Power and parameter estimation of complex segregation analysis under a finite locus model

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Power and parameter estimation of complex segregation

analysis under a finite locus model

P Uimari, Bw Kennedy, Jcm Dekkers

To cite this version:

Original

article

Power and

_parameter

estimation

of

_complex

_{segregation analysis}

under

a

finite locus

model

P

_Uimari,

BW

_Kennedy

JCM Dekkers

Department

of

Animal and

_{Poultry Science,}

Centre

_for

Genetic

_Improvement

of Livestock, University of Guelph, Guelph,

ON N1G

2W8,

Canada

(Received

20 October 1995;

accepted

7

_May

₁₉₉₆₎

Summary - Power and parameter estimation of

_{segregation analysis}

was

investigated

for

independent

nucleus

_family

data on a

quantitative

trait

generated

under a finite locus model and under a mixed model. For the finite locus

_model,

_gene effects at ten loci were

generated

from a

_geometric

series.

_{Additionally, linkage}

between a

_major

locus and other loci was considered. Two different methods of

_segregation

_analysis

were

_compared:

a mixed model and a finite

_polygenic

mixed model. Both statistical methods _gave similar _{power to} detect a

_major

_gene and estimates of _parameters. An

_exception

was a situation where two

major

loci had an

_equal

effect on

_phenotype:

the mixed model had a

higher

_power than the

finite

_polygenic

mixed

model,

but estimates of the parameters from the mixed model were more biased than estimates from the finite

_polygenic

mixed model.

_Segregation

analysis

was more

_powerful

in

_detecting

a

_major

_gene when data were

_generated

under the finite locus model than under the mixed model. When a

major

_gene was linked to another _gene, a

_major

_gene was more difficult to detect than without such

_{linkage. Segregation}

of two

major

genes created biased estimates. Bias increased with

linkage

when parents were not a random

sample

from a

population

in

linkage equilibrium.

parameter estimation

_/

power

/

major gene

/

segregation analysis

Résumé - Puissance et estimation des paramètres dans _l’analyse de _{ségrégation}

com-plexe avec un modèle à nombre fini de locus. La _puissance de

l’analyse

de

_{ségrégation}

et

l’estimation des

_paramètres

ont été étudiées sur des

familles

nucléaires

indépendantes

pour un caractère

quantitatif

déterminé soit _par un nombre

fini

de locus soit selon un modèle d’hérédité

_mixte,

_impliquant

un

_{gène majeur}

et un résidu

_{polygénique infinitésimal.}

Dans

le modèle à nombre

_fini

de

_locus,

le nombre de locus

_supposé

était de dix et leurs

_effets

sui-vaient une loi de distribution

_{géométrique.}

En _outre, la

_possibilité

de liaison

_génétique

entre un locus _majeur et d’autres locus était

_envisagée.

Deux méthodes

_d’analyse

de

_{ségrégation}

ont été

_comparées,

utilisant soit un modèle d’hérédité _mixte, soit un modèle d’hérédité avec un nombre

fini

de locus. Les deux méthodes

statistiques présentaient

des

_puissances

simi-laires pour détecter un

_{gène majeur}

et estimer les

_{paramètres correspondants.}

À

l’exception

toutefois

d’une situation avec deux locus _{majeurs ayant} le même

_effet

sur le

_phénotype.

nom-bre

_fini

de

_locus,

mais les estimées des

_paramètres

à partir du modèle mixte étaient

_plus

biaisées _que celles du modèle à nombre

_fini

de locus.

_L’analyse

de

_{ségrégation}

était

_plus

puissante

pour détecter un

_gène

_majeur dans le cas d’un caractère déterminé _par un nom-bre

_fini

de locus _que dans une situation d’hérédité mixte. Un

_{gène majeur}

lié à un autre

gène

était

_{plus difficile}

à détecter

_qu’en

l’absence de liaison

_génétique.

La

_{ségrégation}

de deux

_gènes

majeurs créait des biais d’estimation. Les biais étaient encore accrus en cas de liaison

_{génétique quand}

les parents n’étaient pas tirés d’une

population

en

équilibre

gamétique

pour les deux locus majeurs.

estimation de _paramètre

_/

_{puissance / gène majeur}

_/

_analyse de _{ségrégation}

INTRODUCTION

Statistical methods used to determine the mode of inheritance of a

_quantitative

trait in detection of

_major

_genes

_rely

on

_phenotypic

information. In

_addition,

methods can utilize information on

_{genetic markers,}

which are now numerous. In both cases, the most common statistical methods to detect a

_major

_gene are based

on maximum likelihood

_theory.

Maximum-likelihood-based

_complex

_segregation

analysis

was introduced

by

Elston and Stewart

(1971)

and Morton and MacLean

(1974).

Complex

segregation

analysis

combines three factors into a mixed model for

analysis

of

_phenotypes

for a

_quantitative

trait: a _gene which

explains

a detectable

part

of

_genetic

variance

_{(major gene);}

residual

_polygenic

_variance,

for which individual _gene effects are not of direct interest or

_detectable;

and environment.

Recently

a finite

polygenic

mixed

model,

which

explains

the

polygenic

_part

of inheritance

_by

a finite number of

loci,

was

_{proposed by}

Fernando et al

₍₁₉₉₄₎

as an alternative formulation for the mixed model. To make the finite

_polygenic

mixed model

_{computationally}

feasible it is assumed that loci which

explain

the

_polygenic

part

of inheritance are

_{unlinked, biallelic, codominant,}

and have

_equal

_gene effects and

_{equal frequencies}

of favourable alleles

_(0.5)

across loci

_(Fernando

et

_al,

_1994).

Power of

_segregation

_analysis

of

_independent

nucleus

_family

data

_(full-sib

fami-lies)

with the mixed model was

_{investigated by}

MacLean et al

₍₁₉₇₅₎

and Borecki

et al

₍₁₉₉₄₎

and for half-sib data

_by

Le

_Roy

et al

₍₁₉₈₉₎

and Knott et al

_(1991).

In all cases, data were simulated

according

to the mixed model of inheritance. The

general

conclusion from these studies was that the best chance to detect a

_major

gene is if it is dominant with moderate to low

frequency

in the

population. By

increasing

data size

_(number

of families and size of the

_families),

major

genes with

smaller effects can be detected.

Many

aspects

that

_might

affect robustness of

_{segregation analysis}

with the mixed model have been studied also

_(MacLean

et

al, 1975;

Go et al

_1978;

Demenais

et

_al,

_1986).

The main concern has been false detection of a

_major

_gene with

skewed data. To overcome this

_problem,

power transformation of the data was

proposed

(MacLean

et

_al,

_1976).

The

_optimal

solution for skewed data is to make the transformation

_{simultaneously}

with estimation of other

_parameters

_(MacLean

et

_al,

_1984).

_Removing

skewness _may,

_however,

lead to reduced _power to detect a

Other common

assumptions

in

_segregation

analysis

include

_homogeneous

vari-ance within

_major

_genotypes,

_independence

between the

_major

_gene and

_polygenic

effects,

no

_genotype

_by

environmental

_correlation,

and no correlation between en-vironment of

_parent

and

_offspring

_(MacLean

et

_al,

_1975).

One basic

_assumption

of

_segregation

_analysis,

which has received less

_attention,

is

normality

of the residual distribution

_(polygenic

+

_{environmental)}

within a

_major

genotype.

This

_assumption

is met if the

_polygenic

_part

is controlled

_by

infinite number of genes that each have

only

a small effect on

_phenotype,

_ie,

the infinitesimal model

_(Bulmer,

_1980),

and if the environmental factor is

_normally

distributed.

However,

the infinitesimal model

_might

not be the best model for the distribution

of gene effects. A model where few genes with a

_large

effect and several _genes

with small effects control a

_quantitative

_{trait may} be closer to the real nature of the distribution _{of gene} effects. Evidence from

Drosophila

melanogaster

supports

this

_hypothesis

_(Shrimpton

and

Robertson, 1988;

Mackay

et

_al,

_1992).

Such a distribution of _gene effects can be

_{approximated by}

a

_geometric

series

_(Lande

and

Thompson,

1990).

If gene effects follow a

_{geometric series,}

the distribution within

_major

_genotype

may not be

normal,

as with the infinitesimal model. This violates the

_assumption

of a

_normally

distributed

_polygenic

_part

of the mixed model

_commonly

used in

segregation

analysis.

Two or more loci with

_large

effects can also lie in a cluster on

a

_chromosome,

which would link the

_major

_{gene to} other genes and thus violate the

assumption

of

_independent

_segregation

of a

_major

_gene and

polygenes.

The

_objective

of this paper was to

_study

the effect of violation of the two

assumptions

of the

_underlying

model in

_segregation

_{analysis, namely}

a skewed

polygenic

distribution and

_linkage

between a

_major

_gene and

_polygenes,

on the

power of

detecting

a

_major

_gene and on

_parameter

estimation. Behavior of the

mixed model of

_segregation

_analysis

_(Morton

and

_MacLean,

₁₉₇₄₎

was

_compared

to the finite

_polygenic

mixed model

_(Fernando

et

_al,

_1994).

The methods were

compared

under an

_independent

nucleus

_family

data structure.

MATERIALS AND METHODS

Balanced data on a

_quantitative

trait were simulated for 25

independent

full-sib

_families,

with a

_{sire, dam,}

and ten

_offspring.

All

_parents

were assumed to

be unrelated and were

_generated

from a

_population

under

_{Hardy-Weinberg}

and

linkage equilibria.

Genotypes

of

_parents

were

generated

under a ten-locus model

(finite

locus

_model)

or under a mixed model

_(from

now on this will be called the

mixed

_generating

_model,

whenever necessary, to

_distinguish

between models used

for

_generating

and for

_analyzing

the

_data).

Under the finite locus

_model,

the _gene with

_largest

effect had a substitution effect of 1.0

_(the

difference between two

_homozygotes

is twice the substitution

_effect)

and the gene with the second

largest

effect had a substitution effect of

_0.25,

0.5 or 1.0. Gene effects of the

_eight

other loci followed the

geometric

series

_{0.25, 0.125, 0.0625,}

where one locus had an effect of

_0.25,

three loci an effect of 0.125 and four loci an

effect of 0.0625. Gene

_frequencies

were 0.5 for all loci

_except

for the

_{major locus,}

for

which

_frequency

of the dominant allele was either

_{0.1, 0.5,}

or 0.9. Two alleles _per

and other loci were additive.

_Genotypes

of _progeny were

_{generated using}

either

independent segregation

of loci or the two loci with the

_largest

effect were linked with a recombination rate of 0.1. In the case

of linkage, linkage

phase

of the

_parents

was either random or all

_parents

were double

_{heterozygotes}

for the two linked loci

(favourable

alleles on same

chromosome).

For _every finite locus

_scenario,

_{corresponding}

_genotypes

were also

generated

with a mixed model. Under the

mixed-generating

model,

a

_major

_gene with a substitution effect of 1.0 was

simulated, along

with a

polygenic

_part,

which was simulated from a normal distribution with 0 mean and

_genetic

variance

_equal

to

the total

genetic

variance

_(additive

+

_dominance)

of the other nine loci in the

corresponding

finite locus model. The

_polygenic

effect _{of progeny} was

generated

from a normal distribution with mean

_equal

to _{the average of}

_polygenic

effects of the

parents

and variance

_equal

to half of the

_polygenic

variance.

Phenotypes

were

generated

for both the finite locus and the

mixed-generating

model

_{by adding}

an environmental effect to the

_genotypic

effects. Environmental effects were simulated from a normal distribution with mean 0 and variance

corresponding

to one minus the broad sense

_heritability

(H

2

,

total

_genetic

variance over

phenotypic

variance),

which was

_equal

to 0.4. A _summary of the

_genetic

Simulated data sets were

_analyzed

_by

two

_computer

_packages.

The

_Pedigree

Analysis Package

(PAP

Rev

4.02, Hasstedt, 1982,

1994)

was used to

_compute

the likelihood of the mixed model and SALP

_(segregation

and

_linkage

_analysis

for

pedi-grees, Stricker et

_al,

₁₉₉₄₎

to

_compute

the likelihood of the finite

_polygenic

mixed model.

_Only

one

_major

locus was fitted in SALP. Mendelian transmission

proba-bilities,

equal

variances within

_genotypes

and no _power transformation were used

in PAP. Downhill

_simplex

method is used for maximization in SALP and Gemini

(Lalouel, 1979)

in PAP. Because Gemini does not allow maximization at boundaries

of the

_parameter

space

(gene

frequency

and

_heritability

have boundaries at 0 and

1)

the _program

_{occasionally stopped.}

In those cases, the

parameter

that reached the

_boundary

was fixed close to the

_boundary

_(0.0001

or 0.9999 for gene

frequency

and 0.0001 for

_{heritability)}

and other

parameters

were maximized conditional on that. Because the

_major

gene was simulated with

_{complete dominance,}

_pAA was

fixed to be

_equal

to _{pAa in} all maximum likelihood

_analyses.

_Input

values for sim-ulation were used as

_starting

values for the maximization process. Likelihood ratio

test statistic was calculated

_{by comparing}

a

_general

model to a model with

equal

means

(fJ

AA

=

_{fJAa =}

_/-

_t

aa)-Because SALP and PAP use different

_{parameterization}

of

_effects,

_parameters

were converted to two

_genotypic

means

(

PAA

and

_Aaa

_),

_gene

_frequency

of the dominant allele

_(p),

and

_polygenic

_(ufl)

and environmental

_(ud)

variances. Instead of

_polygenic

and environmental

_variances,

PAP estimates

_heritability

_(h

₂

₎

and the

_phenotypic

standard deviation conditional on

_major

_genotype;

for the finite

polygenic

mixed model SALP estimates a

_scaling

factor

(=

(Qu!(q(1 -

_q)k)],

where

q is

the allele

_frequency

at

_polygenic

_loci,

which was fixed at

_0.5,

and k is twice the number of

_polygenic

_loci,

which was fixed at

_ten),

and

_phenotypic

variance.

Each simulated

_major

_gene scenario

_{(table I)}

was

_replicated

50 times.

_Empirical

power of the mixed model of

analysis

was measured as the

_proportion

of cases in

which the likelihood ratio test statistic exceeded the

X

Z

distribution with 2

_df

at

5%

_significance

level.

Because the likelihood test statistic is

_only

_{asymptotically}

distributed

_according

to the

X

2

distribution

_(Wilks,

_1938),

200

_replicates

of six data sets without a

_major

gene were

generated

based on the infinitesimal model and the

_proportion

of test

statistics which

_supported

the

_major

_gene

_hypothesis

was calculated for both the mixed model and the finite

_polygenic

mixed model.

_Polygenic

and environmental variances of the

_examples

_corresponded

to sets 2 and 3

_{(table I)}

without a

_major

gene. The

proportion

of false detection is

_expected

to be

5%

when a

5% type

I error

level is used.

Empirical

power of the mixed model was measured as the

_proportion

of cases in

which the

_major

_gene

_hypothesis

was

_accepted.

Under the

_{mixed-generating}

_model,

the power

corresponds

to the

_probability

of

_detecting

the simulated

_major

gene. This is not the case when data are simulated under the finite locus

_model;

instead of

_detecting

the first locus as a

_major

_gene, the _power indicates the

_probability

of

RESULTS

Power of the likelihood ratio test

The

_proportions

of false detection of

_major

gene when no

_major

_gene effect was

generated,

but the likelihood ratio between the mixed model and the

_polygenic

model was

compared

to the

X

2

table value with two

_degrees

of freedom at

5%

significance

level,

were

_4,

3 and

6%

for set 2 distribution of _gene effects

_{(table I)}

and

4,

3 and

5%

for set 3 distribution of gene effects with gene

frequencies

of

0.1,

0.5,

and

_0.9,

_{respectively.}

_Using

the finite

_polygenic

mixed model and its sub-model

the

_{corresponding}

values were

_{4, 3,}

4 and

_{4, 4,}

_3%,

for set 2 and set

_3,

_{respectively.}

Thus the true _power of

_detecting

a

_major

_gene for the data structure used here can be somewhat

higher

for both methods than

_reported

in table II.

When data were

_generated

under the mixed

model,

the

highest

power was achieved when

_frequency

of the dominant allele was low and the lowest _power with a rare recessive allele

_(table

_II).

This

_pattern

was consistent across different

proportions

of

_genetic

variance

_{explained by polygenes}

_(sets

_1,

2 and

_3).

Under the finite locus

_model,

the

_pattern

_changed

when two

_major

loci had an

_equal

effect on the trait

_(table

II,

set

_3);

the

_highest

_power for the mixed model was achieved when one of the genes was almost fixed in the

population,

however,

the difference

between cases of _gene

_frequency

of 0.5 and 0.9 for the finite

_polygenic

mixed model was small

(without linkage).

The effect of the

_proportion

of total

_genetic

variance that a

_major

_gene

ex-plained

on the _power was _very clear under the

_{mixed-generating}

_model;

the _power was

higher

if the

_major

_gene

_explained

a

large proportion

of total

_genetic

power reduced when the effect of the second

_largest

locus increased

_(table

_II,

sets

_1,

2 and

_3).

An

_exception

_was,

_again,

a case when two

_major

loci had an

_equal

effect on the trait and

_frequencies

of favourable alleles at the

_major

loci were 0.5 and 0.9

(table

II,

set

3,

p =

0.9).

In most cases, the

_higher

_power of

_detecting

a

_major

_gene was achieved when data were

_generated

under the finite locus model than under the mixed model.

Violation of the

_assumption

of

_independent

_segregation

of the

_major

_gene and

other genes had a

_negative

effect on the power of the mixed model as well as on the _power of the finite

_polygenic

mixed model

_{(table II).}

Even

_larger

reductions

in the power were observed when all

_parents

were double

_{heterozygotes}

for the

two linked loci with

_largest

effects

_(table

_II).

In this case, not

only

the

_assumption

of

_independent

_segregation

of a

_major

_{gene and}

_polygenes

was violated but also the

_assumption

of

_{Hardy-Weinberg}

_equilibrium

in the

_{parental population;}

true

probabilities

for

_parents

to be

_homozygotes

were _zero, not

_p

₂

and

_{(1 -}

_p)

₂

_,

as was assumed in the

_analysis.

The reduction in the _power due to violation of

Hardy-Weinberg

equilibrium

was confirmed

_by

a simulation where all

_parents

were

heterozygous

for the

_major

locus

_(a

finite locus model similar to set 2 with _p =

_0.5,

no

linkage).

In this _case, the _power of the mixed model was

28%

_compared

to

58%

when the

_parent

_population

was in

_{Hardy-Weinberg}

_equilibrium

_(table

_II,

set

_2,

p = 0.5).

Parameter estimation

Mean estimates of

_parameters,

with their

_empirical

standard deviations based on 50

_replicates,

and true values are

_given

in tables III and IV. The

_expected

variance

components

for

_polygenes

_given

in table III

_(results

for the finite locus

_model)

do

not include dominance variance of the second and the third

_largest

loci

_(smaller

loci were

additive),

because the statistical methods studied here did not take

_polygenic

dominance variance into account. As a

_result,

dominance variance _may be

_partly

confounded with estimates of additive

_genetic

variance and

_partly

with estimates of residual variance.

For the first distribution of _gene effects

_{(set 1)}

and the finite locus

model,

both methods _gave similar estimates

_{(table III).}

In most _cases, estimates

_agreed

well with true

_{values, although}

some

_{discrepancies}

were found for variance

components.

The standard deviation of the estimate of the

_genotypic

mean

_depended

on the

estimated gene

frequency

and was

_larger

for low

_frequencies.

Going

from the set 1 distribution of _gene effects to set

2,

with a

_larger

second

locus

_effect,

variation of estimates increased

_(table

_III).

More bias was also observed. For

_example,

when gene

frequency

was

_0.9,

the difference between

_genotypes

was underestimated

_(by

about

_0.25)

_by

both methods and gene

frequency

was

underestimated at 0.8.

When two

_major

_genes with

equal

effect were

_simulated,

_parameter

estimates were biased

_(table

III,

set

_3).

The difference between

_homozygotes

was inflated

by

as much as

25%

in the case of

equal

gene

frequencies

(0.5).

Gene

frequency

estimates were also

biased;

with a simulated _gene

_frequency

of

_0.1,

the _average

0.9 and the finite

_polygenic

mixed model between 0.5 and 0.9. Overestimation of differences between

_genotypes

led to underestimation of

_polygenic

_variance,

because a

_larger

_proportion

of total

_genetic

variance was attributed to variance between

genotypes.

With

_linkage

between the two loci with

_largest

effect,

a

_significant

inflation was

observed in all estimates when the linked genes were of

_equal

size

(table

_III,

set

_3).

When all base

_population

_parents

were double

_{heterozygotes}

for the two linked loci

of

_{large effect,}

parameter

estimates were

_highly

biased

_{(table III).}

Estimates of the difference between the two

_genotypes

was 0.8 units

_higher

than the true difference

between the

_genotypes

in one locus when the two loci with the

_largest

effect on

phenotype

had

equal

effects. Also in this case, gene

frequency

was

higher

than the

expected

0.5 and the estimate of additive

_genetic

variance was almost zero. Bias

in estimates of the

parameters

was

_larger

for the mixed model than for the finite

polygenic

mixed model.

More consistent estimates over the different

_genetic

scenarios were achieved when data were

_generated

under the mixed model than under the finite locus model

(table IV).

No

important

differences were found between the mixed model and the finite

_polygenic

mixed model. The variance of estimates of all

_parameters

increased when the

_proportion

of

_genetic

variance

_{explained by}

the

_major

gene decreased

(going

from set 1 to set

_3),

but _average values of estimates were still close to

expected

values.

DISCUSSION AND CONCLUSIONS

The purpose of this paper was to

_study

the

_sensitivity

of

_complex

_segregation

analysis

to violation of some of the

_assumptions

of the

_underlying

_model,

in

particular

a normal distribution of

_polygenic

effects and no

_linkage

between a

major

gene and

polygenes. Similarity

in _{the power of both methods of}

_segregation

analysis

(the

mixed model and the finite

_polygenic

mixed

_model)

was

_observed,

except

when data were

_generated

based on the finite locus model with two

_major

genes. Similar results for both methods can be

_expected

because the

_computer

package

(SALP),

which maximized the finite

_polygenic

mixed model used

_equal

allele

frequencies

(0.5)

and additive gene action for all genes

except

the

_major

_gene, which created an

_approximate

normal

_genetic

distribution within

_major

_genotypes.

The finite

_polygenic

mixed model with one

_major

locus is a closer

_{approximation}

of a mixed model

(Fernando

et

_al,

₁₉₉₄₎

than an

_{oligogenic model,}

which

_explains

inheritance

_by

a few

_independent

loci and estimates the effect of the each locus

separately

(Elston

and

_Stewart,

_1971).

Performance of the

_oligogenic

model or a

finite

_polygenic

mixed model with several

_major

loci was not

studied,

but

_might

have been better than the methods studied here when data are

_generated

from a finite number of loci.

The nature of

_polygenic

variance

_(ie,

the finite locus model versus the

mixed-generating

model)

had a

_significant

_impact

on _power of

_major

_gene detection. In the mixed

_model,

the

_polygenic

component

inherited

_by

_{progeny has} an

_expected

value

equal

to _{the average} of the

_polygenic

values of the

_parents

_(or

_midparent

_breeding

value),

which is not _{valid if any} of the genes

contributing

to the

_polygenic

component

are dominant. The

_discrepancy

of _progeny from the

_expected

_{midparent polygenic}

value increases with an increase in the relative

_magnitude

of dominant loci over all

_polygenic

loci. In

_addition,

with

_dominance,

the

_genetic

variance of

_offspring

conditional on

_parental

_polygenotype

is not

_equal

to half of the additive

_genetic

variance but also contains dominance

_variance,

which is

_relatively

_{large compared}

with additive variance when a

_large

recessive _gene with low

_frequency

_segregates

in the

population.

These

_{discrepancies}

from

_assumptions

of the mixed model should have a

_{negative impact}

on its _power in cases where data were simulated under a finite locus model

_compared

with a mixed

_generating

model.

_However,

no

_negative

effect on the _power was observed.

_Instead,

in most cases the _power was

_higher

under the finite locus model than under the

_{mixed-generating}

model

_{(table II).}

In the case of two loci with

_major

effect

_(table

_II,

set

₃₎

and to a lesser extent with

sets 1 and

_2,

the methods had a chance to detect either of the

_major

genes, which may

explain

the

higher

power under the finite locus model. In contrast, when the same situation was

_generated

_using

the mixed

_model,

a

_major

_gene

_explained

_only

a small

_proportion

of the total

_{genetic variance,}

the detection of the

_major

gene was difficult. Which of the _genes was detected as a

_major

_gene under the finite locus model was not

_{investigated,}

but based on intermediate estimates for _gene

frequency,

it seems that in some families the gene from the first locus was detected as a

_major

_gene, and in other families the _gene from the second locus

(or

other

loci)

was detected.

Linkage

between a

_major

_gene and

polygenes

reduced _power but did not have a

_large

_impact

on

_parameter

estimates if _{the linked genes} were not of

_equal

size and if the

parents

were a random

sample

from a

_population

in

_linkage

equilib-rium.

_Furthermore,

based on one simulation

_example,

violation of the

_assumption

of

_{Hardy-Weinberg}

_equilibrium

in the

_parental

_generation

reduced _power

substan-tially.

Therefore,

it is recommended to test a model that assumes

Hardy-Weinberg

equilibrium against

a model with free

_genotypic

_frequencies

for the

_parental

gener-ation.

The results

_given

here are restricted to data from

_independent

nucleus families. Based on results

_by

Fernando et al

_(1994),

the finite

_polygenic

mixed model

is a closer

approximation

of the mixed model under an

_example

data set with three

_generations

than PAP if data are

generated

with a mixed model. How these

methods

perform

under the finite locus model when information from more than

two

_generations

are available or when nucleus families are not

_independent

was not

studied.

_Thus,

the natural area for future studies is the

performance

of methods under

_{multigenerational}

data when data are

_generated

under the finite locus model. In

_conclusion,

both

_segregation

_analysis

_{methods studied here gave similar power}

polygenic

mixed model

_(or

_SALP)

in

_rejecting

the

_{polygenic model,}

but the finite

polygenic

mixed model _{gave estimates} with less bias than the mixed model. The finite locus model did not have a

_negative

effect on the _power

compared

with the mixed

_generating

model.

_Instead,

the power of the methods was often

_higher

under the finite locus model than when data were

_generated

under the mixed model.

Segregation

of two

_major

_{genes in} a

population

caused biased estimates.

_Linkage

had a

_negative

effect on the _power, but

_parameter

estimates remained unbiased if the

_parents

were a random

sample

from a

_large

_population

in

_{linkage equilibrium}

and if the

_major

_gene had a

substantially larger

effect on the trait than the other genes.

ACKNOWLEDGMENTS

This research was funded

_by

the Natural Sciences and

Engineering

Research Council of Canada and the

_Academy

of Finland which are

_{greatly acknowledged.}

We thank two anonymous reviewers for the comments on the _paper.

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