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Distribution of Model-based Multipoint Heterogeneity Lod

Scores

Chao Xing1,*, Nathan Morris2, and Guan Xing3

1Department of Clinical Sciences, McDermott Center of Human Growth and Development,

University of Texas Southwestern Medical Center, Dallas, Texas

2Department of Epidemiology and Biostatistics, Case Western Reserve University, Cleveland,

Ohio

3Bristol-Myers Squibb Company, Pennington, New Jersey

Abstract

The distribution of two-point heterogeneity lod scores (HLOD) has been intensively investigated

because the conventional χ2 approximation to the likelihood ratio test is not directly applicable.

However, there was no study investigating the distribution of the multipoint HLOD despite its

wide application. Here we want to point out that, compared with the two-point HLOD, the

multipoint HLOD essentially tests for homogeneity given linkage and follows a relatively simple

limiting distribution

examine the theoretical result by simulation studies.

, which can be obtained by established statistical theory. We further

Keywords

heterogeneity lod score; distribution; multipoint

Locus heterogeneity represents a form of genetic architecture of complex traits where alleles

at more than one locus lead to the same phenotype. It adversely affects the power of linkage

analysis if the heterogeneous disease genetic background of families is not taken into

account. A natural way to model such heterogeneous data is by a mixture model, as first

suggested by Smith [1963]. Under the mixture model framework one can either test for

homogeneity given linkage [Ott, 1983] or test for linkage allowing for heterogeneity [Hodge

et al., 1983] by a likelihood ratio test. The distribution of two-point heterogeneity lod scores

(HLOD) has been intensively investigated [Abreu et al., 2002; Chernoff and Lander, 1995;

Chiano and Yates, 1995; Faraway, 1993; Huang and Vieland, 2001; Lemdani and Pons,

1995; Liang and Rathouz, 1999] because the conventional χ2 approximation to the

likelihood ratio test is not directly applicable [Davies, 1977, 1987]. However, to our

surprise, there was no study investigating the distribution of the multipoint HLOD despite its

wide application. The multipoint HLOD is reported by popular software packages such as

GENEHUNTER [Kruglyak et al., 1996] and MERLIN [Abecasis et al., 2002] without a P-

value accompanying it. Here we want to point out that, compared with the two-point HLOD,

the multipoint HLOD essentially tests for homogeneity given linkage and follows a

© 2010 Wiley-Liss, Inc.

*Correspondence to: Chao Xing, MC 8591, UT Southwestern Medical Center, 5323 Harry Hines Blvd, Dallas, TX 75390.

chao.xing@utsouthestern.edu.

NIH Public Access

Author Manuscript

Genet Epidemiol. Author manuscript; available in PMC 2011 December 1.

Published in final edited form as:

Genet Epidemiol. 2010 December ; 34(8): 912–916. doi:10.1002/gepi.20535.

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relatively simple limiting distribution, which can be obtained by established statistical

theory. We further examine the theoretical result by simulation studies.

Denote by M the genotype data, by D the phenotype data, by α the admixture parameter,

which indicates the proportion of families linked to the locus tested, and by x the map

position of a putative disease locus. A general format of the likelihood for the ith family is

defined as, Li(M, D; x, α) = αLi(M, D; x)+(1 − α)Li(M, D; x = ∞), and the HLOD is defined

as HLOD = log10[sup L(α, x)/L0(α = 1; x = ∞)], where the subscript 0 denotes the null

hypothesis of no linkage. In two-point analysis, x is parameterized as the recombination

fraction, commonly denoted as θ ∈ [0,0.5], between the marker and disease loci by a map

function. The two-point analysis tests the hypotheses H0: θ = 0.5 versus H1: θ < 0.5. The

likelihood ratio test statistic is Ttwo-point = 2 ln[L(α̂, θ̂)/L(θ = 0.5)]. Note that when θ = 0.5, α

is unidentifiable, thus the conventional χ2 approximation to the likelihood ratio test is not

directly applicable [Davies, 1977, 1987], which causes difficulties in determining the

distribution of Ttwo-point. In multipoint analysis, under the null hypothesis x = ∞, which is

equivalent to θ = 0.5 in the two-point analysis by a map function; however, the alternative

hypothesis is that the putative disease locus is at a specific location c, i.e., x is not a free

parameter. The likelihood ratio test statistic is expressed as Tmultipoint = 2 ln[L(α̂, x = c)/L(α

= 1; x = ∞)], which indicates the multipoint analysis tests the hypotheses H0: α = 0 versus

H1: α > 0. Thus, the multipoint HLOD essentially tests for homogeneity given linkage, in

contrast to the two-point HLOD testing for linkage allowing for heterogeneity. The

multipoint HLOD test statistic corresponds to case 5 of Self and Liang [1987] with one

parameter on the boundary of parameter space, and also corresponds to example 11 of

Lindsay [1995] that tests H0: π = 0 versus H1: π > 0 in the mixture model (1 − π)f+πg, where

f and g are both known. Thus, following the theoretical arguments by both authors, the

asymptotic distribution of Tmultipoint is

at zero with probability one. It is of interest to note that, given linkage, the two-point

homogeneity test statistic 2 ln[L(α̂, θ̂)/L(α = 1, θ̂)] follows the same asymptotic distribution

as Tmultipoint [Ott, 1999].

, where denotes a distribution degenerate

Below we examine the theoretical result by simulation studies under varying simulation

models, analysis models, and sample sizes. We simulated nuclear families consisting of two

parents and four children, and two linked markers 10 cM apart with four alleles of equal

frequency at each locus. Two randomly chosen children were set to be affected and the other

two unaffected. The parental phenotypes were simulated under three models. In model I, one

parent was set to be affected and the other unaffected, which mimicked a dominant trait; in

model II, both parents were set to be unaffected, which mimicked a recessive trait; and in

model III, both parents were set to be unknown, which mimicked a trait with mode of

inheritance unclear. We generated 5,000 replicate samples under each of the four sample

size scenarios—100, 500, 1,000 and 5,000 families in a sample dataset. Denote the

penetrance by fi, where i ∈ {0,1,2} indicates the number of disease predisposing alleles.

Always assuming a sporadic rate of 0.01 and a disease predisposing allele frequency of 0.01,

we performed model-based multipoint linkage analysis on each dataset under 18 different

genetic models: (f0,f1,f2) = (0.01,0.1,0.1), (0.01,0.2,0.2),…,(0.01,0.9,0.9), (0.01,0.01,0.1),

(0.01,0.01,0.2),…, (0.01,0.01,0.9) denoted D1,D2,…,D9, R1,R2,…,R9, respectively, and

recorded the multipoint HLOD at the middle of the two markers. The linkage analysis was

performed using MERLIN [Abecasis et al., 2002], in which the mixture likelihood was

maximized using the Brent’s method [2002].

The simulation results confirmed the limiting distribution of Tmultipoint (= 2 × ln 10 ×

multipoint HLOD) to be . In Table I we estimated some parameters of the

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empirical distribution under analysis models D2, D5, D8, R2, R5, and R8 with varying

sample sizes in each simulation scenario. The proportion of Tmultipoint equal to zero under

different models always approximated to a half. The mean and variance of a random

variable of

are 1 and 2, respectively. We observed the mean and variance of non-zero

Tmultipoint approximated to 1 and 2, respectively, under most dominant models and high

penetrance recessive models when the sample size was large. Under each analysis model the

mean and variance approached their expectations under the theoretical distribution as the

sample size increased. However, both mean and variance were smaller than their

expectations under low penetrance recessive models even if the sample size was 5,000. In

contrast, the maximum likelihood estimate of mean approximated to 1 when sample size

was 500 under any analysis model. Given the sample size and analysis model, the estimated

mean of Tmultipoint for data generated under simulation model I approximated to 1 closer

than that for data generated under models II and III, which was mostly clearly illustrated

when analyzing the data by low penetrance recessive models. We calculated the empirical

type I error rate of the multipoint HLOD under analysis models D2, D5, D8, R2, R5, and R8

with varying sample sizes in each simulation scenario assuming that Tmultipoint follows a

distribution of

distribution as summarized in Table I. The dominant analysis models gave proper type I

error rate. The recessive models were conservative except for high penetrance models with

large sample size. Figure 1 illustrated the empirical distribution of the multipoint HLOD

under analysis models D5 and R5 with the theoretical distribution.

(Table II). The results were consistent with the empirical

The multipoint HLOD method is powerful to detect linkage even when the assumed

heterogeneity model is incorrect [Greenberg and Abreu, 2001; Hodge et al., 2002].

However, the distribution of the multipoint HLOD has remained mysterious, possibly

because it has been obscured by the complexity of the two-point HLOD. Similarly, the

model-based multipoint lod score that is best suited in an evidential paradigm [Hodge et al.,

2008] displays some asymptotic complexity and does not have a limiting distribution [Xing

and Elston, 2006]. We have shown in this paper that, in contrast with the two-point HLOD

and the multipoint lod score, the multipoint HLOD test statistics follows a relatively simple

asymptotic distribution. That is, 2 × ln 10 × multipoint HLOD follows an asymptotic

distribution of

also facilitates further inferences such as multiple testing correction. The rate of

convergence to asymptotic distribution depends on the informativeness of both markers and

the trait. Given data, the pre-specified analysis model defines the informativeness of the

trait. As the model-based linkage statistics generally do, the multipoint HLOD under low

penetrance recessive models is conservative because the phenotype contributes little

information under such trait models, which is reflected as a high proportion of zeros, low

mean and small variance (Table I). Similarly, data simulated under model I contain more

trait information than that simulated under models II and III; therefore, the multipoint

HLOD under simulation models II and III is more conservative than that under simulation

model I. In multipoint analysis, the marker information is relatively constant across a map;

thus, the behavior of Tmultipoint should also be relatively stable across the map given an

analysis model. We note that the proportion of HLODs equal to zero is always greater than,

though close to, one half; thus, a nominal P-value is presumably conservative. In this study

we did not investigate the performance of the multipoint HLOD test statistic in the extreme

tails of its null distribution, which will be crucial in determining genome-wide significance.

Considering the efficiency of the test depends on multiple factors such as the true, yet

unknown, disease model, analysis model employed, and sample size, when a large

. This not only enables evaluating the significance level easily, but

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multipoint HLOD is observed in reality, it would be more appropriate to perform Monte

Carlo simulations to evaluate the significance level [Lin and Zou, 2004].

Acknowledgments

We thank Dr. Robert Elston for critically reading the manuscript and helpful discussions, and thank Dr. Gonçalo

Abecasis for clarification and advice on the maximization procedure in MERLIN. C.X. was partially supported by a

Pilot Award from UL1RR024982 from the National Center for Research Resources.

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Fig. 1.

Empirical distribution of multipoint HLOD compared with the theoretical distribution under

the null hypothesis of no linkage. (A) and (C) correspond to the probability density functions

of HLOD analyzed under models D5 and R5, respectively; (B) and (D) correspond to the

cumulative distribution functions of HLOD analyzed under models D5 and R5, respectively.

Solid line represents the theoretical distribution of HLOD assuming Tmultipoint follows a

distribution of

HLOD based on results from 5,000 replicates, each of which contained 5,000 families

simulated under simulation model III—parental phenotype unknown.

; dashed line and histogram represent the empirical distribution of

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TABLE I

Proportion of zeros among Tmultipoint and empirical distribution of non-zero Tmultipointa

Analysis model

Sample size

D2

D5

D8

R2

R5

R8

Simulation model Ib

100

Zero %

51.0

52.0

52.8

52.1

51.8

52.3

Meanc

1.08

1.08

1.06

0.20

0.80

1.01

Variance

2.28

2.23

2.15

0.03

0.74

1.77

dfc

1.08

1.09

1.08

0.75

1.00

1.08

500

Zero %

50.4

51.1

51.9

51.8

51.8

51.7

Mean

1.02

1.02

1.03

0.39

1.03

1.05

Variance

2.05

1.98

1.97

0.11

1.82

2.23

df

1.06

1.08

1.09

0.92

1.08

1.08

1,000

Zero %

51.4

52.5

53.0

53.2

53.7

53.8

Mean

1.05

1.05

1.04

0.52

1.06

1.03

Variance

2.05

2.09

2.04

0.22

1.95

1.87

df

1.09

1.09

1.07

0.98

1.09

1.08

5,000

Zero %

51.1

51.7

51.8

51.8

51.7

52.0

Mean

1.05

1.04

1.02

0.88

1.08

1.08

Variance

1.95

1.89

1.91

0.90

2.08

2.13

df

1.08

1.08

1.07

1.04

1.09

1.10

Simulation model IIb

100

Zero %

52.1

52.8

53.5

51.6

52.8

53.7

Mean

1.04

1.04

1.03

0.17

0.71

1.00

Variance

1.90

1.93

1.90

0.02

0.50

1.52

df

1.08

1.08

1.08

0.72

0.96

1.09

500

Zero %

51.4

51.7

51.8

52.1

52.0

52.5

Mean

1.03

1.05

1.06

0.35

0.98

1.07

Variance

1.94

2.10

2.24

0.09

1.50

2.26

df

1.08

1.10

1.08

0.89

1.06

1.09

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Analysis model

Sample size

D2

D5

D8

R2

R5

R8

1,000

Zero %

51.8

52.7

53.3

52.9

53.0

53.4

Mean

1.04

1.05

1.04

0.47

1.04

1.05

Variance

2.06

2.08

2.08

0.17

1.85

1.95

df

1.07

1.07

1.06

0.96

1.09

1.10

5,000

Zero %

50.7

51.1

52.3

51.5

51.3

51.8

Mean

1.04

1.03

1.05

0.82

1.08

1.08

Variance

1.97

2.03

2.09

0.74

2.04

2.06

df

1.07

1.07

1.09

1.01

1.07

1.08

Simulation model IIIb

100

Zero %

51.8

52.7

53.7

52.4

53.3

54.0

Mean

1.05

1.05

1.04

0.17

0.71

0.99

Variance

2.26

2.27

2.24

0.02

0.49

1.46

df

1.06

1.05

1.04

0.72

0.97

1.08

500

Zero %

51.0

51.1

52.2

51.6

51.9

52.1

Mean

1.04

1.03

1.03

0.36

0.99

1.04

Variance

2.15

2.13

2.10

0.09

1.54

2.07

df

1.07

1.06

1.06

0.88

1.06

1.07

1,000

Zero %

52.0

52.6

52.9

52.6

53.0

53.3

Mean

1.03

1.02

1.00

0.47

1.04

1.05

Variance

2.11

1.94

1.82

0.17

1.84

1.90

df

1.07

1.06

1.05

0.95

1.09

1.10

5,000

Zero %

50.7

51.1

52.1

51.4

51.1

51.5

Mean

1.03

1.03

1.05

0.81

1.07

1.06

Variance

1.96

2.03

2.09

0.74

2.09

2.10

df

1.06

1.06

1.08

1.00

1.07

1.07

aResults based on 5,000 replicates.

bSimulation models I, II, and III correspond to parental phenotypes of one affected the other unaffected—a dominant model, both unaffected—a recessive model, and both unknown—an ambiguous model,

respectively.

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cAssuming the non-zero Tmultipoint follow a χ2 distribution, “mean” and “df” (degree of freedom) correspond to the mean estimates by the method-of-moments and the maximum likelihood estimation,

respectively.

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TABLE II

Empirical type I error ratesa for multipoint HLOD

Nominal levelb

Analysis model

Sample size

D2

D5

D8

R2

R5

R8

Simulation model Ic

100

0.05

0.0530

0.0530

0.0502

–

0.0004

0.0472

0.01

0.0124

0.0118

0.0100

–

–

0.0064

0.001

0.0012

0.0010

0.0010

–

–

0.0004

0.0001

0.0004

0.0004

0.0002

–

–

–

500

0.05

0.0502

0.0460

0.0470

–

0.0466

0.0498

0.01

0.0110

0.0094

0.0100

–

0.0088

0.0096

0.001

0.0012

0.0012

0.0006

–

0.0004

0.0018

0.0001

–

0.0004

0.0002

–

–

0.0002

1,000

0.05

0.0516

0.0496

0.0500

0.0002

0.0490

0.0498

0.01

0.0096

0.0104

0.0094

–

0.0100

0.0088

0.001

0.0010

0.0008

0.0010

–

0.0004

0.0006

0.0001

0.0002

0.0002

0.0002

–

–

–

5,000

0.05

0.0552

0.0512

0.0506

0.0272

0.0518

0.0502

0.01

0.0080

0.0082

0.0094

0.0002

0.0102

0.0108

0.001

0.0008

0.0006

0.0006

–

0.0012

0.0012

0.0001

0.0004

–

–

–

0.0004

0.0002

Simulation model IIc

100

0.05

0.0470

0.0534

0.0484

–

0.0086

0.0464

0.01

0.0088

0.0092

0.0078

–

–

0.0058

0.001

0.0006

0.0008

0.0004

–

–

–

0.0001

–

–

–

–

–

–

500

0.05

0.0506

0.0506

0.0504

–

0.0500

0.0508

0.01

0.0094

0.0112

0.0112

–

0.0056

0.0118

0.001

0.0010

0.0012

0.0016

–

–

0.0016

0.0001

–

0.0004

0.0006

–

–

–

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Nominal levelb

Analysis model

Sample size

D2

D5

D8

R2

R5

R8

1,000

0.05

0.0508

0.0468

0.0476

–

0.0532

0.0502

0.01

0.0100

0.0114

0.0118

–

0.0084

0.0094

0.001

0.0008

0.0010

0.0014

–

0.0002

0.0008

0.0001

–

–

–

–

–

–

5,000

0.05

0.0500

0.0488

0.0498

0.0172

0.0544

0.0538

0.01

0.0094

0.0102

0.0110

0.0004

0.0128

0.0120

0.001

0.0006

0.0010

0.0012

–

0.0008

0.0008

0.0001

–

0.0002

–

–

0.0002

0.0002

Simulation model IIIc

100

0.05

0.0506

0.0498

0.0468

–

0.0072

0.0452

0.01

0.0132

0.0128

0.0108

–

–

0.0046

0.001

0.0012

0.0012

0.0016

–

–

–

0.0001

0.0002

0.0002

–

–

–

–

500

0.05

0.0518

0.0480

0.0450

–

0.0468

0.0476

0.01

0.0112

0.0114

0.0102

–

0.0066

0.0104

0.001

0.0022

0.0016

0.0018

–

–

0.0010

0.0001

–

–

–

–

–

–

1,000

0.05

0.0500

0.0442

0.0450

–

0.0518

0.0492

0.01

0.0102

0.0112

0.0096

–

0.0086

0.0092

0.001

0.0012

0.0006

0.0006

–

0.0002

0.0008

0.0001

0.0002

–

–

–

–

–

5,000

0.05

0.0492

0.0488

0.0494

0.0170

0.0538

0.0530

0.01

0.0094

0.0102

0.0110

0.0002

0.0120

0.0112

0.001

0.0006

0.0010

0.0012

–

0.0010

0.0008

0.0001

–

0.0002

–

–

0.0002

0.0002

aResults based on 5,000 replicates. “–” means no HLOD greater than the corresponding critical value.

bThe nominal levels of 0.05, 0.01, 0.001 and 0.0001 approximately correspond to the multipoint HLOD of 0.59, 1.17, 2.07, and 3.00, respectively under the assumption that Tmultipoint follows a

distribution of

.

Genet Epidemiol. Author manuscript; available in PMC 2011 December 1.

Page 12

NIH-PA Author Manuscript

NIH-PA Author Manuscript

NIH-PA Author Manuscript

Xing et al.Page 12

cSimulation models I, II, and III correspond to parental phenotypes of one affected, the other unaffected—a dominant model, both unaffected—a recessive model, and both unknown—an ambiguous model,

respectively.

Genet Epidemiol. Author manuscript; available in PMC 2011 December 1.