk fat dropped in the spring, remained at a relatively low level through the summer and climbed slowly to i t s peak value in mid-winter. Percent protein. Percent protein increased in the spring, dropped off in the late summer, climbed to a peak in the autumn and then dropped slowly to a stable winter level. Percent lactose. Percent lactose, which was less variable than either percent milk fat or protein, was lowest in the summer and autumn. Milk shipment weight. The amount of milk shipped per herd was highest in the spring and early summer and dropped to i t s lowest levels in late summer and autumn. The effect of season on the composition and level of production of herd milk can mainly be attributed to two factors. F i r s t l y , to the stage of lactation of the cu Di cd •P C CU V U CO p< •p c cu •P -P (A c o u •H 2 5.0 n 4.0 O Milk weight O % lactose , A- - A • • A. • % milk fat A % protein , A --o o-. • 2400 . 2200 - 2000 £ o iC n U 1800 § co -1600 - 1400 - 1200 3.0 - S f J Apr | May [ June [ July [Aug [Sept [Oct ^ [Nov ^ j Dec ^  | Jan ^  | Feb ^  | Mar ^  [ 1 2 3 4 5 6 7 8 9 10 11 12 13 P-P E R I O D S Figure 4 Period average milk constituent percentages and milk shipment weight for thirteen periods VO 96 cows in a herd in a particular season (i.e. the calving distribution) and secondly to the effect of season of the year on milk production and composition on cows at a l l stages of lactation. These factors can fluctuate from year to year and therefore the seasonal effects may. vary. However, the seasonal trends reported in the current study agree with those reported by Waite and Robertson [ig], Johnson et a l . [12] and Boswell et a l . [3 ]. Transformations Table 17 shows the results of the tests for skewness and kurtosis in the distributions of the transformed and untransformed within herd-period variances of percent milk fat, protein and lactose. In a l l cases the untransformed data showed significant skewness and kurtosis, however, after transformation both skewness and kurtosis were non-significant. Regression Analyses The l o g e transformed within herd-period variances of percent milk fat, protein and lactose, calculated with no strata and with four strata per period, were f i t t e d as dependent variables to regression models 30 to 35. A l l results are presented in the transformed scale, so that regression coefficients measure the change in the TABLE 17 TESTS OF NORMALITY OF THE DISTRIBUTION OF WITHIN HERD-PERIOD VARIANCES BEFORE AND AFTER LOGARITHMIC TRANSFORMATION M i l k C o n s t i t u e n t ,Skewness K u r t o s i s Untransformed Transformed Stan.Dev. Untransformed Transformed Stan.Dev. % M i l k f a t 4.85* 0.20 0.139 40.91* 0.05 0.277 % P r o t e i n 2.01 .12 .139 5.10* .10 .277 % L a c t o s e 2.37* .03 .139 8.61* .01 .277 s i g n i f i c a n t skewness or k u r t o s i s . 98 log e of within herd-period variance of a given milk constituent percentage associated with a unit change in an independent variable. Within Herd-Period Variance of Percent Milk Fat The estimates of the regression coefficients, t-tests of the coefficients and the proportion of the sums 2 of squares (R ) accounted for by the regression equations are shown i n Table 18A for simple linear regressions and multiple linear regression; overall, within period and within herd for the log of the within herd-period variance e of percent milk fat. F-tests of the differences among herds and among periods are also shown in Table 18A. The results for the l o g e of the within herd-period variance of percent milk fat with four strata per period are shown in Table 18B. Milk shipment weight. The overall simple linear regression of the l o g e of the within herd-period percent milk fat variance on the average weight Ckg.) of milk shipped was significant and the regression coefficient - 4 was (-.2 64 ± ,083)xl0 ; the within period regression was -4 also significant (-.249 ± 0.76)xl0 but the within herd regression was non-significant (-.295 ± .318)xl0"~^. These results 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 Within Period Within Herd b±S.E. R2 biS.E. R2 C F b+S .E. R 2 C F SLRa Milk wt.b - .264±.083* 3.29 - .2491.076* 3.65 6.2 8* -.2951.318 0.32 2.14* % Fat 0 .128±.082 0.83 0 .1971.076* 2.31 6.62* - . 7531.238* 3.56 3.01* % Protein .724+.186* 4.91 .7921.177* 6.67 6.73* 0 .3721.380 0.98 1.91* % Lactose .8891.318* 2.58 .5291.378 0.69 5.61* . 8971.406* 1.76 2.46* MLRa Milk wt.b -.2491.086* 2.54 - .1891.080* 1.81 - . 5781.327 1.09 % Fat -.4221.127* 3.37 - . 3771.147* 2.13 - . 6581.240* 2.62 % Protein 1 .1841.282* 5.36 1 .3311.327* 5.34 0 . 6941.385 1.13 % Lactose 1 .0401.325* 3.13 0 .4051.414 0.31 1 .1011.444* 2.14 MLR 11.43 10.09 5.67* 6.56 1.79* equation adegrees of freedom: SLR; 294,282 and 272; MLR; 291, 279 and 269 for overall, within periods and within herds respectively. 3D A regression coefficients xl0~ . significant: regression coefficients by t-tests and differences among levels (within sub-class models) by F-tests. c o °^ R^  calculated on the total within subclass sums of squares TABLE 18B SIMPLE AND MULTIPLE LINEAR REGRESSION COEFFICIENTS PERCENT MILK FAT WITH FOUR STRATA PER PERIOD Overall Within Period Within Herd b±S .E. R2 b+S .E. R 2 c F b+S .E. R 2 C F SLRa Milk wt. b - .3081.080* 4.75 - .288±.074* 5.12 5.72* -.610±.299* 1.51 2. 89* % Fat 0 .282±.078* 4.25 0 .317+. 073* 6.24 6.22* - .154±.229 0.17 2. 77* % Protein .774±.179* 5.97 .878±.171* 8.55 6.43* - .116±.359 .04 2. 48* % Lactose .636±.311* 1.41 .547±.3.70 0.77 5.42* 0 .3391.387 .28 3. 24* MLRa Milk Wt.b - .238±.084* 2.47 -.190±.078* 1.90 - • 756±.315* 2.10 % Fat - . 035±.124 0.02 - .025±.143 0.01 - .1021.232 0.07 % Protein 0 .615±.278* 1.53 0 .776±.320* 1.89 0 .0401.372 .01 % Lactose .5011.319 0.77 .040±.405 0.01 0 .6091.428 .74 MLR equation 9.00 10.49 6.03* 2.48 2. 32* adegrees of freedom: SLR; 294, 282 and 272; MLR; 291, 279 and 269 for overall, within periods and within herds respectively. regression coefficients xlO significant: regression coefficients by t-tests and differences among levels (within subclass models) by F-tests. i-* C o o R^  calculated on the total within subclass sums of squares 101 of milk were associated with low within herd-period variance of milk fat percent. But, increased milk shipments by a particular herd were not associated with a significant reduction in the within herd-period variance of milk fat percent. The range of milk shipment weights was much greater (therefore the standard error of the regression coefficient was much smaller) for both the overall and the within period regressions than for the within herd regression. Milk fat percent. The overall simple linear re-gression of the l o g e of the within herd-period variance of milk fat percent on the average milk fat percent was non-significant; the regression coefficient was 0.128±.082. Both the within period and the within herd regressions were significant; the regression coefficients were 0.197±.076 and-.753±.238 respectively. These results indicated that high percent fat herds tend to have large variances of percent milk fat; but that within herds, periods of low percent milk fat (spring, see Figure 4) were associated with high variance of milk fat percent. The increase in the within herd-period variance of milk fat percent that was associated with periods of low milk percent may be due to the relatively rapid decline of milk fat percent associated with the advent of spring grazing. A consistent directional change in a milk constituent percentage 102 across time would be expected to increase within herd-period variance of the milk constituent percentage. Protein percent. The overall and the within period simple linear regressions were significant and the regression coefficients were 0.724 ± .186 and,0.792 ± .177 respectively. However, the within herd regression was non-significant; the coefficient was 0.372 ± .380. Thus herds with high percent protein had higher than average within herd-period variance of milk fat percent but changes in protein content within a herd were not significantly associated with changes in the variance of milk fat percent. Lactose percent. The simple linear regression coefficients were significant for overall CO.889 ± .318) and within herds (0.897 ± .406) regression equations, but the within period regression coefficient (0.529 ± .378) was non-significant. These results indicated that increases in percent lactose within a herd were associated with an increase in the within herd-period variance of milk fat percentage, but that differences between herds in percent lactose were not significantly associated with differences in the variance of milk fat percent. The F-tests of the differences among levels were significant for both within subclass regression models and for a l l independent variables used. These results 103 indicated that differences among both herd and period means were significant when the independent variables were held constant (i.e. differences exist among herd means even after adjustment for the effects of herd size). Overall multiple linear regression. A l l coefficients differed significantly from zero by a t-test when a l l four independent variables were included in the overall multiple linear regression model. The coefficients for average milk weight and milk fat percent were positive while those for percent protein and lactose were negative (Table 18A). The model accounted for 11.43 percent of the sums of squares of the dependent variable. Within period multiple linear regression. Three of the independent variables were significant; average milk shipment weight, percent milk fat and percent protein when a l l four independent variables were included in the within period multiple linear regression model. These three independent variables were also significant when f i t t e d singly in the simple linear regression model. However, the sign of the coefficient for percent milk fat changed from positive in simple linear regression to negative when the remaining three independent variables were held constant. The within period multiple regression model accounted for 10.09 percent of the total within period sums of squares of the dependent variable. Differences among periods in 104 the dependent variable were significant by the F-test when the independent variables were held constant (Table 18A). Within herd multiple linear regression. Two of the independent variables, percent milk fat and percent lactose, were significant sources of variation when the within herd multiple linear regression model was f i t t e d . These two independent variables were also the only significant sources of variation when f i t t e d i n the simple linear regression models. The within herd multiple regression model accounted for 6.56 percent of the total within herd sums of squares of the dependent variable. Differences among herds in the dependent variable were significant by the F-test when the independent variables were held constant. Within strata variance of milk fat percent. The l o g g of the within herd-period variances of percent milk fat, calculated on a pooled within four strata basis, were f i t t e d as dependent variables to the same regression models. The regression coefficients estimated when the variance was calculated without s t r a t i f i c a t i o n (Table 18A) were not significantly different from the regression coefficients estimated with four strata per period (Table 18B1. However, for the independent variables of percent milk fat and percent lactose the within herd 105 regression coefficients (both multiple and simple) were not significant when the variance was calculated with s t r a t i f i c a t i o n but the regression coefficients were significantly different 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 coefficients were not significant when the variance was calculated without st r a t i f i c a t i o n but were significant when the variance was computed with four strata per period. Differences among herd and period means, tested by the F-test of the difference in levels of the within herd and within period regressions, were significant when the variance was calculated with four strata per period. Within herd-period variance of a milk constituent percentage can be mainly attributed to two factors (Materials and Methods); (1) random day-to-day variations in the milk constituent percent and (2) directional changes in the milk constituent percent across time. The second factor (time trends) would be expected to account for more of the within herd-period variation in long periods than in short periods (strata). The magnitude of the random component would not be expected to change with length of periods. The results of the regression analyses (strata vs. no strata) indicated that the relationships between the within herd-period variance of milk fat percent and the independent variables can be mainly 106 a t t r i b u t e d to the magnitude of the random p a r t o f the 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 m i l k f a t p e r c e n t . 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 P e r c e n t P r o t e i n The l o g e of the 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 p e r c e n t p r o t e i n , c a l c u l a t e d w i t h f o u r s t r a t a per p e r i o d and w i t h o u t s t r a t i f i c a t i o n , were used as dependent v a r i a b l e s i n r e g r e s s i o n models 30 t o 35. The e s t i m a t e s o f the r e g r e s s i o n 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 p r o p o r t i o n o f the sums o f 2 squares (R ) accounted f o r by the r e g r e s s i o n e q u a t i o n s a r e shown i n Tab l e 19A f o r simple l i n e a r r e g r e s s i o n s and m u l t i p l e l i n e a r r e g r e s s i o n ; o v e r a l l , w i t h i n p e r i o d and w i t h i n herd f o r the l o g e of the 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 p e r c e n t p r o t e i n . F - t e s t s of the d i f f e r e n c e s among herds and p e r i o d s are a l s o shown i n T a b l e 19A. The r e s u l t s f o r the l o g e of the 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 p e r c e n t p r o t e i n c a l c u l a t e d with' f o u r s t r a t a per p e r i o d a r e shown i n T a b l e 19B. M i l k shipment weight. M i l k 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 f o r the o v e r a l l and the w i t h i n p e r i o d simple l i n e a r r e g r e s s i o n e q u a t i o n s . 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 (-.306 ± .071)xlO and (-.293 ± .064)xl0 4 r e s p e c t i v e l y . M i l k 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 f o r the w i t h i n herd r e g r e s s i o n . These r e s u l t s i n d i c a t e d t h a t herds 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 Within Period Within Herd b±S .E. R2 b±S.E. R2 - F b± S . E . 2 c R F SLRa Milk wt.b - •306±.071* 5.92 - .293± .060* 7.07 7 .6 0* - .524±.278 1.30 1.50 % Fat 0 .131±.070 1.16 0 .171±.065* 2.41 7 .61* - ,296±.211 0.72 2.11* % Protein 0 .572±.161* 4.11 0 .5121.152* 3.87 7 .14* 0 .697±.330* 1.62 2.05* % Lactose - .362±.278 0.57 - .075±.322 0.02 7 .05* - .636±.357 1.16 2.26* MLRa Milk wt.b - .257±.075* 3.64 - .244+.068* 4.18 - .440±.289 0.83 % Fat - .153±.111 0.60 - • 051± .125 0 .05 - .374±.213 1.10 % Protein 0 •710±.247* 2.58 0 .479± .280 0.96 0 .620±.342 1.18 % Lactose - .383± .284 0.57 - ,304± .354 0.24 - .410±.394 0.39 MLR equation 9 .33 8.86 6 .99* 4 .18 1.38 adegrees of freedom: SLR; 294, 282 and 272; MLR; 291, 279 and 269 for overall, within periods and within herds respectively. regression coefficients xlO * i - 1 significant: regression coefficients by t-tests and differences among levels (within o subclass models) by F-tests. **J  CR 2 calculated on the total within subclass sums of squares TABLE 19B SIMPLE AND MULTIPLE LINEAR REGRESSION COEFFICIENTS PERCENT PROTEIN WITH FOUR STRATA PER PERIOD Overall Within Period Within Herd b±S.E. R2 b±S.E. R2 C F b±S .E. R2 C F SLRa Milk Wt.b - .188±.075* 2.06 - .167±.067* 2.16 7.99* - .729±.292* 2 .25 1.74* % Fat 0 .212± .073* 2.81 0 .194±.066* 2.93 7.99* 0 .1211.224 0.11 1.33 % Protein 0 •477±.169* 2.63 0 .566±.156* 4.48 8.56* - .065±.352 0.01 1.34 • % Lactose 0 .057± .290 0.01 0 .058±.331 0 .01 7.95* - .1161.380 0.03 1.72* MLRa Milk wt.b - .119±.080 0.73 - .090± .072 0 .54 - .776±.309* 2.29 % Fat 0 .123±.119 0.35 - .008±.131 <0.01 0 .129±.228 0 .12 % Protein 0 .192±.264 0.17 0 .571±.293* 1.29 .0241.365 <0.01 % Lactose - .143±.304 0.07 - .341±.370 0.29 0 .2201.421 0.10 MLR equation 3.96 5.50 8 . 3 8 * 2.44 1.47 adegrees of freedom: SLR; 294, 282 and 272; MLR; 291, 279 and 269 for overall, within periods and within herds respectively. b -4 regression coefficients xlO significant: regression coefficients by t-tests and differences among levels (within subclass models) by F-tests. £ c 2 0 0 R calculated on the total within subclass sums of squares 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 significantly associated with changes in the within herd-period variance of protein percent. Milk fat percent. Milk fat percent was a significant source of variation for the within period simple linear regression model only. The regression coefficient was 0.171 ± .065. This result indicated that herds shipping milk high in milk fat percent were associated with high within herd-period variance of protein percent. Protein percent. Percent protein was a significant source of variation for the overall, within period and within herd simple linear regression models. The regression coefficients were 0.572 ± .161, 0.512 ± .152 and 0.697 ± .330 respectively. These results indicated that herds shipping milk high in protein percent were associated with high within herd-period variance of protein percent. The results from the analyses of the within herd regression model indicated that an increase in the level of protein in milk shipped by an individual herd was associated with an increase in the within herd-period variance of protein percent. 110 Lactose percent. Lactose percent was not a s i g n i f i -cant source of variation for any of the three simple linear regression models. Overall multiple linear regression. The parti a l regression coefficients associated with the independent variables of milk shipment weight and percent protein were significantly different from zero by a t-test. The model accounted for 9.33 percent of the total sums of squares of the dependent variable. Within period multiple linear regression. Only the independent variable of milk shipment weight was a significant source of variation when the within period multiple linear regression model was f i t t e d . The -4 regression coefficient was (-.244 ± .068)xl0 . The model accounted for 8.86 percent of the total sums of squares of the dependent variable. Differences in levels were significant by the F-test. Within herd multiple linear regression. When the multiple regression was computed on a within herd basis none of the independent variables was a significant source of variation. This model accounted for 4.18 percent of the total sums of squares of the dependent variable. Differences between herd means were not significant by the F-test of differences of levels. I l l Within strata variance of protein percent. The regression coefficients estimated when the within herd-period variance was calculated without s t r a t i f i c a t i o n were not significantly different from the coefficients estimated with four strata per period (Table 19B). Within Herd-Period Variance of Percent Lactose The within herd-period variances of percent lactose, calculated without st r a t i f i c a t i o n and with four strata per period were used as dependent variables, after l o g e transformation, in the regression models. The estimates of the regression coefficients, t-tests of the coefficients and the proportion of the sums of 2 squares (R ) accounted for by the regression equations are shown in Table 2OA for simple linear regressions and multiple linear regression, both overall, within period and within herd for the l o g e of the within herd-period variance of percent lactose. F-tests of the differences among herds and among periods are also shown in Table 2OA. The results for the l o g e of the within herd-period variance of percent lactose calculated with four strata per period are shown in Table 2OB. Simple linear regression. Milk shipment weight was not a significant source of variation for any of the three simple linear 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 Period Within Herd b±S.E. R2 biS .E. R 2 C F biS.E. R 2 C F SLRa Milk wt.b -.1001.074 0.61 -.1041.059 1.11 16 .49 0.2821.294 0.34 1.22 % Fat -.2331.071* 3.53 -.1111.058 1.27 15.38* -l.481i.205* 16.19 3.32* % Protein 0.0401.167 0.02 -.2591.138 1.24 16.78* 0 .8541.347* 2 .18 1.56 % Lactose -.3311.283 0.46 - . 7 3 3 1.285* 2.30 17.04* -.0561.378 0.01 1.20 MLR Milk wt.b -.1701.076* 1.54 -.1451.063* 1.83 -.2971.279 0.34 % Fat - . 6 4 3 1.112* 10.07 -.0341.115 0.03 -1 . 5 3 8 1 .206* 16.77 % Protein 1.0511.249* 5.43 -.2191.256 0.25 0.9201.330* 2.33 % Lactose 0.1241.286 0.06 -.4931.325 0.79 -.1911.380 0 .08 MLR equation 11.12 4.51 13.39* 19.56 2.66* adegrees of freedom: SLR; 294, 282 and 272; MLR; 291, 279 and 269 for overall, within periods and within herds respectively. regression coefficients xlO * . M significant: regression coefficients by t-tests and differences among levels (within y-> subclass models) by F-tests. M c 2 R calculated on the total within subclass sums of squares TABLE 2OB SIMPLE AND MULTIPLE LINEAR REGRESSION COEFFICIENTS PERCENT LACTOSE WITH FOUR STRATA PER PERIOD Overall Within Period Within Herd b±S.E. R2 biS.E. F biS .E. R 2 C F SLRa Milk wt.b -.080±.077 0.37 -.0911.065 0.68 10.21* -.0981.299 0.04 1.46 % Fat -.1791.073* 1.99 -.0911.065 0.69 9.67* -1.0721.218* 8.23 2.46* % Protein 0.1201.172 0 .16 -.1671.154 0.41 10.19 0 .8981.352* 2.34 1.82* % Lactose - .4441.290 0.79 -.4501.320 0.70 10.08* - .4871.383 0.59 1.48 MLRa Milk wt.b -.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 MLR equation 8 .18 2 .12 7.94* 11.93 2.07* adegrees of freedom: SLR; 294, 282 and 272; MLR; 291, 279 and 269 for overall, within periods and within herds respectively. b -4 regression coefficients xlO * significant: regression coefficients by t-tests and differences among levels (within subclass models) by F-tests. CR 2 calculated on the total within subclass sums of squares 114 Percent milk fat was a significant source of variation when the regression was computed overall and within herds. The regression coefficients were -.233 ± .071 and -1.481 ± .205 respectively. These results indicated that the within herd-period variance of percent lactose increased, for a herd, when the fat content of the milk dropped (spring, see Figure 4). Percent protein was a significant source of varia-tion for the within herd regression only. The regression coefficient was 0.854 ± .347. This result indicated that the within herd-period variance of percent lactose increased when protein content of herd milk increased (Figure 4). Percent lactose was a significant source of variation for the within period model only. The regression coefficient was -.733 ± .285 indicating that herds with low lactose levels were significantly higher in the variance of percent lactose. The simple regression coefficients 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 significantly different from the regression coefficients estimated with four strata per period. Multiple linear regression. For the overall regression milk shipment weight, percent milk fat and percent protein were a l l significant sources of variation. 115 The partial regression coefficients were; (-.170 ± .076) xl 0 ~ 4 , -.643 ± .112 and 1.051 ± .249 respectively. The model accounted for 11.12 percent of the total sums of squares of the dependent variable. On a within period basis only milk shipment weight was a significant source of variation; the p a r t i a l -4 regression coefficient was (-.145 ± .063)xl0 . The model accounted for 4.51 percent of the total within period sums of squares. Period levels were significantly different by the F-test. On a within herd basis two of the independent variables, percent milk fat and percent protein, were significant sources of variation. The partial regression coefficients were -1.538 ± .206 and 0.920 ± .330 respectively. The model accounted for 19.56 percent of the total within herd sums of squares. The F-test of differences in herd levels was significant. The regression coefficients estimated when the variances were calculated without s t r a t i f i c a t i o n were not significantly different from the regression coeffic-ients estimated with four strata per period. The F-test of levels of both periods and herds were significant in both cases. Conclusion of Regression Analyses 116 Although the regression analyses showed that in many cases the variances of milk constituent percentages were significantly associated with the independent variables used, the proportion of the total sums of squares accounted for by the regression equations was relatively low and therefore the regression equations have l i t t l e value for predicting the within herd-period variance of an individual herd-period subclass. The regression analyses also showed that differences among herds and among periods in within herd-period variances of milk constituent percentages were significant. 117 Herd and Period Variation The criterion of precision used in the current study was that random sample estimates of herd-period milk constituent percentages should be at least as precise as composite estimates (i.e. that the level of precision of current sampling methods was acceptable to the industry). The variance of estimates that w i l l meet this criterion were presented in Table 14. By rearrangement of equation 24 to yield: a2 = n(Na- - a 2)/(N - n) (37) w x a ' The maximum value of the within herd period variance of milk constituent percentages that w i l l satisfy this criterion can be calculated for a given sample size. The values presented in Table 14 were substituted in equation 37 for 2 2 a- , values for a (defined in equation 2) were taken from X cl Table 3, to calculate maximum values of within herd-period variance for two, three, four and five random samples per fifteen shipment period for a l l three milk constituents (Table 21). The values in Table 21 were used to calculate the proportion of herds, periods or individual herd-period subclasses that would meet this criterion for various sampling schemes. 118 TABLE 21 MAXIMUM VALUE OF O 2 FOR THE PRECISION OF A RANDOM SAMPLE TO MEET THE SPECIFIED CRITERION Variance CxlO"2) Milk Constituent Sample Size Two Three Four Five % Milk fat 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 V a r i a t i o n The r e g r e s s i o n a n a l y s e s showed t h a t d i f f e r e n c e s among p e r i o d s i n the 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 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 f o r a l l m i l k con-s t i t u e n t s when the independent v a r i a b l e s were h e l d c o n s t a n t . As the data i n the c u r r e n t study were c o l l e c t e d over o n l y one year no comparisons of season e f f e c t s a c r o s s y e a r s are p o s s i b l e . I f seasons are d i f f e r e n t e s t i m a t e s of h e r d - p e r i o d means would be more p r e c i s e i n some seasons than i n o t h e r s under the same random sampling scheme; t h e r e f o r e , i t c o u l d be worthwhile t o take more samples i n some seasons than i n o t h e r s . A l t e r n a t i v e l y the sampling frequency should be g r e a t enough t h a t the c r i t e r i o n of p r e c i s i o n i s s a t i s f i e d f o r the most v a r i a b l e seasons, t h i s would mean t h a t the sampling frequency w i l l be g r e a t e r i n some seasons than n e c e s s a r y . T h i s course of a c t i o n would be w a s t e f u l of r e s o u r c e s and would i n c r e a s e c o s t s a s s o c i a t e d w i t h sampling and t e s t i n g b u l k m i l k . 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 m i l k f a t p e r c e n t . The p e r i o d v a r i a n c e of m i l k f a t p e r c e n t was c a l c u l a t e d f o r each p e r i o d by d i v i d i n g the p o o l e d w i t h i n herd sums of squares by the pool e d degrees o f freedom w i t h no s t r a t a and w i t h two, th r e e and f o u r s t r a t a per p e r i o d (Table 22). The r e s u l t s showed t h a t the 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 TABLE 22 PERIOD AVERAGE WITHIN HERD-PERIOD VARIANCE ( a w ) OF PERCENT MILK FAT WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA Variance (xlO 2) No Strata Two Strata Three Strata Four Strata Period Number a2±S.E. w DF a 2±S w .E. DF a 2±S.E. . w DF c2±S.E. w 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 fat was the highest in the spring and early-summer (Figure 5). These results agree with those re-ported by O'Keeffe [16]; however, Herrmann and Anderson [11] and Boswell et a l . [3 ] found that the variance was the highest in the period October to December, although the work of Boswell et a l . [3] showed a secondary peak in 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 in the second period (second half of A p r i l and the f i r s t half of May) to 0.00614 ± .00049 in the twelveth period (end of February and beginning of March). 'Stratification into four strata resulted in a reduction of the within herd-period variance of milk fat percentage in a l l periods; however, the reduction was, in general, greater in those periods of high variance than in those periods of low variance (Figure 5). The period variances estimated with four strata per period were a l l lower than the maximum values shown in Table 21 for four samples per period. Therefore with four samples per period (one from each of four strata) the criterion of precision would be met in a l l periods. Three samples (possibly two in some months) would be adequate in the winter i f the seasonal trends reported in the current study are consistent across years. The differences among seasons may be due to changes in biological variance or testing variance (Part 1). 123 However, as within herd-period variance was generally large i n those seasons Cspring and autumn) associated with changes in herd feeding and handling, the seasonal differences in within herd-period variance can probably be attributed mainly to differences in biological 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 strata (Table 23). The results (graphed Figure 6) showed two peaks; one in the spring (period two) and one in the autumn (period eight). Stratification resulted in a reduction in the estimates of the within herd-period variance of percent protein in a l l periods. With four samples per period the variance was higher than the maximum allowable for four of the periods (period two, five, eight and ten). However, as estimates of within herd-period variance of percent protein were lower than that of percent milk fat the standard error of the estimate of percent protein would be lower than the standard error of the estimate of percent milk fat. Period means within herd-period variance of percent lactose were calculated (Table 24). The results (graphed in Figure 7) showed that with four samples per period the criterion of precision was met in a l l periods. TABLE 23 PERIOD AVERAGE WITHIN HERD-PERIOD VARIANCE (a 2) OF PERCENT PROTEIN WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA Variance (x!0~^) No Strata Two Strata Three Strata Four Strata Period Number a2±S.E. w DF a2±S.E. w . DF o 2±S.E. DF 02±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 to •ti 2.0 1.5 A CN I O CU CJ c cd •H u > 1.0 H 0.5 H O No strata A Four strata per period [April J May | June | July | Aug | Sept | Oct | Nov | Dec | Jan 1 Feb [March i 2 3 4 5 6 7 8 9 10 11 12 13 Figure 6 P E R I O D S Within herd-period variance of lactose percent for t h i r t e e n periods 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 (xlO~ ) No Strata Two Strata Three Strata Four Strata Period 2 2 o 2 Number a ±S.E. DF a ±S.E. DF O 1S.E. DF a iS.E. DF w w w w 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 thirteen periods 128 Herd Variation The within period regression analyses (Tables 18A to 2OB) showed that large herds were lower in within herd-period variance of milk constituent percent than smaller herds and also, in general, that higher variance was associated with high herd levels of milk fat and protein. Herd means of within herd period variance of milk constituent percent were calculated for the twenty-three herds used in the regression analyses with no strata and with two, three and four strata for the within herd-period variance of milk constituent percent. The within herd-period variance of percent milk fat (Table 25) herd means ranged from 0.02955 ± .00313 for the most variable herd to 0.00590 ± .00063 for the least variable without s t r a t i f i c a t i o n . With four strata per period the range was from 0.01695 ± .00205 to 0.00380 ± .00046. Therefore with four samples and four strata per period the criterion (Table 21) was met for 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 strata per period. For nearly half the herds the criterion (Table 21) of precision w i l l not be met with four samples (one from each of four strata). TABLE 25 HERD AVERAGE WITHIN HERD-PERIOD VARIANCE (aw) OF PERCENT MILK FAT WITHOUT STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA Variance (x!0~ ) No Strata Two Strata Three Strata Four Strata a 2±S.E. w DF a2±S.E. w DF a 2±S.E. w DF a2iS.E. w DF 2 .955±.313 176 1.402±.154 163 1 .215±.139 150 1 .453±.174 137 2 .081± .221 176 1.8581.207 159 1 .714±.197 150 1 .5401.185 137 2 .020±.215 17 4 1.624±.182 157 1 .728±.200 148 1 .6951.205 135 1 .889± .202 173 1.741±.193 160 1 .4841.172 147 1 .5791.192 134 1 .675±.184 163 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 TABLE 25 (continued) Variance (xlO" ) No Strata Two Strata Three Strata Four Strata a 2±S.E. w DF o-2+S.E. w DF a 2±S.E. w DF a 2±S.E. w DF 1 .201± .130 170 1 .138± .129 154 0.997±.117 143 0 .907±.111 131 1 .165± .125 172 0 .983± .111 155 1.003+.117 146 0 .939±.114 133 1 .152± .123 174 0 .792± .088 161 0.750± .087 148 0 .688±.083 135 1 .117± .119 175 0 .912± .102 158 0.759± .087 149 0 •757±.091 136 1 .005± .109 169 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 O J O TABLE 26 HERD AVERAGE WITHIN HERD-PERIOD VARIANCE (a~) OF PERCENT PROTEIN WITHOUT. STRATIFICATION AND WITH TWO, THREE AND FOUR STRATA Variance (x!0~ ) No Strata Two Strata Three Strata Four Strata a 2±S.E. w DF a2 i S . E . w DF a2 i S .E. w DF a2+S.E. w 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 Two Strata Three Strata Four Strata c 2±S.E. w DF a2±S.E. w DF a2±S.E. w DF a2±S.E. W DF 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 st r a t i f i c a t i o n and from 0.00714 ± .00086 to 0.00276 ± .00034 with four strata per period. The herd means are a l l below the maximum value allowed i f the criterion of precision (Table 21) i s to be met and four samples are taken each period. Laboratory determinations were done for a l l herds at approximately the same time; therefore differences among herds can mainly be attributed to differences in biological variance. Distribution of Within Herd-Period Variances Table 28 shows the frequency distribution of the within herd-period variances of milk fat percent calculated with no strata and with two, three and four strata per period. A histogram of the distribution i s shown in Figure 8 for no- strata and for four strata. With four samples taken at random in a period (no strata) 77.57 percent (Table 31) of the individual herd-periods were predicted to meet the specified criterion of precision (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, st r a t i f i c a t i o n w i l l result in 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 Four Strata a2±s.E. w DF a2±s.E. w DF a 2 ± S.E. DF c 2±S.E. w DF 0.836±.089 175 0.786±.087 162 0.744± .086 149 0.714±.086 137 .806±.087 170 .763±.086 154 .615±.072 143 .618± .076 137 .726±.077 175 .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 TABLE 27 (continued) Variance (xlO*" ) No Strata Two Strata Three Strata Four Strata a 2±S.E. DF C 2 i S . E . DF a 2 i s . E . DF cr 2iS.E. DF w w w 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 Two Three Four % Cum.a % Cum.a % Cum.a % Cum.a 0.0 - 0.0049 17.95 24.52 32.06 33.33 .005 - .0099 31.73 49.68 40.13 64.65 38.41 70.47 35.56 68.89 .010 - .0149 21.47 71.15 15.92 80.57 14.60 85.07 15.87 84 .76 .0150 - .0199 10.90 82.05 8 .60 89.17 6.03 91.10 6.98 91.74 .0200 - .0249 5.13 87.18 4.46 93 .63 3.81 94 .91 3.17 94.91 .0250 - .0299 5.13 92.31 2.87 96.50 2.22 97 .13 2.54 97.45 .0300 - .0349 2.56 94.87 1.59 98.09 1.90 99 .03 0.32 97.77 .0350 - .0399 1.28 96.15 0.0 98 .09 0.0 99.03 1.59 99.36 .0400 - .0449 0.96 r 97.11 0.0 98 .09 0 .32 99.35 0.0 99.36 .0450 - .0499 0.96 98.07 1.27 99.36 0.0 99.35 0.32 99.68 .0500 - .0549 0.32 98.39 0.0 99.36 0.32 99.67 0.0 99.68 .0550 - .0599 0.32 98.71 0.32 99.68 0.0 99.67 0.0 99.68 .0600 - .0649 0.32 99.03 0.0 99.68 0.0 99.67 0.3.2 100.00 .0650 — 0.96 99.99 0.32 100.00 0.32 99.99 LO CA TABLE 28 (continued) R e l a t i v e and Cumulative F r e q u e n c i e s Number of S t r a t a None Two Three Four Mean 0.01371 0.01052 0 .00928 0 .00926 Stan. Dev. .01373 .00897 .00785 .00796 L a r g e s t Value .1592 .0685 .06503 .06358 S m a l l e s t Value .00146 .00065 .00133 .00089 Number 312 314 315 315 Cumulative f r e q u e n c i e s . 138 50 40 -1 30 -20 -10 -Four strata No strata 1.0 2 0 _2 Variance (xlO ) 4.0 >6.0 Figure 8 Distribution of the within herd-period variance of milk fat percent (no strata and four strata) 139 per period in order to meet the same criterion for 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 criterion. Table 29 shows the frequency distribution of the individual herd-period variances of percent protein without strata and with two, three and four strata per period. A histogram i s presented in Figure 9. With four simple random samples per period 41.03 percent of the herd-periods were below the limits specified in Table 21. The percentages for st r a t i f i e d random sampling were; 33.97 and 68.89 for three and four strata respectively (Table 31). Table 30 shows the frequency distribution of the within herd-period variances of percent lactose with no strata and with two, three and four strata per period. A histogram i s presented in Figure 10. With four simple random samples per period 89.42 percent of the herd-period were below the limits specified in Table 21. With s t r a t i f i e d random sampling the percentages were 83.81 and 95.87 for three and four strata per period respectively. A l l Possible Samples for Seven Sampling Schemes - Experiment I A l l possible samples for seven random sampling schemes (Material and Methods) were computer generated from the data of Experiment 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 Cumulative Frequencies Number of S t r a t a None Two Three Four C l a s s L i m i t s % t Cum.a % Cum.a % Cum.a % Cum.a 0.0 - 0.0049 37.18 48.41 55.24 64 .13 .0050 .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 6.35 94 .29 3.49 94 .92 .0150 .0199 . 6.41 95.83 3.50 97.45 3.49 97 .78 3 .17 98 .09 .0200 .0249 1.92 97.75 1.27 98.72 1.27 99 .05 0.63 98 .72 .0250 .0299 1.28 99.03 0.32 99.04 0.32 99.37 0.95 99.67 .0300 .0349 0.64 99.67 0.96 100.00 0.32 99.69 0.0 99.67 .0350 .0399 0.32 99 .99 0.32 100.01 0.32 99.99 Mean =00790 0 .00644 0 .00600 0. 00528 Stan. Dev. .00581 .00480 .00485 o 00452 Larg e s t Value .03954 .03261 .03700 • 03655 Smal l e s t Value .00061 .00053 .00054 » 00059 Number 312 314 315 315 Cumulative f r e q u e n c i e s . £ o 1 4 1 7 0 T 6 0 5 0 -4 0 -3 0 -2 0 -1 0 _ Four strata No strata 1 . 0 2 . 0 3 . 0 Variance CxlO"*2) Figure 9 Distribution 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 None Two Three Four Class Limits % Cum.a % _, a Cum. % Cum.a % Cum a . 0.0 0.0049 57.69 68.79 67 .94 70.48 .0050 - . .0099 31.73 89.42 .25.16 93.95 27.94 95.88 25.40 95. 88 .0100 - .0149 7.05 96.47 3.18 97.13 1.59 97.47 3.17 99. 05 .0150 - .0199 2.56 99.03 2.23 99.36 1.90 99.37 0.0 99. 05 .0200 - .0249 0.32 99.35 0.0 99.36 0.0 99.37 0.32 99. 37 .0250 - .0299 0.32 99.67 0.0 99.36 0.32 99.69 0.67 100. 00 .0300 - .0349 0.32 99.99 0.64 100.00 0.32 100.01 Mean 0. 00552 0. 00470 0. 00447 0.00427 Stan. Dev. c 00412 « 00372 <* 00363 .00331 Largest value o 03061 e 03195 Q 03325 .02788 Smallest value • 00054 • 00027 • 00026 .00028 Number 312 314 315 315 Cumulative frequencies. 143 70 -f 1 W O < W U P4 W 60 -50 -40 -30 -20 -10 -Figure 10 Four strata No strata 1.0 2.0 3.0 — I 4.0 -2 Variance CxlO ) Distribution 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 Samples Number of Strata None Three Four Percent milk fat 3 60.26 77.14 4 77.57 89.53 Percent protein 3 19.87 33.97 4 41.03 68.89 Percent lactose 3 73.08 83.81 4 89.43 95 .87 145 sample mean (percent m i l k f a t and p r o t e i n ) from the f r e s h e s t i m a t e o f the h e r d - p e r i o d mean were c a l c u l a t e d f o r a l l p o s s i b l e samples i n each h e r d - p e r i o d s u b c l a s s f o r a l l herd-p e r i o d s . Frequency d i s t r i b u t i o n s o f the a b s o l u t e v a l u e of these 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 each of the seven sampling schemes. These frequency d i s t r i b u t i o n s i n d i c a t e d the expected r e s u l t s i f random sampling had been used f o r the herds i n Experiment I. The r e l a t i v e and cumulative f r e q u e n c i e s of the a b s o l u t e d e v i a t i o n s , the standard e r r o r o f the mean, the l a r g e s t a b s o l u t e d e v i a t i o n and the number o f a l l p o s s i b l e samples f o r each o f the seven sampling schemes are shown i n T a b l e s 32A and 32B f o r p e r c e n t m i l k f a t and i n T a b l e s 33A and 33B f o r p e r c e n t p r o t e i n . Histograms are p r e s e n t e d i n F i g u r e s 11 to 14 o f the d i s t r i b u t i o n of the d e v i a t i o n s . The c o n f i d e n c e l i m i t s f o r the mean pe r c e n t m i l k f a t f o r the seven schemes (Tables 3 2A and 32B) agree w e l l w i t h those p r e d i c t e d from experiment two (Table 9 ) . The c o n f i d e n c e l i m i t s f o r p e r c e n t p r o t e i n are l a r g e r (Tables 33A and 33B) than p r e d i c t e d (Table 9); however the c o n f i d e n c e l i m i t s f o r p e r c e n t p r o t e i n are s m a l l e r f o r each of the schemes than f o r p e r c e n t m i l k f a t . T h e r e f o r e , f o r any sampling scheme, the mean p e r c e n t p r o t e i n would be more p r e c i s e l y e s t i m a t e d than the mean p e r c e n t m i l k f a t . S t r a t i f i c a t i o n reduced the frequency o f l a r g e d e v i a t i o n s and the magnitude of the l a r g e s t d e v i a t i o n . For example: w i t h t h r e e samples and t h r e e s t r a t a 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 Number of Observations per Sample One Two Three Four Class Limits % Cum.a % Cum.a % Cum.a % Cum.a 0.00 - 0.019 16 .43 24 .02 30 .22 35 .78 0.020 - 0.039 15 .45 31.88 21.19 45.21 25 .03 55.25 27 .48 63.26 0.040 - 0.059 14 .08 45.96 17.31 62.52 17.73 72 .98 17 .26 80.52 0.060 - 0.079 12 .21 58.17 12.49 75.01 11.20 84 .18 9 .47 89.99 0.080 - 0.099 10 .00 68.17 8.79 83.80 6.74 90.92 4 .91 94.90 0.100 - 0.119 7 .47 75.64 5.60 89 .40 3.93 94.85 2 .49 97.39 0.120 - 0.139 6 .01 81.65 3.68 93.08 2 .18 97.03 1 .21 98 .60 0.140 - 0.159 4 .72 86.37 2.42 95.50 1.19 98 .22 0 .64 99.24 0.160 - 0.179 3 .44 89.81 1.51 97.01 0.72 98 .94 0 .32 99.56 0.180 - 0.199 2 .86 92.67 1.04 98.05 0.40 99.34 0 .18 99.74 0.200 - 0.249 3 .95 96.62 1.23 99.28 0.43 99.77 0 .18 99.92 0.250 — 0.299 1 .80 98.42 0.42- 99.70 0.15 99 .92 0 .05 99.97 TABLE 32A (continued) Relative and Cumulative Frequencies Number of Observations per Sample One Two Three Four % Cum.a % Cum.a % Cum.a % Cum.a .300 ~ Largest Value . Mean Deviation Stan. Dev.*3 99% C.L.C No. of Samples 1.59 100.01 0.742 0.085 0.113 ±.292 4,511 0.30 100.00 0.623 0.058 0.077 ±.199 30,154 0.08 100.00 0 .605 0.038 0.060 ± .156 124,803 0.03 100.00 0.538 0.045 0.050 ± .129 357,659 Cumulative relative frequencies. ^Standard deviation of the distribution of deviations. °99% confidence limits. TABLE 32B 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 R e l a t i v e and Cumulative F r e q u e n c i e s Number of S t r a t a Two Three Four C l a s s L i m i t s % Cum. a % Cum. a % Cum. a 0.00 - 0.019 26.33 34.92 40.61 .020 - .039 22.78 49.11 27.45 62.37 29.72 70.33 .040 - .059 18.76 67.87 17.87 80.24 16.18 86.51 .060 - .079 11.79 79.66 9.92 90.16 7.57 94.08 .080 - .099 8.16 87.82 5.01 95.17 3.42 97.50 .100 - .119 4.85 92.67 2.49 97.66 1.39 98 .89 .120 - .139 2.89 95.56 1.21 98 .87 0.65 99.54 .140 - .159 1.86 97.42 0.52 99 .39 0.28 99.82 .160 - .179 0.89 98 .31 0.30 99.69 0.09 99.91 .180 - .199 0.73 99.04 0.14 99.83 0.05 99.96 .200 - .249 0.65 99.69 0.13 99.96 0.04 100.00 .250 - .299 0.20 99.89 0.04 100.00 .300 — 0.11 100.00 CO TABLE 32B (continued) Number of Strata Two Three Four Largest Deviation 0.465 0.295 0.236 Mean Deviation .051 .038 .032 Stan. Dev.*3 .0675 .0529 .0416 99% C L . .174 .136 .117 Number of Samples 16,081 34,240 50,469 Cumulative relative frequencies. 'standard deviation of the distribution of deviations. 99% confidence limits. TABLE 33A 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 R e l a t i v e and Cumulative R e l a t i v e F r e q u e n c i e s Number of Observations per Sample One Two Three Four C l a s s L i m i t s % Cum. a % Cum. a % Cum. a % Cum. a 0.0 - 0.019 21.41 30.07 36.95 43.23 .020 - .039 19.53 40.94 25.25 55.32 28.73 65.68 30.44 73.67 .040 - .059 16.47 57.41 18 .30 73.62 17.32 83 .00 15.59 89.26 .060 - .079 12.28 69 .69 11.40 85.02 9.05 92.05 6 .47 95.73 .080 - .099 9.47 79.16 6.71 91.73 4 .25 96.30 2.60 98.33 .100 - .119 7.27 8 6.43 3.75 95.48 1.93 98.23 1.04 99.37 .120 - .139 4.61 91.04 1.94 97.42 0.90 99.13 0.40 99.77 .140 - .159 3.08 94.12 1.08 98.50 0.49 99.62 0.15 99.92 .160 - .179 1.97 96.09 0.59 99.09 0.22 99.84 0.05 99.97 .180 - .199 1.42 97.51 0.34 99.43 0.10 99.94 0.02 99.99 .200 - .249 1.42 98 .93 0.41 99.84 0.06 100.00 • 0.01 100.00 .250 - .299 0.53 99 .46 0.13 99.97 0.01 100.01 .300 — 0.53 99.99 0.03 100.00 TABLE 33A (continued) Number of Obs e r v a t i o n s per Sample One TWO Three Four L a r g e s t D e v i a t i o n 0.598 0.447 0.305 0.249 Mean D e v i a t i o n .064 .044 .035 .029 Stan. Dev.*3 .0856 .0585 .0460 .0381 99% C.L. C .221 .151 .119 .099 Number o f Samples 4,511 30,154 124,803 357,659 Cumulative r e l a t i v e f r e q u e n c i e s . Standard d e v i a t i o n o f the d i s t r i b u t i o n o f d e v i a t i o n s . c 9 9 % c o n f i d e n c e l i m i t s . 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 Cumulative R e l a t i v e Frequencies Number of S t r a t a C l a s s L i m i t s Two Three Four 0.0 0.019 32.36 41.39 50.71 .020 .039 26.85 59 .21 30.47 71.86 30.97 81.68 .040 .059 18.31 77.52 16.07 87 .93 12.56 94.24 .060 .079 11.36 88 .88 7.05 94 .98 3.80 98 .04 .080 .099 5.67 94.55 2.82 97.80 1.18 99.22 .100 .119 2.64 97.19 1.12 98 .92 0.50 99.72 .120 .139 1.20 98.39 0.57 99.49 0.19 99.91 .140 .159 0.66 99.05 0.30 99.79 0.07 99.98 .160 .179 0 .30 99.35 0.12 99.91 0.01 99.99 .180 .199 0.25 99.60 0.05 99.96 0.01 100.00 .200 .249 0.24 99.84 0.02 99.98 .250 .299 0.12 99.96 0.01 99.99 .300 0.03 99 .99 TABLE 33B (continued) Number of S t r a t a Two Three Four L a r g e s t D e v i a t i o n 0.377 0 .309 0.182 Mean D e v i a t i o n .040 .031 .024 Stan. Dev.*5 .0531 .0404 .0317 99% C.L. C .137 .104 .082 Number o f Samples 16,081 34,240 50,469 a 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 . ^Standard d e v i a t i o n o f the d i s t r i b u t i o n o f d e v i a t i o n s . c 9 9 % c o n f i d e n c e l i m i t s . r-1 C n LO 50 T 154 cu rH Cu 40 . CO rd 4-> O tri M H o 4J CD O u CD PH 30 - % Protein 20 " 10 -40 co <D rH §• rd CO •H rd 4-1 o tH m O +J C (D O u CD PH 30 -20 - % Milk fat 10 -Figure 11 0.02 Q.06 0.10 0.14 0.18 0.22 >0.3 Absolute Deviation From Fresh Sample Mean Distribution of absolute deviations of a l l possible single samples (n=l> from the fresh sample estimate-percent milk fat and protein o •H •P • H H •P (0 •rH Q rd -P O <4H 0 -P CU O U (U CO cu • H ft B rd 0) CO c •rH H c cu ^ o u « TJ CD -P rd o •H TJ C CP •H rH g rd co TJ cu •rH <4H •rH •P rd U -P co co cu r H ft MH CO O rd -P O EH UH O -P a cu O M CU PU 50 -, 40 _ 30 -20 _ % P r o t e i n 10 -155 40 n 30 -20 -10 -% M i l k f a t 0.02 0.06 0.10 0.14 0.18 0.22 Absolute D e v i a t i o n From Fresh Sample Mean >0.3 F i g u r e 12 D i s t r i b u t i o n of absolute 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 two (n=2) from the f r e s h sample estimate-percent m i l k f a t and p r o t e i n 50 156 CD c - H fl CD M O M CQ > i X) T3 CD 4-> rd o •rH fl tn fl - H r H Ht rd co cu - r H M H - r H - P id U - P co 01 cu H B fd LO r-H rd • P o E-« MH O • P c CD o >H 0) PH 01 rd M H w fl o • H - P 0 Xi • H H 4J 01 • H Q fd • P O EH M H O - P fl CD O n CD PH 40 . 1 30 -20 -10 . 30 % Protein 40 _ 20 . 10 -% Milk f a t 0.02 0.06 0.10 0.14 0.18 0.22 >0.3 Absolute Deviation From Fresh Sample Mean Figure 13 D i s t r i b u t i o n of absolute deviations of a l l possible samples of siz e three (n=3) from the fresh sample estimate-percent milk f a t and protein 50 T 157 40 " 30 -20 10 -% P r o t e i n 40 i a 30 . 20 . % M i l k f a t 0.02 0.06 0.10 0.14 0.18 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 ~1 r 0.22 >0.3 F i g u r e 14 D i s t r i b u t i o n of a b s o l u t e d e v i a t i o n s o f a l l p o s s i b l e samples o f s i z e f o u r (n=4) 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 158 largest deviation was 0.295 in absolute value but the largest deviation with three simple random samples was 0.605 for percent milk fat; with four samples the largest values were 0.23 6 and 0.538 percent milk fat for s t r a t i f i e d and simple random sampling respectively. In both cases the largest deviation from the fresh mean with s t r a t i f i e d sampling was less than one-half as large as the largest deviation with simple random sampling. Monitoring Random Sampling A milk sampling scheme should contain provisions for resolving a disputed result (i.e. the producer considers that a particular estimate i s too low). As producers receive the results of the analyses after the period to which i t applies i s over, any additional samples taken in order to settle a disputed result 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 to monitor the observed test results as they are accumulated so that the decision to eliminate or replace observations which show large deviations from prior tests could be made before the period i s over. With str 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 misidentifica-tion, equipment malfunction, etc. 2. errors, in 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 strata means. Large deviations between consecutive milk samples due to points 2 and 3 above are expected to occur but are valid unbiased estimates of the true mean and in general the observations should not be replaced or eliminated. Large deviations due to point one above however should be detected and the offending observation should be replaced or eliminated i f the error cannot be corrected. However, i t may not be possible to determine the cause of large deviations; therefore, under practical conditions an additional sample would have to be taken i f large unexplained deviations occurred. The expected distribution 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 strata variance. On this basis 99 percent of the deviations are expected to 2 l i e within the interval, ±2.575 2 , where a i s the 2 a w within strata variance. Values for various sampling 2 schemes were calculated using estimates of a from w Experiments I and II (Table 34). The distribution of 160 TABLE 34 99 PERCENT CONFIDENCE LIMITS OF THE DIFFERENCE BETWEEN TWO RANDOM MILK SAMPLES Milk Constituent Experiment Number Number of Strata None Two Three Four % Fat I . .426 .374 .351 .351 II .420 .394 ,393 .381 % Protein I .323 , .293 .283 .266 II .259 .225 .195 .201 % Lactose I .270 .249 .243 .237 II .253 .246 .237 .232 161 d i f f e r e n c e s u n d e r p o i n t t h r e e w i l l h a v e t h e same v a r i a n c e a s a b o v e ( p o i n t two) b u t w i l l h a v e a mean e q u a l t o t h e d i f f e r e n c e b e t w e e n two a d j a c e n t s t r a t a . The g e n e r a l p r i n c i p l e i n a n y s y s t e m d e s i g n e d t o m o n i t o r a r a n d o m s a m p l i n g scheme i s t o s e l e c t a c r i t i c a l v a l u e ( o f m i l k s a m p l e t o s a m p l e d i f f e r e n c e ) s m a l l e n o u g h so t h a t t e c h n i c a l e r r o r s c a n be d e t e c t e d b u t l a r g e e n o u g h s o t h a t d e v i a t i o n s due t o c h a n c e a r e i g n o r e d . R e a s o n a b l e c r i t i c a l v a l u e s f o r v a r i o u s s a m p l i n g s c h emes a r e t h e 99 p e r c e n t c o n f i d e n c e l i m i t s shown i n T a b l e 34. U s e o f t h e s e v a l u e s w o u l d mean t h a t one s a m p l e i n a h u n d r e d w o u l d b e e x p e c t e d t o be r e p l a c e d due t o c h a n c e o f s a m p l i n g a l o n e i f d i f f e r e n c e s b e t w e e n s t r a t a means w e r e z e r o . I f a d e v i a t i o n e x c e e d s t h e c r i t i c a l v a l u e t h e o b s e r v a t i o n s h o u l d b e c h e c k e d t o d e t e r m i n e i f a t e c h n i c a l e r r o r h a s o c c u r r e d a n d i f a n e r r o r i s d e t e c t e d a n d c a n n o t be c o r r e c t e d a n a d d i t i o n a l m i l k s a m p l e s h o u l d be d r a w n f o r t h i s s t r a t a . I f no t e c h n i c a l e r r o r i s d e t e c t e d t h e n t h e d e c i s i o n t o r e t a i n o r r e p l a c e t h e o b s e r v a t i o n w i l l h a v e t o be b a s e d o n w h e t h e r t h e m a g n i t u d e o f t h e d e v i a t i o n i s r e a s o n a b l e when t h e a v e r a g e d i f f e r e n c e b e t w e e n t h e two s t r a t a ( a c r o s s h e r d s ) a r e c o n s i d e r e d o r when c h a n g e s i n t h e management o r c o m p o s i t i o n o f t h e p a r t i c u l a r h e r d a r e c o n s i d e r e d . The l a t t e r p a r t o f t h e e v a l u a t i o n o f d e v i a t i o n s i s somewhat s u b j e c t i v e ; h o w e v e r , t h e p r o p o s e d m o n i t o r i n g scheme s h o u l d r e d u c e t h e number of observations with real errors and therefore increase the producers' confidence in the random sampling scheme. The monitoring system could be made more objective i f estimates of the expected difference between strata means could be associated with the month or season in which the strata f e l l . Data collected in a random sampling program should be analysed regularly so that the program can be evaluated and modified i f necessary. Computer handling of milk test results make regular analyses relatively simple. Factors that should be considered in such analyses include: 1. the effect of seasonal changes in milk con-stituent percentages on the differences between strata means; 2 . the effect of season on within herd-period variance of milk constituent percentages; 3. the identification of herds with large ship-ment to shipment variation in milk constit-uent percentages; and 4 . the estimation of testing variance by regular replicate testing of milk samples from randomly selected herds. The results of these analyses could be used to modify the sampling program for certain herds or seasons. The results 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 relative to the total within herd-period variances of milk constituent percentages. Sampling and laboratory (including testing) procedures used in this study—except for the formation of composites in Experiment III—were those usually followed in Bri t i s h Columbia and the work was done by the people who are regularly employed to do this 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 relatively large. Milk testing procedures were found to be the main source of variation 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 fat and percent protein were concluded to vary from time to time. 165 I f t e s t i n g v a r i a n c e s v a r y then s t a t i s t i c a l l y v a l i d p r e -d i c t i o n s o f the v a r i a n c e o f est i m a t e s o f p e r c e n t m i l k f a t and p r o t e i n 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 do not 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 o f the expected v a r i a n c e of these e s t i m a t e s ; a r e a s o n a b l e approximation i s s u f f i c i e n t . E s t i m a t e s of t e s t i n g v a r i a n c e s from; the c u r r e n t study, the study by Dunn [6] and the review by Green [10] i n d i c a t e d t h a t the t e s t i n g v a r i a n c e s of p e r c e n t m i l k f a t and p r o t e i n c o u l d u s u a l l y be expected t o be l e s s than 0.007. B i o l o g i c a l v a r i a n c e accounted f o r ap p r o x i m a t e l y h a l f the t o t a l 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 p e r c e n t m i l k f a t and p r o t e i n . B i o l o g i c a l v a r i a n c e i s sampling v a r i a n c e i n the s t a t i s t i c a l sense and i s , t h e r e f o r e , not a component o f the v a r i a n c e of e s t i m a t e s of h e r d - p e r i o d mean m i l k c o n s t i t u e n t percentages o b t a i n e d from sampling schemes i n which a l l shipments a r e sampled. B i o l o g i c a l v a r i a n c e was es t i m a t e d t o be s m a l l e r i n s h o r t p e r i o d s ( s t r a t a ) than i n l o n g p e r i o d s , but the average r e d u c t i o n was r e l a t i v e l y s m a l l . However, i t was concluded t h a t s t r a t i f i e d random sampling was worthwhile as i t would be expected to reduce the frequency of l a r g e d e v i a t i o n s from the t r u e mean. S i g n i f i c a n t r e l a t i o n s h i p s were found between 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 and m i l k shipment weight, p e r c e n t m i l k f a t , p e r c e n t p r o t e i n and p e r c e n t l a c t o s e by simple and m u l t i p l e l i n e a r r e g r e s s i o n t e c h n i q u e s . However, 1 6 6 the proportion of the sums of squares accounted for by the regression equations was relatively small for a l l equations. Therefore, the relationship are not useful for predicting herd-period variances. Two-week composite samples were concluded to yield biased estimates of true means. Random samples are expected to yield 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 in use. The costs associated with the collection 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, expected precision and unbiasedness random sampling is to be preferred to composite sampling. The precision of estimates obtained by st 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, for certain herds or periods the sampling frequency may need to be greater to achieve an acceptable level of precision 167 Alternatively costs could be reduced by taking fewer than four samples for certain herds or periods and s t i l l achieve an acceptable level of precision. The results in this study indicated that in the i n i t i a l stages of a random sampling program four samples should be taken for each herd-period. The program could be assessed and modified, i f necessary, by using the results obtained in 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 in this study) would reduce the probability of obtaining samples with large deviations from the true mean and allow time to accumulate data to assess the program prior 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 in Research. McGraw-Hill Book Co., New York. 2. Biggs, D.A. 1967. Milk Analysis with the Infrared Milk Analyzer. J. Dairy Sci., 50: 799-803. 3. Boswell, R.C, E. Green and D.I. Jenkins. 1967. Daily variation in the compositional quality of ex-farm milk. Brit. 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 certain class of populations. Ann. Math. Stat. 17: 164-177. 5. Dimick, P.S. and H.V. Atherton. 1962. Factors influencing butterfat sampling accuracy in bulk cooled milk. Vermont Agr. Expt. Sta. Bull. 626. 6. Dunn, L.K. 1973. (unpublished data). 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. Fisher, R.A. and F. Yates. 1957. S t a t i s t i c a l Tables. Oliver and Boyd, Edinburgh. 5th ed. 9. Freese, F. 1964. Linear Regression Methods for Forest Research. U.S. Forest Serv. Research Pub. FBL17. 10. Green, E. 1970. Automatic measurement of the fat and protein contents of milk. Jour. Soc. Dairy Technol. 23: 190-193. 11. Herrmann, L.F. and E.D. Anderson. 1965. Butterfat sampling and testing problems. U.S.D.A. Tech. Bull . # 1336. 12. Johnson, K.R., D.L. Fourt, R.A. Hibbs and R.H. Ross. 1961. Effect of some environmental factors on the milk fat and solids-not-fat content of cows milk. J. Dairy Sci. 44: 658. 169 13. Liska, B.J. and H.E. Calbert. 1954. Study of the influence of agitation time on the Babcock test of milk samples from farm bulk holding tanks. J. Milk Food Technol. 17: 14-17. 14. Morris, H.A., S.T. Coulter and C.E. Gates. 1968. Variation within herds in composition of herd milk. J. Dairy Sci. 51: 1207-1209. 15. O'Keefe, M.G. 1967. Factors affecting the design of milk total solids testing schemes. J. Dairy Res. 34: 207-210. 16. • 1968. The use of single or composite milk samples for the determination of fat. J. Dairy Res. 35: 291-294. 17. Preston, H.J. 1954. Developing butterfat sampling and test-ing programs. U.S. Dept. Agr. Farmer Co-op. Serv. Bull . 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 in the chemical composition of milk with particular reference to solids-not-fat. I. The effect of stage of lactation, season of year and age of cow. J. Dairy Res. 23: 65-81. 20. Welch, B.L. 1956. On linear combinations of several variances. J. Amer. Stat. Assoc. 51: 132-148. 

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