Genet. Sel. Evol. 37 (2005) 129–150 129

c INRA, EDP Sciences, 2005

DOI: 10.1051/gse:2004041

Original article

Statistical power for detecting epistasis QTL

eﬀects under the F-2 design

∗Yongcai Mao, Yang Da

Department of Animal Science, University of Minnesota, Saint Paul, MN 55108, USA

(Received 16 April 2004; accepted 9 September 2004)

Abstract – Epistasis refers to gene interaction eﬀect involving two or more genes. Statistical

methods for mapping quantitative trait loci (QTL) with epistasis eﬀects have become available

recently. However, little is known about the statistical power and sample size requirements for

mapping epistatic QTL using genetic markers. In this study, we developed analytical formulae to

calculate the statistical power and sample requirement for detecting each epistasis eﬀect under

the F-2 design based on crossing inbred lines. Assuming two unlinked interactive QTL and the

same absolute value for all epistasis eﬀects, the heritability of additive× additive (a× a) eﬀect is

twice as large as that of additive× dominance (a× d) or dominance× additive (d× a) eﬀect, and

is four times as large as that of dominance× dominance (d× d) eﬀect. Consequently, among the

four types of epistasis eﬀects involving two loci, ‘a× a’ eﬀect is the easiest to detect whereas

‘d× d’ eﬀect is the most diﬃcult to detect. The statistical power for detecting ‘a× a’ eﬀect is

similar to that for detecting dominance eﬀect of a single QTL. The sample size requirements

for detecting ‘a× d’, ‘d× a’ and ‘d× d’ are highly sensitive to increased distance between

the markers and the interacting QTLs. Therefore, using dense marker coverage is critical to

detecting those eﬀects.

epistasis/ QTL/ statistical power/ sample size/ F-2

1. INTRODUCTION

Epistasis refers to gene interaction eﬀect involving two or more genes.

Evidence from studies in several species, including cattle [18, 21], dogs [1],

rat [23], drosophila [8] and humans [10, 20], indicates that epistasis can play a

signiﬁcant role in both quantitative and qualitative characters. Epistasis eﬀects

of quantitative trait loci (QTL) have been found in soybean [17], maize [9]

and tomato [11]. Among the many models to study epistasis eﬀects, the linear

partition of genotypic values into additive, dominance, and epistasis eﬀects by

Fisher [12] is considered a classical model for epistasis [6]. Cockerham [5]

∗ Corresponding author: yda@umn.edu130 Y. Mao, Y. Da

and Kempthorne [15] further partitioned Fisher’s epistasis eﬀects into four

components, additive× additive, additive× dominance, dominance× addi-

tive, and dominance× dominance epistasis eﬀects with the genetic interpre-

tation of allele× allele, allele× genotype, genotype× allele, and genotype×

genotype interactions respectively. The partition of Fisher’s epistasis eﬀect by

Cockerham and Kempthorne provides a necessary tool to understand the pre-

cise nature of gene interactions. The genetic modeling of epistasis by Fisher,

Cockerham and Kempthorne can be applied for testing epistasis eﬀects of can-

didate genes and for mapping interactive QTL using genetic markers. Several

statistical methods for mappingve QTL have been reported, e.g.,the

ANOVA method [22], the randomization test [3], the mixture model likelihood

analysis based on Cockerham’s orthogonal contrast [14], and the Bayesian ap-

proach for an outbred population [24]. However, little is known about the sta-

tistical power and sample size requirement for detecting each epistasis eﬀect.

The purpose of this article is to derive and analyze the statistical power and

sample size requirements of the F-2 design for detecting epistasis eﬀects. For-

mulae for statistical power take into account major factors aﬀecting statistical

power and sample size, including separate testing and estimation of additive×

additive, additive× dominance, dominance× additive, and dominance× dom-

inance eﬀects, various levels of epistasis eﬀects, marker-QTL distances, type-I

error and sample size; formulae for sample size requirements take into account

various levels of epistasis eﬀects, marker-QTL distances, type-I and type-II

errors.

2. MATERIALS AND METHODS

2.1. Assumptions

Throughout this paper, two unlinked quantitative trait loci (QTL) on diﬀer-

ent chromosomes, QTL 1 and QTL 2, are assumed. Let A and B denote codom-

inant marker loci linked to QTL 1 and QTL 2, respectively, and letθ ,andθ1 2

denote the recombination frequencies between marker A and QTL 1, and be-

tween marker B and QTL 2, respectively. Then, the marker-QTL orders are

A-θ -QTL1 and B-θ -QTL2. Two inbred lines with gene ﬁxation for the mark-1 2

ers and QTL are assumed for the convenience of analytical derivations. We

assume Line 1 has AAQ Q BBQ Q genotype and Line 2 has aaq q bbq q1 1 2 2 1 1 2 2

genotypes, where A, a, B and b are marker alleles, and Q , q , Q and q1 1 2 2

are QTL alleles. The cross between these two lines yields the F-1 genera-

tion with AQ/aq BQ/bq individuals. The F-2 design is a result of matings1 1 2 2Statistical power for detecting epistasis 131

among the F-1 individuals. The least squares partitioning of genotypic values

and variances [12, 15] will be used to derive the common genetic modeling

for QTL values, assuming Hardy-Weinberg equilibrium. The statistical test-

ing of epistasis eﬀects using genetic markers will use a least squares model,

because analytical solutions are available from this method. From this least

squares model, elements required for the calculation of statistical power and

sample size requirements will be derived, including the marker contrast for

testing each epistasis eﬀect and the variance of the contrast. Theoretical results

for experimental designs will be compared with simulation studies assuming

various levels of epistasis eﬀects, and marker-QTL distance.

2.2. Genetic modeling and marker contrasts

The purpose of genetic modeling of the QTL genotypic values and indi-

vidual phenotypic values is to establish a theoretical foundation for deﬁning

marker contrasts for testing epistasis eﬀects. The least squares partitioning of

genotypic values by Kempthrone [15] will be used for the genetic modeling.

Letg = genotypic value of individuals with genotype ij at locus 1 and klijkl

at locus 2, (i= Q and j= q of locus 1, k= Q and l= q of locus 2).1 1 2 2

Then, using Kempthrone’s partitioning of genotypic values for the case of two

unlinked loci [15, 19], the genotypic value can be modeled as:

g =µ+ (α+α )+ (α +α )+δ +δ + (αα +αα +αα +αα )ijkl i j k l ij kl ik il jk jl

+ (αδ +αδ )+ (δα +δα )+δδikl jkl ijk ijl ijkl

=µ+α +α +δ +δ +αα +αδ +δα +δδ (1)ij kl ij kl ijkl ijkl ijkl ijkl

whereµ is the population mean of QTL genotypic values,α ,α ,α ,α arei j k l

the additive eﬀects of QTL allele Q , q, Q, q , respectively;δ ,δ are the1 1 2 2 ij kl

dominance eﬀects of locus 1 and locus 2, respectively;αα ,αα ,αα ,ααik il jk jl

are the additive× additive eﬀects accounting for the dependency of the eﬀect

of an allelic substitution at one locus on the allele present at a second locus;

αδ ,αδ are the additive× dominance eﬀects accounting for the interactionikl jkl

of single alleles at locus 1 with the genotype at locus 2;δα ,δα are theijk ijl

dominance× additive eﬀects representing the interaction of the genotype at

locus 1 with single alleles at locus 2; andδδ is the dominance× dominanceijkl

eﬀect representing the interaction between the genotype at locus 1 and the

genotype at locus 2. In equation (1),α =α +α ,α =α +α ,αα =ij i j kl k l ijkl

αα +αα +αα +αα ,αδ =αδ +αδ ,δα =δα +δα .For anik il jk jl ijkl ikl jkl ijkl ijk ijl

F-2 population with equal allele frequencies, it can be shown that the genetic132 Y. Mao, Y. Da

eﬀects in equation (1) have the following symmetry property:

a =α =−α1 i j

a =α =−α2 k l

d =δ =−δ =δ1 ii ij jj

d =δ =−δ =δ2 kk kl ll

i =αα =−αα =−αα =ααaa ik il jk jl

i =αδ =−αδ =αδ =−αδ =αδ =−αδad ikk ikl ill jkk jkl jll

i =δα =−δδ =δα =−δα =δα =−δαda iik ijk jjk iil ijl jjl

i =δδ =−δδ =δδ =−δδdd iikk iikl iill ijkk

=δδ =−δδ =δδ =−δδ =δδ .ijkl i jll jjkk jjkl j jll

This symmetrical property leads to simpliﬁed modeling of equation (1), as

shown in Table I. More importantly, this symmetry property will greatly sim-

plify the marker contrasts for testing epistasis eﬀects, allowing simple analyti-

cal solutions for evaluating statistical power and sample size requirement, as to

be shown later. By combining the nine equations in Table I and solving forµ,

a , a , d , d , i , i , i ,and i , the unique solutions of the eﬀect parameters1 2 1 2 aa ad da dd

in terms of the genotypic values are:

1 (g + 2g +g + 2g + 4g + 2g +g + 2g +g )µ= iikk iikl iill ijkk ijkl i jll jjkk jjkl j jll16

(2)

1a = [(g + 2g +g )− (g + 2g +g )] (3)1 iikk iikl iill jjkk jjkl j jll16

1a = [(g + 2g +g )− (g + 2g +g )] (4)2 iikk ijkk jjkk iill i jll j jll16

1d = [(g + 2g +g )− 2(g + 2g +g )+ (g + 2g +g )]1 iikk iikl iill ijkk ijkl i jll jjkk jjkl j jll16

(5)

1d = [(g + 2g +g )− 2(g + 2g +g )+ (g + 2g +g )]2 iikk ijkk jjkk iikl ijkl jjkl iill i jll j jll16

(6)

1= [(g −g )− (g −g )] (7)iaa iikk jjkk iill j jll16

1i = (g − 2g +g −g + 2g −g)(8)ad iikk iikl iill jjkk jjkl j jll16

1i = (g − 2g +g −g + 2g −g9)da iikk ijkk jjkk iill i jll j jll16

1i = (g − 2g +g − 2g + 4g − 2g +g − 2g +g ).dd iikk iikl iill ijkk ijkl i jll jjkk jjkl j jll16

(10)

In equations (2–10), a = additive eﬀect of QTL 1, d = dominance eﬀect1 1

of QTL 1, a = additive eﬀect of QTL 2, d = dominance eﬀect of QTL 2,2 2

i = additive× additive epistasis eﬀect, i = additive× dominance epistasisaa adStatistical power for detecting epistasis 133

Table I. Representation of genotypic values in terms of additive, dominance, and epis-

tasis contributions for the case of two loci with two equally frequent alleles.

Genotype Frequency Genotypic Value

Q Q Q Q 1/16 g =µ+ 2a + 2a + d + d + 4i + 2i + 2i + i1 1 2 2 iikk 1 2 1 2 aa ad da dd

Q Q Q q 1/8 g =µ+ 2a + d − d − 2i − i1 1 2 2 iikl 1 1 2 ad dd

Q Q q q 1/16 g =µ+ 2a − 2a + d + d − 4i + 2i − 2i + i1 1 2 2 iill 1 2 1 2 aa ad da dd

Q q Q Q 1/8 g =µ+ 2a − d + d − 2i − i1 1 2 2 ijkk 2 1 2 da dd

Q q Q q 1/4 g =µ− d − d + i1 1 2 2 ijkl 1 2 dd

Q q q q 1/8 g =µ− 2a − d + d + 2i − i1 1 2 2 i jll 2 1 2 da dd

q q Q Q 1/16 g =µ− 2a + 2a + d + d − 4i − 2i + 2i + i1 1 2 2 jjkk 1 2 1 2 aa ad da dd

q q Q q 1/8 g =µ− 2a + d − d + 2i − i1 1 2 2 jjkl 1 1 2 ad dd

q q q q 1/16 g =µ− 2a − 2a + d + d + 4i − 2i − 2i + i1 1 2 2 j jll 1 2 1 2 aa ad da dd

eﬀect, i = dominance× additive epistasis eﬀect, and i = dominance×da dd

dominance epistasis eﬀect between QTL 1 and QTL 2. Equations (2–10) are

foundations for marker contrasts for QTL detection under the F-2 design, and

can be used for testing candidate genes in an F-2 design.

When a QTL genotypic value is to be predicted by linked markers, as is the

case in QTL detection, the QTL genotypic value can be modeled as:

g = m + r (11)ijkl ijkl ijkl

where m is the eﬀect of markers, and r is the genotypic residual value dueijkl ijkl

to recombination between the markers and QTL. Note that the common genetic

mean (µ) term in equation (11) is dropped for convenience of derivations. The

two marker models with or without theµ term are equivalent models [13, 19]

that achieve the same result for statistical testing. In matrix notation, the QTL

genotypic value modeled by genetic markers can be expressed as:

g= Xm+ r (12)

where g is the column vector of QTL genotypic values, X is the design matrix

for the marker eﬀects, r is the recombination residual of the QTL value not

explained by the common mean and the markers, and m is the vector of marker

eﬀects, i.e.,

m= (m , m , m , m , m , m , m , m , m ).iikk iikl iill ijkk ijkl i jll jjkk jjkl j jll

The normal equations for equation (12) in matrix notation are X Xm= X g,

−1 −1and the solution to this normal equation is m= (X X) X g,where (X X) is

the inverse of X X,and X is the transpose of X.134 Y. Mao, Y. Da

A phenotypic value is modeled as the summation of a QTL genotypic value

2(g ) and a random residual (e) with N(0,σ ) distribution, i.e.,ijkl e

y =g + e = m + (r + e )= m +ε (13)ijkl ijkl ijkl ijkl ijkl ijkl ijkl ijkl

withε = phenotypic residual value not explained by the marker eﬀectsijkl

due to the recombination and random residuals. Using matrix notation, equa-

tion (13) can be expressed as:

y= Xm+ (r+ e)= Xm+ε. (14)

The normal equations are X Xm= X y, and the estimator of m is given by

−1 mˆ = (X X) X y. (15)

Let mˆ = (ˆm , mˆ , mˆ , mˆ , mˆ , mˆ , mˆ , mˆ , mˆ ) be the leastiikk iikl iill ijkk ijkl i jll jjkk jjkl j jll

squares estimate of m= (m , m , m , m , m , m , m , m , m )iikk iikl iill ijkk ijkl i jll jjkk jjkl j jll

deﬁned in equation (15), then the four marker contrasts for testing epistasis

eﬀects are:

1L = mˆ − mˆ − mˆ + mˆ (16)aa iikk iill jjkk j jll16

1L = mˆ − 2ˆm + mˆ − mˆ + 2ˆm − mˆ (17)ad iikk iikl iill jjkk jjkl j jll16

1L = mˆ − mˆ − 2ˆm + 2ˆm + mˆ − mˆ (18)da iikk iill ijkk i jll jjkk j jll16

1L = mˆ − 2ˆm + mˆ − 2ˆm + 4ˆm − 2ˆm + mˆdd iikk iikl iill ijkk ijkl i jll jjkk16

−2ˆm + mˆ (19)jjkl j jll

where L , L , L and L are the contrasts for testing additive× additive,aa ad da dd

additive× dominance, dominance× additive, and dominance× dominance

eﬀects, respectively.

2.3. Variances of recombination and phenotypic residuals

Following the approach of Bulmer [2] (Eq. (5.1) on page 58), Table I can be

expressed more succinctly as:

2 2g =µ+ 2(z − 1)a + 2(z − 1)a + 1− 4z + 2z d + 1− 4z + 2z dijkl 1 1 2 2 1 1 2 21 2

2+ 4(z − 1)(z − 1)i + 2(z − 1) 1− 4z + 2z i1 2 aa 1 2 ad2

2 2 2+ 2 1− 4z + 2z (z − 1)i + 1− 4z + 2z 1− 4z + 2z i1 2 da 1 2 dd1 1 2

(20)Statistical power for detecting epistasis 135

where z is the number of Q alleles in a particular individual at the ﬁrst1 1

putative QTL, and z is the number of Q alleles in a particular individ-2 2

ual at the second putative QTL, (z , z = 0, 1, or 2 respectively). Equa-1 2

tion (20) provides a convenient model for deriving variance and covariance

2of the QTL genotypic values (App. A). Letσ = the total QTL genotypic vari-g

2 2ance,σ = additive variance of QTL 1,σ = additive variance of QTL 2,

A1 A2

2 2σ = dominance variance of QTL 2,σ = dominance variance of QTL 2,

D1 D2

2 2σ = additive× additive variance,σ = additive× dominance variance,

AA AD

2 2σ = dominance× additive variance, andσ = dominance× dominance

DA DD

variance of the two QTLs, then

2 2 2 2 2 2 2 2 2σ =σ +σ +σ +σ +σ +σ +σ +σg A1 A2 D1 D2 AA AD DA DD

2 2 2 2 2 2 2 2= 2a + 2a + d + d + 4i + 2i + 2i + i (21)1 2 1 2 aa ad da dd

with

2 2 2 2 2 2 2 2σ = 2a,σ = 2a,σ = d,σ = d,A1 1 A2 2 D1 1 D2 2

2 2 2 2 2 2 2 2σ = 4i ,σ = 2i ,σ = 2i ,σ = i . (22)AA aa AD ad DA da DD dd

Derivations for equations (21, 22) are given in Appendix A. The total genetic

variance is partitioned into eight independent components, and each variance

component is a function of one eﬀect only. This property greatly facilitates

the evaluation of the contribution of an eﬀect to the total genetic variance.

Note that equivalent partitions can be obtained under alternative models, but

they have diﬀerent meanings in interpreting gene eﬀects, diﬀerent structures

of variance components, and diﬀerent properties in statistical estimation that

may aﬀect the study of QTL [14].

The population variance of recombination residuals for equation (12) is

2 1 σ = (g g− m X g). (23)r n

Applying equation (23) to the F-2 design, and utilizing equation (22), the re-

combination residual variance of QTL genotypic values is found to be:

2 2 2 2 2= 1− (1− 2θ ) σ + 1− (1− 2θ ) σσ 1 2r A1 A2

4 2 4 2+ 1− (1− 2θ ) σ + 1− (1− 2θ ) σ1 2D1 D2

2 2 2 2 4 2+ 1− (1− 2θ ) (1− 2θ ) σ + 1− (1− 2θ ) (1− 2θ ) σ1 2 1 2AA AD

4 2 2 4 4 2+ 1− (1− 2θ ) (1− 2θ ) σ + 1− (1− 2θ ) (1− 2θ ) σ .1 2 1 2DA DD

(24)136 Y. Mao, Y. Da

Derivation of equation (24) is given in Appendix B. The residual variance of

phenotypic values for F-2 design under equation (14) is:

2 2 2σ =σ +σ.ε r e

3. RESULTS

3.1. Mathematical formulae for statistical power and sample size

Statistical power (π) is the probability that an eﬀect is detected when the

eﬀect is present, commonly denoted byπ= 1−β,whereβ is the type II error,

i.e., the probability of false ‘negatives’. A standardized normal distribution

denoted by N(0,1) is assumed for deriving the statistical power. The normal

distribution is chosen because the calculation of the exact residual degrees of

freedom is unnecessary, providing analytical simplicity. Since the residual de-

grees of freedom are suﬃciently large for the sample sizes discussed in this

article (N≥ 200), the normal distribution practically yields identical results

as the t-distribution that is often used in QTL analysis. The general expression

forπ is:

π= 1−β= 1− Pr(Z< z )= 1−Φ(z ) (25)i i

where Z is a N(0,1) normal variable, z is the ordinate of the standardizedi

normal curve corresponding to the type II error ofβ,andΦ is the cumulative

distribution function of standard normal random variable. The application of

equation (25) to QTL mapping designs requires two key elements: a marker

contrast for detecting each epistasis eﬀect, and the variance of the contrast.

Let c be a contrast vector that deﬁnes an estimable function of m,then

ˆE(c m) = c m. Based on this result and the m vector in Appendix B, the

mathematical expectation of each contrast given by equations (16–19), denoted

by E(L ), i = aa, ad, da, dd, are functions of markers-QTL recombinationi

frequencies and the QTL eﬀects being tested, i.e.,

E(L )= (1− 2θ )(1− 2θ ) i (26)aa 1 2 aa

2E(L )= (1− 2θ )(1− 2θ ) i (27)ad 1 2 ad

2E(L )= (1− 2θ ) (1− 2θ ) i (28)da 1 2 da

2 2E(L )= (1− 2θ ) (1− 2θ ) i . (29)dd 1 2 dd

) in place of L for i = aa, ad, da, dd as deﬁned by equa-Using E(Li i

tions (16–19), the z value in equation (25) can be expressed as:i

√

E(L ) N E(L )i i i

z = z −√ = z − √i α/2 α/2

var(L ) Vi iStatistical power for detecting epistasis 137

where N is the sample size and V = N var(L ), for i= aa, ad, da, dd.Fori i i i

convenience, V will be referred to as the ‘kernel’ of the contrast variance,i

meaning that V diﬀers from var(L ) only by a constant of N .Leti i i

2 2σ +σr e2h =ε 2σy

2 2 2 2 4 2 4 2= 1− (1− 2θ ) h − (1− 2θ ) h − (1− 2θ ) h − (1− 2θ ) h1 2 1 2a a1 2 d d1 2

2 2 2 2 4 2− (1− 2θ ) (1− 2θ ) h − (1− 2θ ) (1− 2θ ) h1 2 1 2aa ad

4 2 2 4 4 2− (1− 2θ ) (1− 2θ ) h − (1− 2θ ) (1− 2θ ) h (30)1 2 1 2da dd

2 2 2 2 2 2 2 2 2 2 2 2where h = σ /σ , h = σ /σ , h = σ /σ , h = σ /σ ,a y a y y y1 A1 2 A2 d D1 d D21 2

2 2 2 2 2 2 2 2 2 2 2 2h =σ /σ , h =σ /σ , h =σ /σ ,and h =σ /σ . For con-aa y y y yAA ad AD da DA dd DD

2 2 2venience, we will refer to h and h as the additive heritabilities, h anda a1 2 d1

2 2 2 2 2h as the dominance heritabilities, and h , h , h and h as the additive×aad ad da dd2

additive, additive× dominance, dominance× additive and dominance× dom-

inance heritabilities, respectively. Then, the expressions of V in terms of QTLi

parameters are given as follows:

1 2 2V = σ h (31)aa 4 y ε

1 2 2V = σ h (32)ad y ε2

1 2 2V = σ h (33)da y ε2

2 2V =σ h. (34)dd y ε

The derivations of equations (31–34) are given in Appendix B.

√

Lettingλ = E(L )/ V,then z in equation (25) can be expressed in termsi i i i

of QTL parameters as:

z = z − Nλ (35)i α/2 i i

where

(1− 2θ )(1− 2θ )h1 2 aa

λ = (36)aa hε

2(1− 2θ )(1− 2θ ) h1 2 ad

λ = (37)ad

hε

2(1− 2θ ) (1− 2θ )h1 2 da

λ = (38)da

hε

2 2(1− 2θ ) (1− 2θ ) h1 2 dd

λ = · (39)dd

hε138 Y. Mao, Y. Da

In equations (36–39),θ ,θ are the marker-QTL recombination frequencies.1 2

Theoretical predictions of statistical power for various parameters using equa-

tion (25) and equations (35–39) are shown in Figures 1–4. The implications of

these results will be discussed along with the results of simulation studies.

Using the above results, the minimum sample size required for given levels

of type I and type II errors can be expressed as:

2 2V (Z + Z ) (Z + Z )i α/2 β α/2 β

N = = (40)i 2 2E (L ) λi i

where Z and Z are the ordinate of the standardized normal curve corre-α/2 β

sponding to the probabilities ofα/2andβ. The sample size given by equa-

tion (40) is an increasing function of marker-QTL recombination frequencies,

as well as type-I and type-II errors, and a decreasing function of heritabil-

ity. Sample size requirements obtained from equation (40) for two type-I er-

rors corresponding to the “suggestive” and “signiﬁcant” linkages proposed by

Lander and Kruglyak [16], and diﬀerent levels of epistasis heritabilities and

marker-QTL recombination frequencies are given in Table II, assuming a 95%

statistical power.

3.2. Simulation studies on statistical power

Simulation studies were conducted using the Monte Carlo method to eval-

uate the theoretical results on statistical power for detecting epistasis eﬀects

under the F-2 designs. Markers and QTL genotypes were generated such that

the true recombination frequencies and each QTL eﬀect used to generate these

genotypes can be obtained reversely from the data. A total of 100 sets of

marker-QTL genotypes were generated. The phenotypic value of each indi-

vidual is obtained as the summation of the individual QTL genotypic value

and a random residual with N(0,1) distribution. For each set of data,

10 000 replicates were generated for Figures 1–4. Two interactive QTL with-

out linkage and all epistasis eﬀects have the same absolute value are assumed,

thus the heritability of additive× additive (a× a) eﬀect is twice as large as that

of additive× dominance (a× d) or dominance× additive (d× a) eﬀect, and is

four times as large as that of dominance× dominance (d× d) eﬀect. Heritabil-

ities of additive× additive eﬀect are used in the range of 0.025 to 0.25. Sample

sizes of 200–2000 individuals resulting from crossing between inbred lines

were generated. The signiﬁcant levels (type I errors) used were those corre-

sponding to “suggestive linkage” and “signiﬁcant linkage” proposed by Lander

and Kruglyak [16] with type-I errors of 0.0034 and 0.00072 respectively.