The estimation of genetic parameters for categorical traits : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Animal Science at Massey University

private study only. The thesis may not be reproduced elsewhere without

the permission of the Author.

THE ESTIMATION OF GENETIC PARAMETERS

FOR CATEGORICAL TRAITS

A thesis presented in partial fulfilment of the requ irements for the degree of

Doctor of Philosophy in Animal Science at Massey University.

ARTHUR RICHARD GILMOUR

1983

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ABSTRACT

The estimation of heritabilities, genetic and environmental variances and covariances, and the prediction of breedina values, is a major concern among animal

breede r s . This s tudy

adapts existing statistical methods to provide a new method of estimating these parameters for categorical traits.

The problems associated with the analysis of categorical data arise because of the relationship between the mean and

variance of the sampling distribution. The parameters of the sampling distribution are assumed to be a non-linear function of values on an underlying scale. It is further assumed that fixed and random effects are additive on the underlying

scale. This scale cannot be observed and information about it must be deduced from the observed categorical trait.

A common practice has been to estimate parameters (fixed

effects and variances) on the categorical variable itself and then to transform these estimates to values applicable to the underlying scale. This procedure is theoretically invali�

except for a fully random model in which the only fixed effect is the mean. The method developed in this thesis attempts to estimate parameters directly in the underlying scale by transforming the data before calculating estimates of the parameters. It is a synthesis of mixed model

procedures (Henderson et

?1., 1959) and generalized linear models (N elder and Wedderburn, 1972) and is called the

generalized linear mixed model. The general method is for analysing data presumed to arise from a two-stage sampling procedure when the second sampling has an error distribution belonging to the single parameter exponential family.

The detailed algebra for applying the new method to binomial and multinomial traits for the estimation of fixed effects is presented. The logit transformation is used in this

application and the resulting system of equations is called the logistic linear mixed model. A procedure for estimating

iii

variances, covariances and the intraclass correlation on the underlying scale is also developed.

The logistic linear mixed model is evaluated by comparing parameter estimates from the method with true values used to generate the data being analysed. Biases appear to be small except for some extreme combinations of parameters when

assumptions made while developing the algebra break down.

The logistic linear mixed model is applied to two real problems for which fixed and random effects and variance

components are estimated and comparisons made with parameters estimated by other methods. The first problem is the

analysis of data on the feet characteristics of

2 5 1 3

^lambs,

the second is the analysis of

1 3 96

lambing performance records.

The conclu ding discussion considers the general use of the logistic linear mixed model and its relationship with other models.

ACKNOWLEDGEMENTS

I gratefully acknowledge the guidance of my supervisors, Professor R. D. Anderson, Professor A. L. Rae and

Dr. G. A. Wick ham.

The Australian Wool Corporation provided the post-graduate scholarship which enabled this study to be undertak en. It is thank ed, along with the New South Wales Department of

Agriculture which provided study leave and financial assistance.

The Animal Science Department has provided a fine work ing environment and the opportunity to broaden my outlook .

My gratiude is extended to the friends we have made,

particularly in Milson. Their willingness to include us in their community has made life both pleasant and rewarding for me and especially for my wife, Ellen, and our three

daughters. In particular, I ack nowledge the friendship of Graham and Lynette Toms and the members of Milson Christian F ellowship.

To Ellen and the three girls,

thank you fo r b e ing there when

needed and bearing with me when times were busy.

To my God,

thank you fo r p r o v id ing a l l our ne eds -

friends, a home, work and recreation, health, finance and the opportunity to serve - through Jesus Christ.

Table of Contents

Section

Chapter

1 .

Chapter

2 . 2 . 1 .

2 . 2 . 2 . 2 . 1 . 2 . 2 . 2 . 2 . 2 . 3 . 2 . 2 . 4 . 2 . 3 . 2 . 3 . 1 . 2 . 3 . 2 . 2 . 4 .

Chapter

3 .

3 . 1 .

3 . 1 . 1 . 3 . 1 . 2 . 3 . 1

^•

3 . 3 . 2 . 3 . 2 . 1 . 3 . 2 . 2 .

Heading

Abstract

Ack nowledgments Table of Contents List of Tables List of Figures

Introdu ction

Literatu re review Categorical data

Analysis on the binary

( 0 , 1 )

^scale

Binomial traits Multinomial traits

Correlation between variables

Selection involving categorical traits Analysis on the u nderlying scale

F alconer's liability method

Generalized linear models and other proposals

Scope of the thesis

Review of statistical procedures requ ired in later chapters

Some statistical procedu res u sed in animal breeding

The selection index

Best linear u nbiased prediction Estimation of variance components O ther statistical procedures

Logistic distribu tion

Log likelihood expectations

V

page

ii iv V ix xiv

1

3 4 6 6 1 1 1 3 1 5 1 7 1 7

1 9 2 1

2 2

2 2

22

2 3

2 4

26

26

2 8

Generalized linear models (GLM)

3 0 3 . 2 . 3 . 1 .

Derivation of the general expressions for members

of the single parameter exponential family

3 0 3 . 2 . 3 . 2 .

Testing the 'goodness of fit' of a proposed

3 . 2 . 4 .

model

Maximum lik elihood solution for parameters of the multinomial distribution by generalized linear models

Solution of re-weighted least squares equations

Chapter

4 .

Generalized linear mixed model

4 . 1 .

The joint-maximization method

4 . 2 .

The maximization-expectation method

3 5

3 5 3 9 4 0 4 1 4 4

Chapter

5 .

Deriving the logistic linear mixed model

4 8 5 . 1 .

5 . 1 . 1 . 5 . 1 . 2 .

5 . 1 . 2 . 1 . 5 . 1 . 2 . 2 .

5 . 1 . 2 . 3 . 5 . 2 . 5 . 2 . 1 . 5 . 2 . 1 . 1 . 5. 2 . 1 . 2 .

5 . 2 . 2 . 5 . 2 . 2 . 1 . 5 . 2 . 2 . 2 .

Mixed model analysis of binomial data

4 8

Solution by joint-^·

rnax imiza t ion me thod 4 9

Solution by the maximization-expectation method

Using the N ormal distribution Using the moments of a symmetric

distribution

Using the logistic distribution Analysis of multinomial data Analysis of extremal characters

Sol ut ion by joint-maximization method Solution by the maximization-expectation

method

Analysis of multiple threshold characters Fixed effects generalized linear model

Mixed model threshold analysis by joint-maximization method

4 9 5 0 5 2 5 5 5 9 5 9 5 9 60 66 68

7 0 5 . 2 . 2 . 3 .

Mixed model threshold analysis by

maximization-expectation method

7 1

vii

Chapter

6 .

Application of the logistic linear mixed model

to animal breeding

7 5

6 . 1 .

The simu ltaneou s analysis of categorical

6 . 1 . 1 . 6 . 2 . 6. 3 . 6 . 3 . 1 . 6 . 3 . 2 .

and continu ou s characters by the same model

7 5

Covariance between a binomial character

and a normal character

7 7

Estimation of variances

7 9

Implementation of the logistic linear mixed model in a generalized linear models programme

8 3

Measu rement of 'lack of fit'

Estimation of variance components

8 3 8 4

� Chapter

7 .

Performance of the logistic linear mixed model

·1

in simu lation stu dies.

8 7

7 . 1 .

The behaviou r of estimates of mean and variance

u nder the logistic linear mixed model

8 7

1 . 2 .

The precision of estimates of the threshold and

the intraclass correlation

7 . 3 .

The estimation of breeding valu es u nder the

logistic linear mixed model

Chapter

8 .

A stu dy of foot ailments associated with Merino-cross sheep grazing damp conditions

8 . 1 .

Repeatability of assessment of foot-shape

8 . 2 .

Estimation of heritability of incidence of foot

9 2 97

1 1 0 1 1 1

ailments

1 1 2

Chapter

9 . 9 . 1 .

9 . 2 .

Reprodu ctive performance of Perendale ewes

1 2 5

Analysis of Perendale data by ordinary mu ltivariate

least squ ares

1 29

Analysis of binary traits on the probit scale

u nder a fixed effects model

1 3 0

9 . 3 .

Analysis of the Perendale data u sing the

logistic linear mixed model

1 3 5

Chapter

1 0o

General Discu ssion

1 4 7

1 0o 1o

How relevant is an estimate of heritability on the

1 0o 2o

1 0

⁰

3

⁰

1 0o 4o

1 0o 5o 1 0

⁰

6

⁰

Appendix A

Appendix B

Appendix C

u nderly ing scale ?

1 4 7

What relationship exists between the

logistic linear mixed model and the ordinary

mixed model analy sis?

1 47

What can be done in very large problems where the iterative procedures would be

prohibitively expensive?

1 4 9

When might it be more efficient to u se

equations based on the joint-maximization method rather than those based on the

maximization-expectation method?

What fu rther work is requ ired?

Conclu sion

Some matrix sy mbols and operations

More expectations for section

5o 1o2o 1

^{u sing}

the normal distribu tion

Derivation of the basic expressions requ ired to apply the generalized linear mixed model to mu ltinomial data u sing the

mu ltinomial logit

1 5 0 1 52 1 53

1 5 4

1 56

1 5 9

Appendix D

Some additional tables relating to chapter

8

⁰

Bibliography

1 76

1 8 1

L is t o f Tabl e s T ab l e

3 . 1 Th e r e l at i o n s h i p b e t w e en t h e s tand a r d norm a l a n d sta n d a r d log i s t i c d i s t r i but i on s whe n u s e d as p robab i l i ty t r a n s form a t i o n s .

7 . 1 E f fects o f d i ffer i n g fam i l y s i z e a n d numb e r s o f fam i l i es on e s t i m at e s o f the i n t r a c l a s s c o r r el a t i o n and t h e thr e s h o l d i n b i n om i a l s amples w i th e x t r a - b i n o m i a l v a r i a t i o n .

p a g e

2 8

9 4 7 . 2 S t a t i st i c s for 1 0 s amp l e s o f 1 0 0 N ( O, . 1 0 ) r a n d om

v ar i a b l e s used t o d e f i n e t h e 1 0 f l ocks . 8 . 1 A n a l ys i s o f var i a n c e o f foo t - s h a p e sco r e s . 8 . 2 A n al y s i s o f d ev i a n c e o f foot- s h ape sc o r e s wh e n

a n a l ysed a s a d ou b l e- thr e s h o l d t r a i t w i th an i n tr ac l a s s cor r e l a t i on o f 0 . 7 3 1 .

8 . 3 I t e r a t i v e s equ e n c e for e s t i m a t i n g the i n t r a c l a s s c or r el a t i o n fo r t h e doub l e- thr e s h o l d t r a i t , L 5 4

9 8 1 1 1

1 1 1

( l amb foo t sha pe s co r e 5 , 4 a n d l e s s t h a n 4 ) . 1 1 7 8 . 4 A n a l ys i s o f d e v i a n c e f o r t he d o ub l e- th r e s h o l d t r a it

L 5 4 ( l am b foo t - shape s c o r e 5 , 4 and l e s s th a n 4 ) . 1 1 7 8 . 5 E s t imat e s o f i n t r a c l as s c o r r e l a t i o n and

h e r i t ab i l i t y ( 4 t im e s i n t r a c l a s s c o r r e l a t i o n )

f o r 1 2 s h e e p fo o t t r a i t s . 1 1 8

i x

T a b l e p a g e 8 . 6 C o r r e l at i o n betw e e n e st i m a t es o f b r e ed i n g v a l u e for

f oot- s h a pe s c o r e , ob t a i n e d by v a r iou s m e thod s . 1 18 8 . 7 C or r e l a t i o n bet w e e n e s t i m a t e s o f b reed i n g v a l u e

o f 2 8 s i r e s in fo u r mat i n g - g r o u p s f o r p r e s e n c e o f foo t - s c a l d a n d pr e s e n c e o f foot- ro t .

8 . 8 A n a l ys i s o f dev i a n c e for f oo t - s h ape t r a i t s L5 4 a n d H 5 4 , u s i ng 3 4 s i r e s .

8 . 9 A n a l ys i s o f de v i a n c e fo r foot- sh ape t r a i t s L 5 a nd H5 , u s ing 3 4 s i r e s .

8 . 1 0 A n a l ys i s o f dev i a n c e for p r e s e n c e o f f o o t - s c a l d ( LS ) a n d pre s e n c e o f f o o t - r o t ( L R ) i n l am b s

119

1 1 9

1 2 0

u s i n g 2 8 s i res . 1 2 0

8 . 1 1 A n a l y s i s o f Va r i a n c e f o r foo t - sh a p e s co r e ( L C )

u s i ng 3 4 s i res .

^K

co e f f i c i e n t i s 7 3 . 4 3 8 8 . 1 2 1

8 . 1 2 A n a l ys i s o f va ri a n c e f o r foot- s h a p e t r a i t L 5

u s i n g 3 4 s i re s .

^K

co e f f i c i e n t i s 7 3 . 4 3 8 8 . 1 2 2 8 . 1 3 A n a l y s i s o f Va ri a n c e f o r pre s e n c e of fo o t - s c a l d

( LS ) u s i ng 2 8 s i re s .

^K

coe f f i c i e n t i s 6 9 . 7 0 2 4 . 1 2 3 8 . 1 4 A n a l ys i s o f Va r i a n c e f o r pre s e n c e o f foo t - r o t ( L R )

u s i n g 2 8 s i re s .

^K

c oe f f i c i en t i s 69 . 7 0 2 4 . 1 24 8 . 15 I n c i d e n c e o f foot - s c a l d in l am b s ( 5'mo n t h s ) o n

p e r c ent a g e and l o g i t s c a l e s . 1 2 4

T a b l e p a g e 9 . 1 S u mma r y o f the n u m ber o f d au gh t e r s s i r e d b y e a c h o f

t h e 6 3 r am s u s e d i n the f l oc k f rom 1 9 6 1 t o 1 97 2 . 1 26 9 . 2 D a m age m e an s fo r w ean i n g w e i gh t , hogg e t w e ight

a n d num b e r of l am b s we a n e d .

9 . 3 Y e a r me a n s for w e a n i ng w e i g ht , h o g get we i g ht and num b e r o f l am b s we an e d .

1 2 6

1 2 7

9 . 4 V a r i anc e a n d cov a r i a n c e com pone n t s est i m a t e d b y H e nder s o n ' s ( 1 9 5 3 ) m e t hod 3 . C o v a r i a n c e compo n e n t s a r e bel o w a nd cor r e l at i o n s a r e a b o v e t h e d i a gon a l 1 2 9

9 . 5 H e r itab i l i t i es o f f iv e t r a i t s by H e nder so n ' s ( 1 9 5 3 )

m e thod 3 . 1 2 9

9 . 6 H e r i t a b i l i t y on t he p r o b i t sc a l e by F a l c o n e r ' s

( 1 9 65 ) m e thod . 1 3 2

9 . 7 H e r i t ab i l i t y on t h e p r o b i t s c a l e u s i n g F a l c o n e r ' s m ethod a f t er inc l u d i n g h o g g e t w e i ght a s a cov a r i a t e a nd as s u m i ng g e n e t i c c o r r e l at i o n s o f 1 .

9 . 8 S i r e s i n o rder o f a v e r a g e r an k for l ambe d - o r - no t a n d twi n s - or- not .

9 . 9 F i x ed e f f e c t s for a l l t r a i t s e s t im ated b y m a x imum l i kel i h o o d s in g l e t r a i t m e thod s .

1 3 3

1 34

1 3 9

x i

T a b l e p a g e 9 . 1 0 G e n e t i c a n d env i r o n ment a l v ar i a n c e s , h e r i t a b i l i ­

t i es and b r eed i n g v alue s e s timat e d b y m a x i mum

l i kel i ho o d s i n g l e t r a i t metho d s . 1 4 0 9 . 1 0 c o n t i n u ed . B r e e d i n g v a l u e s for a l l t r a i t s e s t i m at e d

b y ma x i mum l i k e l ihoo d s i ng l e t r a i t m e t h o d s . 1 4 1 9 . 1 1 S ome r e s u l ts f r o m u s ing v a r ious t r i a l v a l u e s for

the v a r i a n c e s a n d c o v a r i a n c e s a s s o c i a t e d w i th an

e x trem a l tra i t . 1 4 2

9 . 1 2 G e n e t i c v a r i a n c e s a n d cov a r i a n c e s betwe e n two

b i nar y t r a i t s , l ambed - or - n o t a nd twi n s - o r - not , a n d h o gget w e i ght e s t i mat ed f o r thr e e env i r o n ment a l c o r r e l a t i o n s , 0 . 0 , 0 . 3 and - 0 . 3 .

9 . 1 3 G en et i c v ar i a n c e s a n d c o v ar i a n c e s b e t w e e n two c a t egor i c a l tr a i t s , l amb- t and l amb- e , a n d

h o gget w e ight , e s t ima t e d for th r e e env i r o nmen t a l

1 43

c o rrel a t i o n s , 0 . 0 , 0 . 3 and - 0 . 3 . 1 44

9 . 1 4 C ompar i s o n of b r e ed i n g v a l ue s f o r l am b e d - or - n o t a n d ho g g e t we i g h t obt a i n e d assum i n g e n v i r o nme n t a l c o rrel a t i o n of 0 . 0 and g enet i c c o r r el a t i o n s o f 0 . 0

a n d 0 . 62 75. 1 45

9 . 1 5 C o r r e l a t i o n s between b r eed i n g v a l u e s i n t a b l e 9 . 1 4 a n d the r e a r ing r a n k o f the s i r e . 1 4 6 1 0 . 1 T h e pr o b a b i l i t y t h a t a g r o u p o f n om i n a t e d s i ze

a nd nom i n a t ed i n c i d en c e has ob s e r v a t i o n s a l l i n

t h e sam e c a tegor y . 1 5 2

T ab l e

0 . 1 F o o t - sh a p e scor e s r ecor d e d a t t wo t imes b y t wo obs e r v e r s o n 9 7 l am b s .

0 . 2 F o o t- s c a l d sco r e s r ec o r d ed a t two t im e s b y t wo obs e r v er s o n 9 7 l am b s .

0 . 3 F o o t - r o t s co r e s r e c o r d e d a t two t im e s b y t wo ob s e r v er s o n 9 7 l am b s .

0 . 4 a E x per i m ent d a t a for 1 9 8 0 l amb i n g 0 . 4 b E x per i m ent d at a for 1 9 8 1 l amb i n g

0 . 5 E s t i mat e s o f br e e d i n g v a l u e s for t h e t r a i t

x i i i p a g e

1 76

1 7 6

1 7 7

1 77 1 78

foot- s h a p e s c o r e - o bt a i n ed b y v a r i ou s m ethod s . 1 7 9 0 . 6 E s t im a t e s of b r e e d i n g v a l u e s f o r the t r a i t s

p r esen c e o f foo t - s c a l d and pr e s e n ce o f f o o t - r o t ,

o b tai n e d b y two m e thod s . 1 8 0

L i st o f F i gu r e s

F i g u r e p a g e

7 . 1 C om par i so n o f t h e s c a l e f a c to r s a s s o c i at e d w i th

the t h r e e e x pr e s s i on s f o r E [ p ] . 9 0 7 . 2 C om p a r i so n o f the s c a l e f a c to r s a s s o c i a t ed w i th

t h e t h r e e e x pr e s s i on s fo r E [ p ( 1 -p ) ] . 9 1

7 . 3 R e l at i o n sh i p be t w e e n ac t u a l and e s t i m a t e d v a l u e s o f t h e i n t r a c l as s c o r r e l a t i o n

c o effi c i e n t for v a r i ous t h r e sho l d v a l u e s .

7 . 4 R e l at i o n s h i p between a c t u a l and e s t im a t ed v a l u e s o f the th r e s ho l d f o r v ar i o u s i n t r a c l a s s

95

c o rr e l a t i o n s . 96

7 . 5 R e l a t i o n sh i p between B ( . 5 0 ) and f am i l y s i z e . 1 0 3 7 . 6 R e l at io n s h i p be t w e en G ( . 5 0 ) a n d fam i l y s i z e . 1 0 3

7 . 7 R e l at i o n s h i p b e t w e en B ( . 1 0 ) a n d fam i l y s i z e . 1 0 4

7 . 8 R e l at i o n s h i p between G ( . 1 0 ) a n d fam i l y s i z e . 1 0 4

7 . 9 R e l ati o n s h i p between B ( . 0 2 ) a n d fam i l y s i z e . 1 0 5

7 . 1 0 R e l at i o n s h i p between G ( . 0 2 ) a n d f am i l y s i z e . 1 0 5

XV

F i gu r e p a g e

7 . 11 R e l at i o n s h i p be t w e en cor r e l at i o n ( U, U X) a n d

fam i l y s i ze . 106

7 . 12 R e l at i o n s h i p b e t w e e n c or r e l a t i o n ( u ' u p) and

f am i l y s i ze . 107

7 . 1 3 R e l at i o n sh i p between cor r e l a t i on ( u

^x'

u P) a nd

famil y s i ze . 108

7 . 1 4 R e l at i o n sh i p b e tween d e v i an c e ( 4 9 9 d f ) and

f ami l y s i ze . 1 09

•