Quantitative Genetic Models for Describing Simultaneous and Recursive Relationships Between Phenotypes



DOI: 10.1534/genetics.103.025734

Quantitative Genetic Models for Describing Simultaneous and

Recursive Relationships Between Phenotypes

Daniel Gianola*

,†,1

_{and Daniel Sorensen}

‡

*Departments of Animal Sciences, Dairy Science and Biostatistics and Medical Informatics, University of Wisconsin, Madison, Wisconsin 53706,†_{Department of Animal and Aquacultural Sciences, Agricultural University of Norway, N-1432 A}˚ s, Norway and

‡_{Department of Animal Breeding and Genetics, Danish Institute of Agricultural Sciences, 8830 Tjele, Denmark}

Manuscript received December 11, 2003 Accepted for publication March 5, 2004

ABSTRACT

Multivariate models are of great importance in theoretical and applied quantitative genetics. We extend quantitative genetic theory to accommodate situations in which there is linear feedback or recursiveness between the phenotypes involved in a multivariate system, assuming an infinitesimal, additive, model of inheritance. It is shown that structural parameters defining a simultaneous or recursive system have a bear-ing on the interpretation of quantitative genetic parameter estimates (e.g., heritability, offspring-parent regression, genetic correlation) when such features are ignored. Matrix representations are given for treating a plethora of feedback-recursive situations. The likelihood function is derived, assuming multivari-ate normality, and results from econometric theory for parameter identification are adapted to a quantita-tive genetic setting. A Bayesian treatment with a Markov chain Monte Carlo implementation is suggested for inference and developed. When the system is fully recursive, all conditional posterior distributions are in closed form, so Gibbs sampling is straightforward. If there is feedback, a Metropolis step may be embedded for sampling the structural parameters, since their conditional distributions are unknown. Extensions of the model to discrete random variables and to nonlinear relationships between phenotypes are discussed.

M

ULTIVARIATE models are of great importance mental or residual effects (E1,E2). The genetic and

en-vironmental effects are assumed to be independently in applied, evolutionary, and theoretical

quanti-distributed random vectors, following the bivariate nor-tative genetics. For example, in animal and plant

breed-mal distributions N(0, G0) and N(0, R0), respectively.

ing, the value of a candidate for selection as a

prospec-Here, tive parent of the next generation often is a function of

several traits,e.g., protein yield, milk somatic cell count,

fertility, and life span in dairy cattle or yield and resis- G0⫽

冤

␴2 u1 ␴u12

␴u12 ␴2u2

冥

(1)

tance to disease in wheat. In evolutionary genetics, the

effects of natural selection on mean fitness depend on and

the values of elements of the genetic variance-covariance

matrix between quantitative characters (e.g., Cheverud _R

0⫽

冤

␴2 e1 ␴e12

␴e12 ␴2e2

冥

(2)

1984).Walsh(2003) and B.Walshand M.Lynch

(un-published results) give a discussion of the dynamics of _{are genetic and residual variance-covariance matrices,}

quantitative traits under multivariate selection. _{respectively. For example,␴}

u1

2 _{is the variance between}

A schematic of the standard multivariate model used _{additive genetic effects affecting trait 1, and␴}

e12is the

in quantitative genetics is displayed in Figure 1, where _{residual covariance between traits 1 and 2.}

a two-trait system is represented; for simplicity, all other _{The standard model depicted in Figure 1 does not}

explanatory variables are omitted. The diagram depicts _{allow for feedback or recursive relationships between}

a 2⫻1 vector of phenotypic values (Y1,Y2) expressed _{phenotypes, which may be present in many biological}

as a function (typically linear) of genetic effects (U1,U2), _{systems. A classical example of feedback (that is, when}

usually taken to be of an additive type, and of environ- _{changes of a quantity indirectly influence the quantity}

itself) is given byHaldaneandPriestley(1905) and

retaken byTurnerandStevens(1959) and byWright

(1960). These authors modeled feedback relationships

This article is dedicated to Arthur B. Chapman, teacher and mentor

of numerous animal breeding students and disciple and friend of _{between the concentration of CO2in the air (A) and} Sewall Wright.

in the alveoli of the lungs (C) and the depth of

respira-1_{Corresponding author:Department of Animal Sciences, 1675}

Obser-tion (D). As shown in Figure 2,Turnerand Stevens

vatory Dr., University of Wisconsin, Madison, WI 53706.

E-mail: [email protected] (1959) posit thatAaffects C; in turn, Cand Dhave a

Figure 1.—Standard bivariate model used in quantitative genetics:Y1andY2are the phenotypic values;U1andU2are additive genetic effects act-ing on the traits;E1andE2are residual effects. A single-headed arrow (e.g.,A→B) indicates that variableAaffects variableB.

feedback relationship.Wright(1960) introduces resid- place. Ignoring the actual biology of the problem, a

model such as that ofTurnerandStevens(1959) or

uals V and W for the C and D variables, respectively,

and makes a further extension of the model. The exten- of Wright (1960) implies the following: if A is

in-creased and the relationship between C andAis such

sion consists of including a variable X for the actual

concentration of CO2in the lungs; in Figure 2,Urepre- thatCincreases as well, thenDwill increase provided␭DC

is positive. Further, if␭CDis positive, thenCwill increase

sents “random” errors of technique; this is a

“measure-ment error” model (Warrenet al.1974;Joresko¨ gand further. If all the loops go in the same direction, there

is positive feedback, which might lead to some equi-Sorbo¨ m2001). In the Turner-Stevens model, the effect

ofConDis through a coefficient␭DC, whereas␭CDgives librium or to an eventual breakdown of the system (

Tur-nerandStevens1959). A second example of reciprocal

the rate of change ofCwith respect toD. Suppose that

these two coefficients are not null, so that feedback takes interaction is the classical supply-demand problem of

Figure 2.—Haldane and Priestley (1905)

respiration relations. Models for describing feed-back relationships between concentrations of CO2 in the respired air (A), in the alveoli of the lungs (C), and depth of respiration;V, W, and U are residuals.Wright(1960) introduces the variable

X, the actual concentration of CO2in the alveoli;

econometrics (Wright 1925; Johnston 1972; Judge sive and exhaustive models for describing the relation-ships between phenotypic values. Formulas pertinent to

et al.1985). Also, the existence of feedback inhibition

is well known in genetic regulation. For instance, the multivariate selection (e.g., best prediction of genetic

values) are given as well. Then, thelikelihood

func-product of a metabolic pathway may bind to a gene

product (enzyme) catalyzing a previous step, to prevent tionandidentification of parameters sections are

presented.bayesian model addresses statistical

infer-the channeling of additional molecules through infer-the

pathway (Lewin1985). A discussion of the implications ence in a structural equations model from a Bayesian

perspective. It is shown in thefully conditional

pos-of interactive enzyme systems in genetics is inKacser

andBurns(1981). They write: terior distributions that, under normality

assump-tions, most conditional posterior distributions arising

In vivoenzymes do not act in isolation, but are kinetically

in the multivariate system are in recognizable form. The

linked to other enzymes via their substrates and products.

implication is that software for standard multiple-trait

These interactions modify the effect of enzyme variation

on the phenotype, depending on the nature and quantity analysis of quantitative traits via Gibbs sampling (a

Mar-of the other enzymes present. An output Mar-of such a system, _{kov chain Monte Carlo method) can be adapted to} han-say a flux, is therefore a systemic property, and its response _{dle simultaneity and recursiveness in a fairly direct} man-to variation at one locus must be measured in the whole

ner. The article concludes with suggestions on how the

system.

approach can be extended to a wider class of models.

It has been long recognized in economics (e.g.,

Haa-velmo1943) that the existence of lagged or of

instan-GENETIC CONSEQUENCES OF SIMULTANEITY

taneous feedback (often referred to as “simultaneity”)

IN A TWO-TRAIT SYSTEM

and of recursiveness between variables has implications

Let yi1andyi2be measurements on traits 1 and 2

ob-on the understanding of multivariate systems, and that

served in individuali. For example, in an animal

breed-special statistical techniques are required for inference.

ing setting,yi1may represent the milk yield of dairy cow

Curiously, Sewall Wright’s work on feedback mechanisms

iand yi2 may be a proxy for the level of some disease

has received scant attention in population/quantitative

(e.g., milk somatic cell count as an indicator of mastitis, a

genetics, in spite of his influence on the aforementioned

bacterial-related inflammation of the mammary gland). fields and the pervasiveness of such mechanisms in

regu-Suppose that biological knowledge admits that produc-lation, as noted. An explanation may reside in the fact

tion affects disease and, in turn, that disease affects that even though path analysis was “extremely powerful

production. As noted earlier, we refer to this as a

simul-in the hands of Wright” (Kempthorne1969), the lack

taneous or instantaneous feedback system, following of matrix representations in his writings hampered a

econometric terminology (Zellner 1979;Judge et al.

general understanding of the method. This is especially

1985). Assume that the relationship between produc-true of Wright’s treatment of reciprocal interaction with

tion and disease can be represented by the two-equation

lags (Wright 1960), which is difficult to follow. Also,

linear system,

Goldberger (1972, p. 988) noted: “when there are

more estimating equations than unknown parameters,

yi1⫽ ␭12yi2⫹x⬘i1␤1⫹ui1⫹ei1, (3)

path analysis gives no systematic guide to efficient

esti-mation,” a situation known as overidentification. At any and

rate, social scientists eventually embedded path analysis

yi2⫽ ␭21yi1⫹x⬘i2␤2⫹ui2⫹ei2. (4)

into the general framework of simultaneous systems and

gave it a formal statistical structure (Goldberger1972; Here,␭12is the rate of change of level of production

GoldbergerandDuncan1973; Duncan1975; Jore- with respect to a disease index and␭21is the gradient

sko¨ gandSorbo¨ m2001). of disease with respect to production.A priori, one might

Our concern here is with the consequences of the expect␭12to be negative and␭21to be positive, since high

existence of simultaneous (“feedback”) and recursive output may be associated with “stress” in the dairy cow.

relationships between phenotypic values on quantitative It is assumed that␭12and␭21are homogeneous across

genetic parameters, as well as with statistical methods individuals, but this can be relaxed. The vectors␤1and

for appropriate inference. The outline of the article is ␤₂, often called fixed effects in the statistical literature

as follows. genetic consequences of simultaneity in a (Searle 1971), are some location parameters such as

two-trait systemandgenetic consequences of recur- age of the cow, parity, or breed affecting production

siveness illustrate effects of simultaneity or recursive- (net of disease) and disease (net of production),

respec-ness on simple two-trait systems. In particular, formulas tively. Further,x⬘i1andx⬘i2are known incidence row

vec-are presented for heritability, for the offspring-pvec-arent tors; whenever the same factors affect the two traits, the

regression, and for the genetic and environmental cor- covariates are such thatxi1⬘ ⫽xi2⬘ ⫽xi⬘, say. Finally,ui1and

relations between traits. Next,matrix representations ui2 are additive genetic effects (Fisher 1918; Bulmer

shows that if four phenotypes are involved in a multivari- 1980) intervening in the system andei1andei2are model

exclu-It is important to note that ui1 (ui2) above can be _h2 1⫽

␴2 u*₁

␴2 u*₁⫹ ␴2e*₁ construed as an additive genetic effect affecting only

production (disease) if and only if ␭12(␭21) ⫽ 0; it is _⫽ (1/(1⫺ ␭12␭21))2(␴2u1⫹2␭12␴u12⫹ ␭212␴2u2)

(1/(1⫺ ␭12␭21))2[␴2u1⫹ ␴2e1⫹2␭12(␴u12⫹ ␴e12)⫹ ␭122(␴2u2⫹ ␴2e2)]

shown subsequently thatui1 andui2may affect each of

the two traits. On the other hand, it is legitimate to view _⫽ ␴2

u1⫹2␭12␴u12⫹ ␭212␴2u2

[␴2

u1⫹ ␴2e1⫹2␭12(␴u12⫹ ␴e12)⫹ ␭212(␴2u2⫹ ␴2e2)]

.

(8)

ui1, say, as the additive genetic component of the linear

combinationyi1⫺ ␭12yi2. This is suggested by a rewriting _{Observe that the apportionment of variance into genetic}

of (3) and (4) as _{and environmental components depends nontrivially}

on the structural parameter ␭12, but not on ␭21 (the

yi1 ⫺ ␭12yi2⫽xi1⬘␤1 ⫹ui1 ⫹ei1 (5)

opposite being true for trait 2). If␴u12⫽ ␴e12⫽0, the

yi2 ⫺ ␭21yi1⫽x⬘i2␤2 ⫹ui2 ⫹ei2. (6) linear combinations or “composite” traitsyi1⫺ ␭12yi2and

yi2 ⫺ ␭21yi1 are statistically independent (by virtue of

More generally,ui1andui2are genetic effects “control- _{normality); however,h}

1

2 _{would still depend on␭}

12, as

ling” system (5)–(6).

When a specification such as system (5)–(6) holds,

h2 1⫽

␴2

u1⫹ ␭212␴2u2

␴2

u1⫹ ␴2e1⫹ ␭212(␴2u2⫹ ␴2e2)

. generalized least-squares estimates (or

maximum-likeli-hood estimates under standard normality assumptions)

The corresponding expression for the heritability of

of the structural parameters␭12(or␭21) are biased and

trait 2 is inconsistent if obtained from a subset of equations

(Johnston1972). For instance, if a univariate analysis

h2

2⫽ ␴

2

u2⫹ ␭221␴2u1

␴2

u2⫹ ␴2e2⫹ ␭221(␴2u1 ⫹ ␴e12)

.

of trait 1 is conducted includingyi2as a covariate (but

ignoring the submodel for trait 2),␭12is estimated with

an upward bias (provided␭12⬍1). The effects on model Regression of offspring on parent:Letyi⬘1be a record

parameters are more cryptic as the dimensionality of for trait 1 measured on the offspring of an individual

the system increases, for example, when a system of five with phenotypeyi1. The offspring-parent covariance is

mutually interacting response variables is analyzed with

Cov(yi⬘1,yi1)⫽Cov

冦

(x⬘i⬘1␤1⫹ ␭12x⬘i⬘2␤2)⫹(ui⬘1⫹ ␭12ui⬘2)⫹(ei⬘1⫹ ␭12ei⬘2)

1⫺ ␭12␭21

, a three-trait model.

The implications of system (5)–(6) on the interpreta- _{(x⬘i1␤1⫹ ␭12x⬘i2␤2)⫹(ui1⫹ ␭12ui2)⫹(ei1⫹ ␭12ei2)}

1⫺ ␭12␭21

冧

. tion of some parameters of interest in quantitative

ge-netics analysis are considered next.

Assuming zero covariances between environmental

ef-Heritability:Use (4) in (3) and solve foryi1, to obtain

fects affecting records taken on different individuals, the “reduced” model (a term from econometrics),

under additive inheritance one has that

yi1⫽ ␮1*⫹ ui1* ⫹ei1* , (7)

Cov(yi⬘1,yi1)⫽ 1_⁄

2(␴2u1⫹ ␭212␴u22)⫹ ␭12␴u12

(1⫺ ␭12␭21)2

, where

which is half of the numerator of the expression leading

␮*_{1 ⫽} x⬘i1␤1 ⫹ ␭12x⬘i2␤2

1⫺ ␭12␭21

,

to (8). In the absence of feedback (␭12⫽ 0), the

off-spring-parent covariance is always positive and equal to

␴2

u1/2. Under simultaneity, however, this covariance

u*_{i1 ⫽}ui1⫹ ␭12ui2

1⫺ ␭12␭21

,

could be negative, provided that

and

␴u12⬎

(␴2

u1⫹ ␭212␴2u2)

2␭12

.

e*_{i1 ⫽}ei1⫹ ␭12ei2

1⫺ ␭12␭21

,

The implication is that a variance component analysis based on the reduced model would, necessarily

(be-with the random effects u*i1 ⵑ N(0,␴2u*

1) and e*i1 ⵑ _{cause of the parameterization), return a positive fitted}

N(0,␴2 e*

1) being independently distributed. Note that _{value of the offspring-parent covariance. On the other}

both␤1and␤2intervene in␮1*, and thatui1* is a linear _{hand, this may not be so under a simultaneous equations}

combination of the system genetic effects ui1 and ui2. _{model. If the observed covariance is negative, this}

An implication of this is that estimates of location

pa-should be construed as evidence against a specification rameters and of predictions of random effects from

failing to accommodate simultaneity, although there

standard univariate or multivariate analyses can be inter- _{may be other reasons (}

e.g., maternal effects) for a

nega-preted differently if simultaneity holds. Suppose data _{tive offspring-parent covariance.}

are missing at random (i.e., that selection is ignorable). _{The regression of offspring on parent is}

In this case the fraction of the variance of trait 1 that can be attributed to additive genetic effects, or coefficient of

bOP⫽

1_⁄2(␴2

u1⫹ ␭212␴2u2)⫹ ␭12␴u12

␴2

u1⫹ ␴2e1⫹2␭12(␴u12⫹ ␴e12)⫹ ␭212(␴2u2⫹ ␴2e2)

, ₍₉₎

yi1. An example is the maternal-effects model proposed

yielding the usual␴2

u1/[2(␴2u1⫹ ␴2e1)] in the absence of

byFalconer(1965) and examined by Koerhuis and

feedback (␭12⫽ ␭21⫽ 0).

Thompson(1997). This model postulates that the

phe-Regression of one variable on another: If x⬘i1⫽x⬘i2⫽

notype of an offspring is affected by the phenotype of

x⬘i, the reduced models as in (7) become

its dam. In pigs, for instance, it is known that females born in larger litters tend to produce smaller litters

yi1⫽x⬘i

冢

␤1⫹ ␭12␤2

1⫺ ␭12␭21

冣

⫹

冢

ui1⫹ ␭12ui2

1⫺ ␭12␭21

冣

⫹

冢

ei1⫹ ␭12ei2

1⫺ ␭12␭21

冣

_{(leading to negative values of the}␭_{coefficients);}

con-versely, females in smaller litters are expected to

pro-⫽x⬘i␤*1 ⫹ u*i1⫹ e*i1 (10)

duce larger litters, etc. A recursive specification can be

and _{obtained from the “full model” by setting␭}

12⫽0 in (3)

or (5), so that the system is now

yi2⫽ x⬘i

冢

␤2⫹ ␭21␤1

1⫺ ␭12␭21

冣

⫹

冢

ui2⫹ ␭21ui1

1⫺ ␭12␭21

冣

⫹

冢

ei2⫹ ␭21ei1

1⫺ ␭12␭21

冣

yi1⫽xi1⬘␤1⫹ui1⫹ei1, (13)

⫽ x⬘i␤*2 ⫹ u*i2⫹e*i2. (11) and

Under normality, the regression function of trait 2 _{yi2⫽ ␭21yi1⫹ x⬘i2␤2⫹ ui2⫹ei2. (14)}

on trait 1 is

Assuming that the incidence vectors are such that E(yi2|yi1)⫽E(yi2)⫹

Cov(yi2,yi1)

Var(yi1)

[yi2⫺E(yi1)]⫽x⬘i␤*2 x⬘_i1⫽x⬘_i2 ⫽x⬘_i, use of (13) in (14) gives a reduced model

foryi2,

⫹(1⫹ ␭12␭21)(␴u12⫹ ␴e12)⫹ ␭21(␴u21⫹ ␴2e1)⫹ ␭12(␴2u2⫹ ␴2e2) ␴2

u1⫹ ␴2e1⫹2␭12(␴u12⫹ ␴e12)⫹ ␭212(␴2u2⫹ ␴2e2)

(yi1⫺x⬘i␤*1).

yi2⫽ x⬘i(␤2⫹ ␭21␤1)⫹ (ui2⫹ ␭21ui1)⫹ (ei2⫹ ␥21ei1)

In the absence of simultaneity, this reduces to the usual

⫽ x⬘i␤*2 ⫹u*i2 ⫹e*i2,

formula,

where

E(yi2|yi1)⫽ x⬘i␤2⫹ ␴

u12⫹ ␴e12

␴2 u1⫹ ␴2e1

(yi1⫺ x⬘i␤1).

u*_i2ⵑ N(0,␴2

u2⫹ 2␭21␴u12⫹ ␭221␴2u1)

and

Genetic and environmental correlations:The reduced models (10) and (11) lead to

e*_i2ⵑ N(0,␴2

e2 ⫹2␭21␴e12⫹ ␭221␴2e1),

Cov(yi1,yi2)⫽Cov(u*i1,u*i2)⫹ Cov(e*i1,e*i2),

so that where

Var(yi2)⫽ ␴2u2⫹ ␴2e2⫹2␭21(␴u12⫹ ␴e12)⫹ ␭221(␴2u1⫹ ␴2e1).

Cov(u*_i1,u*_i2)⫽(1 ⫹ ␭12␭21)␴u12⫹ ␭21␴

2

u1 ⫹ ␭12␴2u2

(1 ⫺ ␭12␭21)2

. _{Heritability: The heritability of trait 1 is the usual}

h2⫽ ␴2

u1/(␴u12 ⫹ ␴2e1). Here,␴2u1and␴e12 are the additive

genetic and residual variances of trait 1, respectively, Similarly

contrary to the simultaneity situation, where these dis-persion parameters pertain to the variation of genetic Cov(e*_i1,e*_i2)⫽(1⫹ ␭12␭21)␴e12⫹ ␭21␴

2

e1 ⫹ ␭12␴2e2

(1 ⫺ ␭12␭21)2

.

and residual effects affecting the distribution ofyi1 ⫺

␭12yi2. The coefficient of heritability of trait 2 has the

The genetic and environmental covariances depend on

form of (8), but with␭21instead of␭12:

the␭coefficients and on the appropriate variances and

covariances of each of the two composite traits.

h2 2⫽

␴2

u2 ⫹2␭21␴u12⫹ ␭221␴2u1

␴2

u2⫹ ␴2e2⫹ 2␭21(␴u12⫹ ␴e12)⫹ ␭221(␴2u1⫹ ␴2e1)

. The genetic correlation is

(15) Corr(u*i1,u*i2)⫽

(1⫹ ␭12␭21)␴u12⫹ ␭21␴2u1⫹ ␭12␴2u2

√(␴2

u1⫹2␭12␴u12⫹ ␭212␴2u2)(␴2u2⫹2␭21␴u12⫹ ␭221␴2u1) ,

Regression of offspring on parent:The regression of (12)

offspring on parent depends on the trait or pairs of traits

and the expression for the residual correlation is similar. _{involved. Using the same notation as in the section for}

If␴u12⫽0 (i.e., when the composite traits are genetically simultaneity, the offspring-parent covariance for trait 1 is

uncorrelated), (12) becomes

Cov(yi⬘1,yi1)⫽Cov(x⬘i⬘1␤1⫹ui⬘1⫹ei⬘1,x⬘i1␤1⫹ui1⫹ei1)⫽

1 2␴

2

u1.

Corr(u*_i1,u*_i2)⫽ ␭21␴

2

u1⫹ ␭12␴2u2

√(

␴2

u1⫹ ␭212␴2u2)(␴2u2⫹ ␭221␴2u1)

.

Hence the regression of offspring on parent for trait 1

is simplyh12/2, the standard result from assuming

addi-tive inheritance.

GENETIC CONSEQUENCES OF RECURSIVENESS _{Let y}

i⬘2 be a record measured for trait 2 on the

off-IN A TWO-TRAIT SYSTEM

spring of an individual with phenotypeyi2. The

offspring-parent regression, assuming that betwegeneration en-A recursive specification postulates, for instance, that

to (9). Consider now the covariance betweenyi⬘2, a re- Likewise, the environmental covariance and

correla-tion are cord for trait 2 measured on the offspring of an

individ-ual with phenotypeyi1. The offspring-parent covariance

Cov(ei1,e*i2)⫽Cov(ei1,ei2⫹ ␭21ei1)⫽ ␴e12⫹ ␭21␴2e1

between such records is now

and Cov(ui⬘2⫹ ␭21ui⬘1,ui1)⫽ ␴

u12⫹ ␭21␴u12

2

Corr(ei1,e*i2)⫽

␴e12⫹ ␭21␴2e1

√

␴2

e1(␴2e2⫹ 2␭21␴e12⫹ ␭221␴2e1)

, ₍₂₁₎

and the regression coefficient is

respectively.

␭O2P1⫽ 1 2

冢

␭21h

2 1⫹

␴u12

␴2

u1 ⫹ ␴2e1

冣

. (16)

MATRIX REPRESENTATIONS

Conversely, withyi⬘1being now a record for trait 1

mea-sured on the offspring of an individual with phenotypic Many possible models:A multivariate system may

in-valueyi2 the offspring-parent regression is volve many response variables, as well as different levels

of simultaneity and recursiveness. When the models comprise more than two traits, the issues and principles

␭O1P2⫽

(␴u12⫹ ␭21␴2u1)

2[␴2

u2⫹ ␴2e2⫹2␭21(␴u12⫹ ␴e12)⫹ ␭221(␴2u1⫹ ␴2e1)]

.

are as discussed above but the algebra is awkward. For

(17) _{example, consider the simultaneous-equation model for}

three traits given in Figure 3. Here, the three response

Regression of one variable on another: Recall that

variablesY1,Y2, andY3have mutually reciprocal effects,

yi1⫽x⬘i1␤1 ⫹ui1⫹ ei1and that

so that there are six␭coefficients or structural

parame-ters in the model.

yi2⫽ x⬘i(␤2⫹ ␭21␤1)⫹(ui2⫹ ␭21ui1)⫹(ei2⫹ ␭21ei1)

Several different models can be derived as special

⫽ x⬘i␤*2 ⫹u*i2⫹ e*i2.

cases of the specification given in Figure 3. There are 64 models that can be viewed as “nested” within the dia-Then

gram depicted. In general, forKresponse variables there

E(yi1|yi2)⫽x⬘i␤1⫹ ␴

u12⫹ ␴e12⫹ ␭21(␴2u1⫹ ␴2e1) ␴2

u2⫹ ␴2e2⫹2␭21(␴u12⫹ ␴e12)⫹ ␭221(␴2u1⫹ ␴2e1) areK(K⫺1) structural coefficients (␭’s) in a fully

simulta-⫻[yi2⫺xi⬘(␤2⫹ ␭21␤1)]. (18) neous model. Since, in a given model, each coefficient

can take the value␭ijor be constrained to be 0 (when

Conversely, the regression function ofyi2 onyi1is _{there is no “effect” of variablejon variableiin the latter}

case), there can be as many as 2K(K⫺1)_{possible models}

E(y21|yi1)⫽x⬘i(␤2⫹ ␭21␤1)⫹

冢

␭21⫹ ␴

u12⫹ ␴e12

␴2

u1⫹ ␴2e1

冣

(yi1⫺x⬘i␤1). _{for explaining relationships between the phenotypic}

(19) _{variables; in practice, however, many of the models can}

be discarded on mechanistic grounds. For example, if The two preceding expressions reduce to the usual

for-all ␭’s are set equal to 0 in Figure 3, this yields the

mulas under bivariate normality by letting␭21⫽ 0.

standard trivariate model used for quantitative genetic

Genetic and environmental correlations:The genetic

analysis of three traits. Some other models that can arise covariance between the two traits is

are illustrated in Figures 4–6. A “cyclically recursive” model is depicted in Figure 4. Here, the causal

relation-Cov(ui1,u*i2)⫽ Cov(ui1,ui2 ⫹ ␭21ui1)⫽ ␴u12⫹ ␭21␴2u1,

ship modeled isY1→Y2→Y3→Y1→. . . . For instance,

so that the genetic correlation is _{consider a hypothetical situation whereY1,Y2, andY3are}

phenotypic values in sibships of size 3, with the subscript indicating birth order. It may be that the chain of

influ-Corr(ui1,u*i2)⫽ ␴u12⫹ ␭21␴

2 u1

√

␴2

u1(␴2u2⫹ 2␭21␴u12⫹ ␭221␴2u1)

. ₍₂₀₎

ences is such that the older sib (with phenotypeY1) affects

the second sib (Y2) and so on, with the loop closing via

If␴u12⫽ 0

an influence of the youngest on the oldest sib.

In Figure 5, it is hypothesized thatY1has an effect of

Corr(ui1,u*i2)⫽

␭21

√

(␭2

21⫹(␴2u2/␴2u1))

,

the recursive type on bothY2 andY3, but that there is

simultaneity betweenY2andY3; this is referred to as a

recursive-simultaneous model. Here, Y1 might be the

in which case the sign of the genetic correlation depends

on the sign of␭21. Further, if the traits are scaled such concentration of a hormone regulating the production

of two metabolites that are involved in a feedback

rela-that␴u12 ⫽ ␴u22 ⫽ 1, the correlation is␭21/√(1⫹ ␭221) . It

tionship. In Figure 6, there is simultaneity betweenY2

is interesting to observe that when genotypes are

ex-pressed in units of standard deviation, the genetic corre- and Y3, with these two variables affecting Y1; this is a

simultaneous-recursive model: two biochemical

prod-lation is driven entirely by ␭21, which is a gradient

Figure3.—Simultaneous-equations model for three variables: Y1, Y2, and Y3are the phenotypic values;U1,U2, andU3are addi-tive genetic effects acting on the system;

E1,E2, andE3are residual effects. A single-headed arrow (e.g.,A→B) indicates that variableAaffects variableB. Double-headed arrows denote correlations between pairs of variables.␭ijindicates the rate of change of variableiwith respect to variablej.

interact reciprocally toward the establishment of some The preceding discussion illustrates that a general

representation is needed for describing the full range of equilibrium. Clearly, there is a constellation of

model-ing alternatives. possibilities. For example, the two-variate

simultaneous-Figure 4.—Fully recursive model

for three variables:Y1,Y2, andY3are the phenotypic values;U1,U2, andU3 are additive genetic effects acting on the system;E1,E2, andE3are residual effects. A single-headed arrow (e.g.,

Figure 5.—Recursive simultane-ous model for three variables:Y1,Y2, andY3are the phenotypic values;U1,

U2, andU3are additive genetic effects acting on the system; E1, E2, and E3 are residual effects. A single-headed arrow (e.g.,A→B) indicates that vari-able A affects variable B. Double-headed arrows denote correlations be-tween pairs of variables.␭ijindicates the rate of change of variableiwith respect to variablej.

equations system of Equations 3 and 4 can be put in of generality, it is assumed thatXihas full-column rank.

Further, the location vector␤is such that

matrix form as

冤

1 ⫺␭21

⫺␭21 1

冥冤

yi1

yi2

冥

⫽

冤

x⬘i1 0

0 x⬘i2

冥冤

␤1 ␤2

冥

⫹

冤

ui1

ui2

冥

⫹

冤

ei1

ei1

冥

. (22)

This representation embeds four models [K ⫽ 2, so ␤⫽

冤

␤1 ␤2

. ␤K⫺1

␤K

冥

,

2K(K⫺1)_{⫽4], including the simultaneous one. The other}

three models are the standard bivariate specification

(␭12⫽ ␭21⫽0) and two recursive models (␭12⫽0 when

where␤j(j⫽1, 2, . . . ,K) ispj⫻1. The vectoruicontains

y1 “affects” y2, but without a reciprocal effect;␭21⫽ 0

additive genetic effects of individual i for the K traits

wheny2 affectsy1).

and, similarly,eiis a vector of residual effects, distributed

Statistical structure:Let there beKtraits observed on

independently ofui.

individual or “cluster” (e.g., a family)i(i⫽1, 2, . . . ,N),

If⌳has full rank, the reduced form of the model is

and write the system as

yi⫽⌳⫺1Xi␤⫹⌳⫺1ui⫹ ⌳⫺1ei ⌳yi⫽Xi␤⫹ ui⫹ ei, (23)

⫽␮*i ⫹u*i ⫹e*i, (24)

whereyiis aK⫻1 vector of phenotypic measurements

on theKtraits of individuali;⌳is aK⫻Kmatrix con- _{where ␮*}

i ⫽ ⌳⫺1Xi␤,u*i ⫽ ⌳⫺1ui, ande*i ⫽ ⌳⫺1ei. For

taining at mostK(K⫺1) unknown␭coefficients (all diag- _{example, in a two-trait simultaneous model, the}

ele-onal elements are equal to 1); and Xi is a K⫻兺Kj⫽1pj _{ments of} ␮_*

i , u*i, and e*i take the form given in (7).

known incidence matrix with the form _{Assume now that}

ui|G0ⵑN(0,G0); i⫽ 1, 2, . . . ,N, (25)

and

Xi⫽

冤

x⬘i1 0 . 0 0

0 x⬘i2 . 0 0

. . . . .

0 0 . x⬘i(K⫺1) 0

0 0 . 0 x⬘iK

冥

,

ei|R0ⵑN(0,R0) (26)

are mutually independent. Hence, the vectors of genetic and residual effects involved in the correlations shown

Figure 6.—Simultaneous-recursive model for three variables:Y1,Y2, and

Y3are the phenotypic values;U1,U2, andU3are additive genetic effects act-ing on the system;E1,E2, andE3are residual effects. A single-headed arrow (e.g.,A→B) indicates that variable

Aaffects variable B. Double-headed arrows denote correlations between pairs of variables.␭ijindicates the rate of change of variablei with respect to variablej.

in Figures 3–6 follow multivariate normal distributions E(u*i|yi)⫽ E(u*i )⫹Cov(ui*,y⬘i)Var⫺1(yi)(yi⫺ ⌳⫺1Xi␤)

with covariance matricesG0 andR0, respectively, each _⫽

H(yi⫺ ⌳⫺1Xi␤). (31)

having order 3⫻3. This implies that

In multiple-trait selection, animal and plant breeders

u*i|⌳,G0ⵑ N(0,⌳⫺1G0⌳⬘⫺1), (27)

are often interested in improving a linear combination

and _{of genetic values; e.g., T}

i ⫽v⬘u*i , where v is a known

K⫻1 vector of relative economic values (Smith1936;

e*i|⌳,R0ⵑ N(0,⌳⫺1R0⌳⬘⫺1) (28)

Hazel1943), andu*i contains the “true” genetic values

are also independently distributed. Further, the marginal _{affecting the traits. Recall that the genetic values are}

distribution of the phenotypic values for individualiis _u

i only in the absence of feedbacks or recursiveness.

Suppose the N candidates are independently

distrib-yi|␤,R0,G0 ⵑN(⌳⫺1Xi␤,⌳⫺1(R0⫹G0)⌳⬘⫺1). (29)

uted, so that the density of the joint distribution of all

Genetic parameters and functions thereof:The “mul- _{genetic and phenotypic values is given by} tivariate heritability and coheritability” matrix can be

defined as

p(u*,y|parameters)⫽

兿

N

i⫽1

p(u*i ,yi|parameters).

H⫽⌳⫺1_G

0⌳⬘⫺1[⌳⫺1(R0⫹G0)⌳⬘⫺1]⫺1

This is whatHenderson(1963, 1973) termed an “equal

⫽⌳⫺1_G

0(R0⫹G0)⫺1⌳. (30)

information” situation. The best predictor of the “merit

In the absence of simultaneity or recursiveness, H ⫽ _function”

Ti is

G0(R0 ⫹G0)⫺1, since⌳would be an identity matrix of

orderKin this case. Note that the trace of (30), _Tˆ_{i ⫽}E(v⬘u*i |yi)⫽v⬘H(yi ⫺⌳⫺1Xi␤)

tr(H)⫽ tr[G0(R0⫹G0)⫺1], ⫽b⬘(y_i ⫺⌳⫺1X_i␤), (32)

is free of the␭ coefficients. Now, using the

measure-where

ments taken on individuali, the best predictor of u*i,

in the sense of minimizing the mean square error of _{b ⫽H⬘v⫽⌳⬘(R0⫹G0)}⫺1_G

0⌳⬘⫺1v (33)

prediction among all possible functions of the data

is the classical “selection index” solution to the Smith-(Henderson1973), is given by the conditional

Var(yi)b ⫽Cov(yi,v⬘u*i ).

Suppose that selection of a truncation type is based

冤

⌳y1 ⌳y2

. ⌳yN

冥

⫽

冤

X1 X2

.

XN

冥

␤⫹ Z

冤

u1 u2

.

uN

冥

⫹

冤

e1 e2

.

eN

冥

onTˆ_iin (32), such that a proportion␣of the candidates

is kept as parents of the following generation. From the

forms of (32) and (33), it follows that the mean of the _⫽

X␤⫹ Zu⫹ e, (35)

distribution ofTˆiin the unselected individuals is 0, since

E(yi)⫽⌳⫺1Xi␤. Under normality assumptions, standard whereucomprises additive genetic effects for all

individ-uals and all traits (umay include additive genetic effects

theory (e.g., Bulmer 1980; Falconer and Mackay

1996) gives a mean of the selected individuals, of individuals without records), and Zis an incidence

matrix of appropriate order. If all individuals have

re-ES(Tˆi)⫽ i

√

Var(Tˆi), (34) _{cords for all traits,Zis an identity matrix of orderNK}⫻

NK; otherwise, columns of 0’s for effects of individuals

wherei⫽z/␣is called “selection intensity” andzis the

without phenotypic measurements would be included ordinate of the standard normal distribution at a point

in Z. In view of the normality assumptions (25) and

at the right of which there is a probability mass equal

(26), one can write

to␣; S stands for selection. Under additive genetic

ac-tion, the expected genetic value of the progeny of

se-u|G0ⵑN(0,A丢 G0)

lected parents is equal to the expected value of the

and selected parents. Hence, the expected response to

selec-tion is given directly by (34). For example, consider

e|R0 ⵑN(0,I丢R0),

single-trait selection and the merit functionTi ⫽u*i1(the

whereAis a matrix of additive genetic relationships (or

additive genetic value of individuali), and suppose that

of twice the coefficients of coancestry) between

individu-the only source of information is yi1, the phenotypic

als in a genealogy, and丢indicates Kronecker product.

value for trait 1. In this case, and from the form of (32),

Note thatI丢R0reflects the assumption that all

individu-it follows that

als with records possess phenotypic values for each of

theKtraits. This is not a requirement, but it simplifies

Tˆi⫽

Var(u*i1)

Var(u*i1)⫹Var(e*i1)

[yi1⫺E(yi1)].

somewhat the treatment that follows.

Given u, the vectors ⌳yi are mutually independent

For a two-trait simultaneous system, it was seen earlier

(since alleivectors are independent of each other), so

that

the joint density of all⌳yiis

Var(u*i1) Var(u*i1)⫹Var(e*i1)

⫽ ␴2u1⫹2␭12␴u12⫹ ␭212␴2u2

␴2

u1⫹ ␴2e1⫹2␭12(␴u12⫹ ␴e12)⫹ ␭212(␴2u2⫹ ␴2e2) p(⌳y1,⌳y2, . . . ,⌳yN|⌳,␤,u,R0) ⬀ 1

|R0|N/2

exp

冤

⫺1 2

兺

N

i⫽1

(⌳yi⫺Xi␤⫺Ziu)⬘R⫺01(⌳yi⫺Xi␤⫺Ziu)

冥

,

and

(36)

E(yi1)⫽ ␮*i ⫽ ␮1⫹ ␭12␮2

1⫺ ␭12␭21

.

where Zi is an incidence matrix that “picks up” theK

breeding values of individuali(ui) and relates these to

Hence

its phenotypic recordsyi. Making a change of variables

from⌳yitoyi(i⫽1, 2, . . . ,N), the determinant of the

ES(Tˆi)⫽

i(␴2

u1⫹2␥12␴u12⫹ ␭212␴2u2)

√␴2

u1⫹ ␴2e1⫹2␥12(␴u12⫹ ␴e12)⫹ ␭212(␴2u2⫹ ␴2e2)

.

Jacobian of the transformation is |⌳|. Hence, the density

ofy⫽[y⬘1,y⬘2, . . . ,y⬘N]⬘is

When ␭12 ⫽ 0, this reduces to the usual ES(Tˆi)⫽i␴u₁

p(y|⌳,␤,u,R0)⬀

|⌳|N |R0|N/2

h1⫽ih21␴y₁, provided selection is based onTˆi⫽ h21[yi1⫺

E(yi1)].

⫻exp

冦

⫺1 2兺

N i⫽1

[yi⫺⌳⫺1(Xi␤⫹Ziu)]⬘⌳⬘R0⫺1⌳[yi⫺⌳⫺1(Xi␤⫹Ziu)]

冧

The covariance matrix between additive genetic

val-ues of related individuals iandi⬘is

⬀ 1 |⌳⫺1_R

0⌳ⴕ⫺1|N/2 Cov(u*i ,u*i⬘⬘)⫽ Cov(⌳⫺1ui,u⬘i⌳ⴕ⫺1)⫽aii⬘⌳⫺1G0⌳⬘⫺1,

⫻exp

冦

⫺1 2兺

N i⫽1

[yi⫺⌳⫺1(Xi␤⫹Ziu)]⬘⌳⬘R0⫺1⌳[yi⫺⌳⫺1(Xi␤⫹Ziu)]

冧

.

whereaii⬘is twice the coefficient of coancestry between

(37)

iandi⬘.

This is the density of the product of theNnormal

distri-butions

LIKELIHOOD FUNCTION

yi|⌳,␤,u,R0ⵑ N(⌳⫺1(Xi␤⫹Ziu), ⌳⫺1R0⌳⬘⫺1),

Consider system (23) in conjunction with the

normal-ity assumptions (25) and (26), and regard the vector highlighting that the data generation process can be

represented in terms of the reduced model (24), with

⌳yias “data.” The model for the entire data vector can

inci-dence matrixZi, with the latter being aK⫻ Kidentity Consider the system ofKresponse variables (23), and

reorganize it as matrix in (24). Hence, the entire data vector can be

modeled as

⌳yi⫹Xi␤ →

⫽ εi, (41)

where␤→ ⫽ ⫺␤andεi⫽ui⫹eiis a residual. It is

conve-冤

y1 y2

. yN

冥

⫽

冤

⌳⫺1_X 1

⌳⫺1_X 2

. ⌳⫺1_X

N

冥

␤⫹

冤

⌳⫺1_Z

1 0 . 0

0 ⌳⫺1_Z

2 . 0

. . . .

0 0 . ⌳⫺1_Z

n

冥冤

u1 u2

. uN

冥

⫹

冤

e*1 e*2

.

e*N

冥

nient to lump the sum of the two random effects into

a single residual for the treatment that follows. Rewrite

⫽X⌳␤⫹Z⌳u⫹e*, (38)

whereX⌳is anNK ⫻兺K

j⫽1pjmatrix (again, assuming that

each of theN individuals has measurements for theK

traits), andZ⌳ has orderNK ⫻ (N ⫹ P)K, whereP is _Xi␤→ ⫽

冤

x⬘i1 0 . 0 0

0 x⬘i2 . 0 0

. . . . .

0 0 . x⬘i(K⫺1) 0 0 0 . 0 x⬘iK

冥冤

␤→1 ␤→2

. ␤→K⫺1

␤→K

冥

the number of individuals in the genealogy lacking

phe-notypic records (the corresponding columns ofZ⌳being

null). Observe that (38) is in the form of a standard multiple-trait mixed-effects linear model, save for the fact that the incidence matrices depend on the

un-known structural coefficients contained in⌳. Hence

⫽

冤

␤→⬘1 0 . 0 0 0 ␤→⬘2 . 0 0

. . . . .

0 0 . ␤→⬘K⫺1 0

0 0 . 0 ␤→⬘K

冥冤

xi1 xi2

.

xi(K⫺1) xiK

冥

p(y|⌳,␤,u,R0)

⬀ 1

|R⌳|1/2exp

冤

⫺ 1

2(y⫺X⌳␤⫺Z⌳u)⬘R ⫺1

⌳(y⫺X⌳␤⫺Z⌳u)

冥

,

(39) _⫽Bxi,

where _{where xi now is a column vector of order 兺}K

j⫽1pj⫻1,

Var(e*)⫽R⌳ ⫽IN丢 ⌳⫺1R0⌳⬘⫺1 andB

→

isK⫻兺K

j⫽1pj. In practice, it suffices to keep the

distinct explanatory variables inxi;e.g., if herd effects

affect all traits in the system, only a single set of

inci-is a block-diagonal matrix consinci-isting ofNblocks of order

dence variables needs to be considered. With this

nota-K⫻ K, and all such blocks are equal to ⌳⫺1_R

0⌳⬘⫺1. It

tion, (41) can be put as

follows thaty|⌳,␤,u,R0ⵑN(X⌳␤⫹Z⌳u,R⌳). Hence,

if simultaneity or recursiveness holds, the estimator of the residual variance-covariance matrix from a reduced

model analysis is actually estimating⌳⫺1_R

0⌳⬘⫺1; this has

a bearing on the interpretation of the parameter esti-

冤

␭⬘1yi⫹ b⬘1xi ␭⬘2yi⫹ b⬘2xi

. ␭⬘K⫺1yi⫹b⬘K⫺1xi

␭⬘Kyi⫹ b⬘Kxi

冥

⫽

冤

εi1

εi2

.

εi(K⫺1)

εiK

冥

, (42)

mates.

Since it is assumed that u|G0 ⵑ N(0, A 丢 G0), the

likelihood function is given by

where␭⬘j andb⬘j (j⫽1, 2, . . . ,K) are the jth rows of

l(⌳,␤,R0,G0)⬀

冮

N(X⌳␤⫹Z⌳u,R⌳)N(0,A丢G0)du _{⌳andB, respectively. The specification}

⬀N(X⌳␤,R⌳⫹Z⌳(A丢 G0)Z⬘⌳). (40) ␭⬘

jyi⫹ b⬘jxi⫽ εij; j⫽1, 2, . . . ,K, i⫽1, 2, . . . , N,

This likelihood has the same form as that for a standard _{constitutes thejth equation of the system. Compactly,}

multivariate mixed-effects model, except that, here, ad- _{the system is}

ditional parameters (the nonnull elements of⌳) appear

⌳yi⫹Bxi ⫽εi. (43)

in both the location and dispersion structures of the reduced model (38). A pertinent issue, then, is whether

The reduced model is expressible as

or not all parameters in the model, that is, ⌳, ␤, R0,

andG0, can be identified (i.e., estimated uniquely) from yi⫽ ⫺⌳⫺1Bxi⫹ ⌳⫺1εi

the likelihood. This is discussed in the following section.

⫽ ⌸xi⫹ ε→i, (44)

where ⌸ ⫽ ⫺⌳⫺1_{B is a K} ⫻ 兺K

j⫽1pj matrix of reduced

IDENTIFICATION OF PARAMETERS

model parameters, and→εi⫽ ⌳⫺1εi. The system in (43)

This is dealt with only briefly here as extensive treat- contains, at least potentially, the following number of

ments can be found in econometrics treatises such as parameters:K2_{(all elements of}⌳_{, including the 1’s in}

Johnston (1972) andJudge et al.(1988); a readable the diagonal),K兺K

j⫽1pj(all elements ofB, including the

null ones), plusK(K⫹ 1) (the distinct elements ofR0

andG0). It is assumed that these two variance-covariance normalization restrictions set all diagonal elements of

⌳as equal to 1, and then␭jj⫽1 (␭jjis thejth element

matrices can be separated in the estimation procedure,

which depends on the genetic structure of the data set. of␭j), this implies that (47) must provideK⫺1 linearly

independent relationships, so that one can arrive at the

Lettingp⫽兺K

j⫽1pj/K, the total number of parameters in

the system isS⫽K2₍₂⫹_p)⫹_{K. In the reduced model, Krestrictions needed. Now, combine (46) and (47), to}

arrive at the system on the other hand, the number of potential parameters

isK2_{p (the order of}⌸_{), plusK(K}⫹ _{1) (the elements}

of G*0 ⫽ ⌳⫺1G0⌳⬘⫺1 and those of R*0 ⫽ ⌳⫺1R0⌳⬘⫺1),

冤

⌸

⬘

兺K

j⫽1pj⫻K I兺Kj⫽1pj⫻兺Kj⫽1pj

RjJ␭⫻K R jb

J⫻兺K_j⫽1p_j

冥

冤

␭j bj

冥

⫽0, (48)

yieldingR⫽ K2₍₁ ⫹_p)⫹_{Kas the total number of}

pa-rameters. To obtain unique estimates of the parameters

in⌳,B,G0, andR0,S⫺R⫽K2restrictions are needed whereRj⫽[Rj␭ Rjb] is given in partitioned form, and

for uniqueness. These can be of four types (Judgeet al. the coefficient matrix must have rank K⫺1⫹兺K

j⫽1pj

1988), as follows. _{to obtain unique estimates of ␭j and bj. Johnston}

(1972) and Judge et al. (1988) state that the rank of

1. “Normalization” restrictions: set the diagonal

ele-the coefficient matrix is K⫺1⫹兺K

j⫽1pj if and only if

ments of⌳to 1, so that the parameters in equation

the rank of

j are expressed relative to this constant of

propor-tionality. This yieldsKrestrictions, so an additional

[Rj␭ _Rjb_]

冤

␭1 ␭2 . ␭K

b1 b2 . bK

冥

⫽ Rj

冤

⌳⬘

B⬘

冥

(49)

K2⫺_K⫽ _K(K⫺_{1) restrictions are still needed.}

2. Exclusion restrictions: some of the␭coefficients may

isK⫺1. Note that the preceding matrix has orderJ⫻K

be 0, as in a recursive model, or the elements of␤

and that columnjis null by virtue of (47). Hence, for

may not appear in each of the equations.

(49) to possess rankK⫺1, it must be that JⱖK⫺ 1;

3. Restrictions in the form of a linear combination of

i.e., a condition for identification of equationjis that

parameters in the same equation or across equations.

the number of restrictions J must be at least K ⫺ 1

4. Restrictions on the variance-covariance matricesG0

(recall that K is the number of traits in the system).

andR0(typically, such restrictions are not employed

However, this is not sufficient: as stated, the rank of in quantitative genetic analysis).

(49) must beK⫺1.

Formal procedures for evaluation of identification of _{In short, if the rank of (49) is K ⫺ 1, equation j is}

equations are described byJohnston(1972) andJudge _{just identified, meaning that the relationship between}

et al.(1988). Suppose that⌸is given and that one wishes _{the reduced model parameters and the␭’s and ␤’s in}

to estimate uniquely (identify) the parameters in⌳and _{the equation is unique. If the rank is larger thanK⫺1,}

inB. Briefly, note that the parameters of the reduced _{the equation is overidentified, meaning that there are}

model,⌸⫽ ⫺⌳⫺1_{B, satisfy⌳⌸⫹B⫽0or, equivalently,}

many ways in which the structural model parameters

can be expressed as a function of the elements of ⌸.

[⌳ B]

冤

⌸

I

冥

⫽0. (45) In these two cases, the ␭’s and ␤’s may be inferred

efficiently, using methods that employ all information

available in the data,e.g., maximum-likelihood or

Bayes-Consider now rowjof (45) and write it as

ian procedures. Finally, if the rank of (49) is smaller

␭⬘j⌸ ⫹b⬘j ⫽ 0. _thanK⫺_{1, equationjis underidentified, and the}

struc-tural parameters cannot be solved as a function of the Transposing, this yields

reduced model parameters (DrezeandRichard1983).

The preceding developments are illustrated with a

关

⌸⬘ I兺K

j⫽1pj⫻兺Kj⫽1pj

兴

冤

␭j bj

冥

⫽ 0. (46)

two-trait simultaneous model. Suppose thatyi1andyi2are

measurements of systolic and diastolic blood pressure,

This defines a system of equations on K⫹兺K

j⫽1pj

un-respectively, taken on individuali; assume that

physio-knowns in which the rank of the known coefficient

ma-logical knowledge postulates a feedback between the

trix is兺K

j⫽1pj. Hence,Krestrictions are needed to identify

two variables. Let the models be

the unknown parameters␭jandbjof equationjof the

system. The restrictions can be denoted (Judge et al. yi1⫽ ␭12yi2⫹ ␤11⫹ ␤12Agei⫹ ␤13Smokingi⫹ui1⫹ei1

1988) as

and

Rj

冤

␭j bj

冥

⫽

0, (47) yi2⫽ ␭21yi1⫹ ␤21⫹ ␤22Agei⫹ ␤24Drinkingi⫹ ␤25Exeri

⫹ ui2⫹ei2,

where isRjis a J⫻(K⫹ 兺Kj⫽1pj) matrix of rankJ⬍K⫹

兺K

j⫽1pj. For example, an exclusion restriction can be indi- where Age is the age ofiin years; Smoking is a binary

variable (0 represents no smoking during the year prior

cated by filling the appropriate row ofRjwith 0’s, save

for a 1 in the position corresponding to the element to measurement and 1 represents smoking); Drinking

is an estimate of the amount of alcoholiconsumed in

the year previous to the blood pressure test, ignoring a timates of the structural model parameters ⌳, ␤, R0,

andG0is not an easy matter, with a main difficulty being

possible error of measurement, and Exer measures the

extent to which i exercises. The u ande variables are the fact that⌳is unknown. On the other hand, if the

elements of this matrix were given, the setting would additive genetic and residual effects, as before, and the

␭’s and␤’s are the structural model parameters. Here, be as in a multivariate mixed-effects linear model, so

standard procedures, such as the

expectation-maximiza-K⫽2 and the number ofxvariables is 5, since the two

intercepts␤11and␤21are related to the measurements tion (EM) algorithm, could be employed for computing

the likelihood-based estimates. Another complication is via the same incidence variate, which takes the value 1

for alli. The first equation has three “beta coefficients” that, typically, highly nonlinear functions of the

parame-ters must be inferred. For example, see the forms of

(␤11,␤12, and␤13) and the second has four (␤21,␤22,␤24,

and␤25). Before normalization the mean ␮*1 in model (7) and of the coefficient of

heritability in (8). Intuitively, asymptotic approxima-tions to the sampling distribution of the maximum-like-⌳2⫻2⫽

冤

␭11 ⫺␭12

⫺␭21 ␭22

冥

,

lihood estimates may be relatively less accurate at a given sample size when the parametric function of interest is and

nonlinear than when it is linear. Note, however, that

␮*1 and (8) may be inferred from the reduced model,

via the standard multivariate parameterization. In spe-cial circumstances, one can form estimators of the

struc-B2⫻5xi⫽

冤

⫺␤11 ⫺␤12 ⫺␤13 0 0

⫺␤21 ⫺␤22 0 ⫺␤24 ⫺␤25

冥

冤

1

Agei

Smokingi

Drinkingi

Exeri

冥

.

tural parameters from statistics derived from the re-duced model. These are called “indirect” procedures

in econometrics (Johnston1972).

Also, inferring random effects is of great importance

Equation 1 of the system uses the two exclusions␤14⫽

in applied quantitative genetics (e.g., animal, plant, or

␤15⫽0. Hence, (49) is

tree breeding), and their best predictor would take a form such as in (32). In practice, however, calculations require replacing the unknown structural parameters by their maximum-likelihood estimates, that is, computing

Eˆ(u*i |yi)⫽⌳ˆ⫺1Gˆ0(Rˆ0⫹Gˆ0)⫺1⌳ˆ(yi⫺ ⌳ˆ⫺1Xi␤ˆ). (50) R1

冤

⌳⬘ B⬘

冥

⫽

冤

0 0 0 0 0 1 0

0 0 0 0 0 0 1

冥

        

␭11 ⫺␭21

⫺␭12 ␭22

⫺␤11 ⫺␤21

⫺␤12 ⫺␤22

⫺␤13 0

0 ⫺␤24

0 ⫺␤25

        

If interest focuses on the “system” genetic effects, the statistic would be

Eˆ(ui|yi)⫽ Gˆ0(Rˆ0⫹Gˆ0)⫺1⌳ˆ(yi⫺ ⌳ˆ⫺1Xi␤ˆ).

⫽

冤

0₀ ⫺␤_⫺␤24

25

冥

. The finite sample properties of the resulting empirical

predictors are unknown. A common Bayesian criticism

(Box and Tiao 1973; Gianola and Fernando 1986;

The rank of this matrix is 1 (which isK⫺1), so that

SorensenandGianola2002) is that (50) does not take the equation is identified. Equation 2 of the system

the uncertainty (error of estimation) of the estimates

employs the exclusion␤23⫽0 so that (49) is

of the parameters into account.

An alternative is to adopt a Bayesian approach, where inferences about structural parameters, random effects, or functions thereof are made from their marginal

pos-terior distributions (Zellner1971, 1979;BoxandTiao

关

0 0 0 0 1 0 0

兴

        

␭11 ⫺␭21

⫺␭12 ␭22

⫺␤11 ⫺␤21

⫺␤12 ⫺␤22

⫺␤13 0

0 ⫺␤24

0 ⫺␤25

        

⫽[⫺␤13 0].

1973;Gelmanet al.1995;CarlinandLouis2000;

Sor-ensen and Gianola 2002). A review of some of the

issues in simultaneous models from an econometric

per-spective is in Zellner (1979), Dreze and Richard

(1983),Judgeet al.(1985), andKoop(2003). A salient

Since the rank of this matrix is 1, the second equation _{feature of the Bayesian analysis is its ability to produce}

is identified as well. Hence,␭12, ␭21, and the elements _{exact finite sample inference, as well as to override}

po-of ␤1⫽[␤11 ␤12 ␤13]⬘and of ␤2⫽[␤21 ␤22 ␤24 ␤25] _{tential underidentification of parameters. If proper}

pri-can be estimated uniquely. _{ors are adopted for all parameters in a model, all}

poste-rior distributions are proper as well (Bernardo and

Smith1994;O’Hagan1994). However, unless the

pa-BAYESIAN MODEL

rameters are identifiable in the likelihood, the influence of the prior does not dissipate asymptotically. An

exam-General: The form of the likelihood function given