Copyright � 2009 by the Genetics Society of AmericaDOI: 10.1534/genetics.108.098913

Binary Trait Mapping in Experimental Crosses With Selective Genotyping

Ani Manichaikul*,1 and Karl W. Broman†

*Department of Biomedical Engineering, University of Virginia, Charlottesville, Virginia 22908 and †Department of Biostatistics andMedical Informatics, University of Wisconsin, Madison, Wisconsin 53706

Manuscript received November 19, 2008Accepted for publication April 21, 2009

ABSTRACT

Selective genotyping is an efficient strategy for mapping quantitative trait loci. For binary traits, wherethere are only two distinct phenotypic values (e.g., affected/unaffected or present/absent), one mayconsider selective genotyping of affected individuals, while genotyping none or only some of theunaffecteds. If selective genotyping of this sort is employed, the usual method for binary trait mapping,which considers phenotypes conditional on genotypes, cannot be used. We present an alternativeapproach, instead considering genotypes conditional on phenotypes, and compare this to the morestandard method of analysis, both analytically and by example. For studies of rare binary phenotypes, werecommend performing an initial genome scan with all affected individuals and an equal number ofunaffecteds, followed by genotyping the full cross in genomic regions of interest to confirm results fromthe initial screen.

WE consider the problem of mapping genetic locicontributing to a binary trait in an experimental

cross with selective genotyping. There are two clearapproaches for linkage analysis with a binary trait.Typically, we compare the proportion of affectedindividuals across genotype groups (Xu and Atchley

1996). Alternatively, we can compare genotype fre-quencies between affected and unaffected individuals,similar to Henshall and Goddard (1999). Beyondthese two basic approaches, binary trait mapping hasseen fundamental advances in regression models(McIntyre et al. 2001; Deng et al. 2006), extensionsto multiple-QTL mapping (Coffman et al. 2005; Chen

and Liu 2009), and the development of Bayesianalgorithms (Yi and Xu 2000; Huang et al. 2007).However, the original data structure and approach haveremained intact. Existing methods for binary traitmapping largely require the availability of genotypeand phenotype data for a representative sample of bothaffected and unaffected individuals, and we have notyet seen a well-developed framework for binary traitmapping in the presence of selective genotyping.

It is not uncommon to see genotype data on affectedindividuals only, in which case the above methodscannot be used. Instead, we can compare observedgenotype frequencies to the expected segregation ratiosgiven the cross type, in a test for segregation distortion(see Faris et al. 1998; Lambrides et al. 2004). Forexample, the expected segregation proportions for anintercross are 1:2:1. The observed genotypes can then

be described by a multinomial model, and statisticallysignificant deviation from the expected segregationratios among the genotyped affected individuals wouldsuggest genotype–phenotype association. Gene map-ping approaches that model genotypes rather thanphenotypes have been developed extensively in theanalysis of affected human relative pairs (see, forexample, Risch 1990; Holmans 1993; Hauser andBoehnke 1998). In the analysis of experimental crosses,however, this type of approach has been developedprimarily for the identification of monogenic mutants(Moran et al. 2006).

Once all affected individuals are genotyped, an in-vestigator may go on to genotype unaffected individuals.With this genotyping strategy in mind, we presentseveral potential methods of analysis that might beapplied in this context. First, we consider a standardanalysis of the genotyped individuals, with diseaseproportions compared across genotype groups (Xu

and Atchley 1996). Having omitted ungenotypedindividuals, this method of analysis appears invalidbecause the estimated disease proportions are biasedupward, reflecting an overrepresentation of affecteds inthe set of genotyped individuals under consideration.As an alternative, we develop a reverse approach withgenotype frequencies compared across phenotypegroups. Because selective genotyping does provide arepresentative sample of genotypes for each phenotypegroup, this reverse approach does not face the bias inparameter estimation seen with the standard approach.We further extend the reverse approach to incorporatea segregation assumption, as is necessary for an affec-teds only analysis. Finally, we present a full-likelihoodanalysis accounting for selective genotyping, similar to

1Corresponding author: Department of Biomedical Engineering, Univer-sity of Virginia, Box 800759, Health System, Charlottesville, VA 22908.E-mail: amanicha@virginia.edu

Genetics 182: 863–874 ( July 2009)

that suggested by Lander and Botstein (1989) forquantitative traits. We develop the full-likelihood ap-proach both with and without incorporating an assump-tion on the genotype segregation proportions.

Having put forth each of these methods, we deriveanalytic relationships among them. These relationshipsprovide important insight regarding application of thepresented methods under selective genotyping. Mostnotably, we find that making a segregation assumptioncan lead to spurious evidence of a QTL, but is necessaryto treat the case of affecteds only genotyping. Wedemonstrate properties of the methods in an analysisof recovery from infection by Listeria monocytogenes inintercross mice and further compare power of themethods through computer simulations. Finally, wesynthesize our analytical and simulation results to offermore general suggestions for the analysis of binary traitdata with selective genotyping.

METHODS

For simplicity, we present methods for the case of abackcross. For analysis, we consider backcross dataconsisting of binary phenotypes (affected or unaf-fected) for all individuals and marker genotypes (AAor AB) on all affecteds and all, some, or none of theunaffecteds. We present methods of analysis to addressthese three genotyping strategies for binary trait data.The observed data are represented in Table 1.

Throughout our description of the methods, we usethe following notation. Let N be the total number ofindividuals in the cross, with nobs genotyped individualsand nmis¼ N� nobs ungenotyped individuals. We assignthe indexes i ¼ 1, . . . , N such that individuals i 2 {1, . . . ,nobs} are genotyped, and the remaining individuals areungenotyped. Let Di ¼ 1 or 0 according to whetherindividual i is affected or unaffected. We take Gi 2 {AA,AB} to denote the underlying unobserved genotype atthe putative QTL of interest, while Omi 2 {AA, AB, � }denotes the observed genotype at marker m, with ‘‘ � ’’indicating a missing value.

Standard approach: In dealing with selective geno-typing, one possible approach is to ignore individuals

without genotype data, performing an analysis ofgenotyped individuals only. Once we have omittedindividuals with missing genotypes from our analysis,we can use the approach of Xu and Atchley (1996) tolook for genotype–phenotype association by testing fora difference in disease probability across genotypes.

Let pAA ¼ Pr(Di ¼ 1 j Gi ¼ AA) and pAB ¼ Pr(Di ¼1 j Gi ¼ AB) denote the penetrance values (the condi-tional phenotype probabilities given the genotype at aputative QTL), and let p� ¼ Pr(Di ¼ 1) denote themarginal phenotype probability (i.e., the prevalence ofdisease).

Similar to standard interval mapping for quantitativetraits (Lander and Botstein 1989), the approach of Xu

and Atchley (1996) makes use of the conditional QTLgenotype probabilities given the full set of multipointmarker data for the ith individual, pig ¼ Pr(Gi ¼ g jO�i).By convention, evidence against the null hypothesis ofgenotype–phenotype independence, Pr(D jG)¼ Pr(D),in favor of the alternative hypothesis of a QTL, Pr(D jG) 6¼Pr(D), is presented as the log10-likelihood ratio

LODF¼ log10

maxpAA ;pABPrðD jO��; pAA;pABÞ

maxp� PrðD; p�Þ

� �

¼ log10

Qi

Pg2fAA;ABg½pig � ðp̂g ÞDi � ð1� p̂g Þ1�Di �Q

iðp̂�ÞDi � ð1� p̂�Þ1�Di

( );

where we model affection status as a Bernoulli randomvariable with a common probability under the null andwith genotype-specific probabilities under the alterna-tive hypothesis. Assuming no missing genotype data forreasons other than the selective genotyping, and nogenotyping error, the maximum-likelihood estimates(MLEs) at the markers are simply sample proportions.Between markers, we can perform interval mapping byan EM algorithm (Dempster et al. 1977), which hasbeen previously described for this application (Xu andAtchley 1996; Broman 2003).

The forward approach using LODF is appropriate inthe case that we have genotyped all individuals. How-ever, if we have done selective genotyping with regard tophenotypes, the approach will yield biased and incon-sistent estimates of pAA and pAB. As a result, the validityof this approach for selective genotyping is not imme-diately apparent.

Reverse approach, conditioning on phenotypes: Asan alternative, we can also look for genotype–phenotypeassociation by reversing the standard approach andinstead modeling genotypes conditional on pheno-types. This approach is technically quite similar to thelogistic regression model presented by Henshall andGoddard (1999), but we present it in a framework thatelucidates its relationship with the standard approach ofXu and Atchley (1996). Placing the reverse approachin this likelihood framework also allows it to be easilyadapted for analysis of affecteds only, as will be seen

TABLE 1

Data at a single genetic marker from a backcross experimentwith binary phenotypes

Genotype

AA AB Missing Total

PhenotypeAffected nAA,D nAB,D nmis,D(¼ 0) nD

Unaffected nAA;D̃ nAB;D̃ nmis;D̃ nD̃

Total nAA nAB nmis ð¼ nmis;D̃Þ N

All affecteds are genotyped, while some or all unaffectedsmay be left ungenotyped.

864 A. Manichaikul and K. W. Broman

below. Again, we consider only genotyped individualsand omit the rest from our analysis.

Let fD ¼ Pr(Gi ¼ AA j Di ¼ 1) and fD̃ ¼ PrðGi ¼AA j Di ¼ 0Þ denote the affection-status-specific proba-bilities of the AA genotype at the putative QTL ofinterest, and let f� ¼ Pr(Gi ¼ AA) denote the marginalprobability of the AA genotype. (For an intercross,we must handle the three possible genotypes {AA, AB,BB}, and so we would consider the vector f� ¼PrðGi ¼ AAÞPrðGi ¼ ABÞ½ �T and analogous vectors for

fD and fD̃.)We calculate the LOD score measuring support for a

QTL as the log10-likelihood ratio comparing evidencefor the alternative hypothesis, Pr(D j G) 6¼ Pr(D) [orequivalently, Pr(G j D) 6¼ Pr(G)], in favor of the nullhypothesis of independence. Here, we model genotypesat the putative QTL using a Bernoulli process (ormultinomial for an intercross) with a common proba-bility under the null and with disease-status-specificprobabilities under the alternative hypothesis (seeTable 2).

To allow analysis at both marker and nonmarkerlocations, we perform interval mapping using an EMalgorithm to calculate the necessary MLEs. Analogousto the pig for standard interval mapping, we make use ofthe reverse quantities, qig¼ Pr(O�i jGi¼ g), probabilitiesof the full set of observed marker data given a specifiedvalue of the underlying QTL genotype, g. We havedeveloped hidden Markov models to obtain the qig;details are provided in appendix a.

Reverse approach, modified: Having presented thereverse approach above, a simple modification allows usto incorporate knowledge regarding the structure of thecross. In particular, we may specify the null hypothesisvalue of f� to be 1

2 for a backcross (or f� ¼ 14

12

� �Tfor

an intercross). This prior knowledge is crucial in theanalysis of affecteds only, for which it is infeasible tosimply check for a difference in genotype proportionsacross phenotype groups, as was done above.

Here, the modified LOD score, LODR;seg ¼ log10

maxfD ;fD̃PrðO�� j D; fD ; fD̃Þ=PrðO��; f� ¼ 1

2 Þ� �

, quanti-fies evidence for segregation distortion, i.e., deviation ofobserved genotype counts from their expected distri-bution. In an affecteds only analysis, this view ofevidence suggests genotype–phenotype associationand so indicates the presence of a QTL.

Since the alternative hypothesis remains the same asin the original reverse approach, LODR, the MLEs forfD and fD̃ may be obtained in the same way as describedabove. Note that a reasonable approach to take withthis method is to constrain the conditional geno-type probabilities such that their weighted average,fD � p�1 fD̃ � ½1� p��, is equal to the marginal genotypeprobability, f�. However, we use the unconstrained valuein calculating LODR,seg, incorporating the constraintthrough a full-likelihood analysis developed furtherbelow.

Full-likelihood analysis: Performing a full-likelihoodanalysis allows us to forgo conditioning on eithergenotypes or phenotypes. Conceptually, this modelmakes complete use of the available data in assessingevidence of a QTL. An apparent advantage of thisapproach is that ungenotyped individuals can be in-cluded in the likelihood. However, careful examinationin the results below shows that full-likelihood analysisyields results quite similar to those of both the forwardand reverse approaches.

The full-likelihood function models the joint proba-bility of disease status and observed genotypes. We writethe full likelihood of a QTL at the putative site ofinterest in terms of parameters fD, fD̃, and p� as

likðfD ;fD̃;p�Þ ¼YNi¼1

PrðDi ;O�i ; fD ;fD̃;p�Þ

¼ likðfD ;fD̃ jDÞ � likðp�Þ: ð1Þ

Since the full likelihood can be decomposed intoorthogonal components to separate fD and fD̃ from p�

TABLE 2

Summary of forward and reverse approaches and likelihood functions

Hypothesis MLEs Likelihood

ForwardAlternative pAA 6¼ pAB p̂AA ¼ nD;AA

nAAlikðp̂AA; p̂AB j O��Þ ¼

Qi

Pg2fAA;ABg½pig � ðp̂g ÞDi � ð1� p̂g Þ1�Di �

p̂AB ¼ nD;AB

nAB

Null p� ¼ pAA ¼ pAB p̂� ¼ nD;AA 1 nD;AB

nobslikðp̂�Þ ¼

Qiðp̂�Þ

Di � ð1� p̂�Þ1�Di

ReverseAlternative fD 6¼ fD̃ f̂D ¼

nD;AA

nD;AA 1 nD;ABlikðf̂D ; f̂D̃ j DÞ ¼

Qi

Pg2fAA;ABg½qig � PrðGi ¼ g j Di ; fD ;fD̃Þ�

f̂D̃ ¼nD̃;AA

nD̃;AA 1 nD̃;AB

Null f� ¼ fD ¼ fD̃ f̂� ¼ nAA

nobslikðf̂�Þ ¼

Qi

Pg2fAA;ABg �PrðGi ¼ g ; f�Þ

Modified Null f� ¼ fD ¼ fD̃ f̂� ¼ 12 likðf̂� ¼ 1

2 Þ

Analytical maximum-likelihood estimates (MLEs) are stated for marker locations.

Binary Trait Mapping 865

(see appendix b for details), the resulting MLEs aresimply those obtained by performing the maximizationseparately. At the markers, these are again the appro-priate sample averages as specified in the previoussections above. Estimating the position of a QTL doesnot depend on the value p� since this parameterestimate is fixed across the genome.

Constrained full likelihood: Analogous to the modifiedreverse approach that incorporates evidence for segre-gation distortion, we can also perform full-likelihoodanalysis under the assumption that marginal genotypeprobabilities should follow their null segregation values.The resulting full-likelihood function is the same as thatspecified above, but subject to the constraint thatdisease-status-specific genotype probabilities averageto a marginal value of f� ¼ 1

2 . We write this constraintin terms of the overall probability of disease, p�:

f� ¼ fD � p�1 fD̃ � ½1� p�� ¼1

2: ð2Þ

(For the case of an intercross, f� ¼ 14

12

� �T, so the

equation above becomes a two-component constraint.)Analysis is performed using all N individuals in the cross,both genotyped and ungenotyped. The constrainedfull-likelihood LOD score is written as

LODNFull;seg ¼ log10

maxfD ;fD̃;p� jf�¼1=2 likðfD ;fD̃;p�Þmaxp� likðf� ¼ 1=2;p�Þ

� �:

Under the null hypothesis, the MLE p̃� ¼ nD=N is thesame as in the absence of the constraint, while f̃� ¼ 1

2according to the segregation assumption. To maximizethe constrained likelihood under the alternative hy-pothesis, we use an EM algorithm, described in appen-

dix c.Significance thresholds: After performing a genome

scan using any of the methods presented above, we canmake use of significance thresholds in reporting statis-tical significance for genomic regions of interest. Thesignificance thresholds must account for multiplecomparisons arising in the complete genome scan. Atypical way to perform this adjustment while controllingthe rate of detecting false positive QTL is to examine thedistribution of the genomewide maximum LOD scoreunder the global null hypothesis of no QTL.

For standard interval mapping, significance thresh-olds conditioning on observed genotypes and pheno-types may be obtained empirically by permutation(Churchill and Doerge 1994), shuffling phenotypeswhile keeping genotypes fixed to approximate the nulldistribution of the genomewide maximum LOD score.

In the case of methods incorporating a segregationassumption, such as LODR,seg, it may not make sense tocondition on observed genotypes. For example, in anaffecteds only analysis, the observed genotypes contain

all of the information we use to test for linkage. If wecondition on those observed genotypes in calculatingthe null distribution, we effectively condition out anyevidence in the data. Put more simply, we cannot shufflephenotypes in an affecteds only analysis because allindividuals have the same phenotype. Instead, we canestimate the null distribution by simulation using agene-dropping approach (MacCluer et al. 1986) tosimulate new genotypes preserving the cross structureand pattern of missing genotypes from the original dataset. The resulting significance thresholds are reportedconditional on observed phenotypes, while averagingacross possible sets of genotypes given the cross used togenerate the data. Since the simulation-based signifi-cance thresholds do not condition on the observedgenotypes, they are appropriate for analyses in which wehave incorporated evidence for segregation distortionor deviation from expected segregation ratios. Wedescribe this form of evidence more explicitly in theresults below.

RESULTS

We have presented approaches to calculating like-lihoods and corresponding likelihood ratios to assessgenotype–phenotype association for binary trait map-ping in the presence of selective genotyping. Since all ofthese methods are likelihood based, they are closelyrelated. In this section, we highlight key relationshipsbetween the various approaches. We summarize allpresented relationships at the end of this section.

Reverse approach vs. modified reverse approach:There is a direct relationship between LODR, in whichwe use the MLE f̂� to calculate the null likelihood, andLODR,seg, in which we assume f� ¼ 1

2 according to theexpected genotype frequencies in a backcross,

LODR;seg ¼ log10

likðf̂D ; f̂D̃ jD;Om�Þlikðf� ¼ 1

2 jOm�Þ

( )

¼ log10

likðf̂D ; f̂D̃ jD;Om�Þlikðf̂� jOm�Þ

3likðf̂� jOm�Þ

likðf� ¼ 12 ; Om�Þ

( )

¼ LODR 1 LODseg:dist:;

where LODseg:dist: ¼ log10 likðf̂� jOm�Þ=likðf� ¼ 12 jOm�Þ

� �quantifies evidence for segregation distortion or de-viation of the observed genotypes from the assumedsegregation proportion, f� ¼ 1

2 .Full likelihood vs. reverse approach: Let LODnobs

Full bethe full-likelihood LOD score based on genotypedindividuals only and LODN

Full be the correspondingLOD score obtained from all individuals whether geno-typed or not. The LOD score to test for genotype–phenotype association based on the reverse approach isclosely related to the corresponding LOD based on full-likelihood analysis. Recall that the full likelihood can be

866 A. Manichaikul and K. W. Broman

decomposed as a product of the likelihood for parame-ters fD ; fD̃ and the likelihood for p�, as shown in (1)above. So, we can factor out the likelihood for phenotypeprobability from the full likelihood to relate it back to thereverse approach. Because this factorization applies atboth marker and nonmarker locations, the relationshipLODnobs

Full ¼ LODNFull ¼ LODR holds quite generally even in

the presence of missing genotype data or genotypingerror (see appendix d for details).

Forward vs. reverse approach: The forward andreverse methods of analysis are closely related as theyare both likelihood-ratio-based tests of independence.For the case of no missing genotypes or genotypingerror, we can examine the relationship analytically atmarker locations

LODF ¼ log10

PrðD jO��; p̂AA; p̂ABÞPrðD; p̂�Þ

� �

¼ log10

PrðD jO��; p̂AA; p̂ABÞ � PrðO��; f̂�ÞPrðD; p̂�Þ � PrðO��; f̂�Þ

� �ð¼ LODnobs

FullÞ

¼ LODR;

where the estimated parameters p̂AA; p̂AB , and f̂� are theMLEs as specified in the methods section above. Notethat these relationships apply only at marker locations,in the case of no missing genotypes. It is in this specialcase that all MLEs are obtained as sample averages, andso the computed likelihood ratio is the same whetherconditional on genotypes or phenotypes. At nonmarkerlocations, we must employ the relationships in (A6) toobtain the full-likelihood MLEs, p̂AA; p̂AB , and f̂�, sothese values could differ from those obtained by thestandard approach, LODF.

Overall relationships: Here, we summarize the rela-tionships among the proposed methods presentedhere, together with those shown in appendix d. At themarkers we have the following relationships:

LODnobsFull ¼ LODF ¼ LODR

LODR;seg ¼ LODF 1 LODseg:dist::

The following relationships hold more generally atboth marker and nonmarker locations:

LODnobsFull ¼ LODN

Full ¼ LODR

LODNFull;seg # LODR;seg ¼ LODR 1 LODseg:dist:

That LODR agrees with full-likelihood analysis, whetheror not we include ungenotyped individuals in theanalysis, suggests it is unnecessary to perform full-likelihood analysis, since we can get the exact sameresults using the simpler reverse approach. However, wealso see that the modified reverse approach incorpo-rates evidence for segregation distortion, which can be

irrelevant to linkage if we have genotyped both affectedsand unaffecteds. Hence, full-likelihood analysis may stillbe necessary to incorporate the segregation assumptionwhile avoiding spurious evidence, as in LODN

Full;seg.

APPLICATION

To demonstrate features of our proposed methods,we perform analysis of recovery from L. monocytogenesinfection in 116 mice from an intercross of the resistantstrain C57BL/6ByJ and the susceptible strain BALB/cByJ (Boyartchuk et al. 2001), using the data setavailable in the R/qtl package (Broman et al. 2003).In our analysis, we make use of genotypes at 131 geneticmarkers on the 19 autosomes. Although phenotypeswere recorded as survival times in the original study, weconverted them to binary values to demonstrate appli-cation to our proposed methods. (Note that analyzingsurvival data as binary values is only one possible strategyin handling survival data; see Broman 2003 for a morecomplete treatment of this particular data set.) Accord-ingly, binary phenotypes were calculated to indicatewhether or not mice survived to 264 hr followinginfection. Among the 116 phenotyped mice in this dataset, 35 survived and 81 died within 264 hr. Since survivalis the rarer phenotype in this cross, we refer to the 35survivors as affected individuals. With the full data, anappropriate analysis would use the standard approach,LODF, with the available full genotypes. To explore theset of possible genotyping strategies using real data, wesubset the available genotypes to create two additionalversions of this data set for analysis: one data set withgenotypes on affecteds only and another with equalnumbers of affecteds and unaffecteds genotyped. Ineach case, we apply methods of analysis appropriate forthe genotyping strategy at hand.

Since some of the presented methods are sensitive tosegregation distortion, we note in advance that chro-mosome 13 showed the strongest evidence of overalldeviation from the expected genotype proportions. Inparticular, the marker D13M233 had genotype segrega-tion proportions of 40:41:21 rather than the expected1:2:1, giving a segregation distortion LOD score of 2.97(P ¼ 0.097 by gene-dropping simulation with 10,000replicates). In contrast, the marker D2M365 on chro-mosome 2 showed little segregation distortion, withsegregation proportions of 24:59:27 and a segregationdistortion LOD score of 0.12.

Standard analysis: We first consider a standardanalysis with full genotypes. In this case, it is appropriateto perform standard interval mapping using the forwardapproach, LODF, conditioning phenotypes on geno-types. In calculating our LOD curve, we use an EMalgorithm, as implemented in R/qtl (Broman et al.2003), with genotype probabilities calculated every1 cM. We use a permutation test (Churchill and

Binary Trait Mapping 867

Doerge 1994) to assign statistical significance andobtain (with 10,000 permutation replicates) a 5%significance threshold of 3.57.

The LOD curve (Figure 1) has statistically significantpeaks on chromosomes 5 and 13, with corresponding P-values of ,0.001 and 0.042, respectively. The evidenceseen on chromosome 13 is not influenced by segrega-tion distortion and shows that genotype proportionsdiffer significantly by affection status.

Affecteds only analysis: If only the 35 affectedindividuals were genotyped, the forward and reverseapproaches, LODF and LODR, respectively, are notappropriate because they will always produce LODscores of strictly zero. Instead, we calculate the LODcurve using the reverse approach, LODR,seg, making useof the intercross segregation assumption, with geno-types CC, CB, and BB segregating according to the ratios1:2:1. A 5% significance threshold is calculated bysimulation to be 3.57.

The LOD peak on chromosome 5 (Figure 1) isstatistically significant with a LOD score of 4.07 (P ¼0.019). To follow up on this result, it is appropriate toobtain genotypes for the full cross at the peak positionon chromosome 5. In our follow-up analysis of D5M357,

we test for a difference in genotype proportions acrossaffected and unaffected individuals. Among affectedindividuals, the observed genotype distribution was18:16:1, compared to 12:39:30 among unaffecteds,yielding an exact pointwise P-value of 2.9 3 10�6. Onthe basis of this analysis, we conclude there is evidencefor a QTL on chromosome 5, and the significant resultobtained using affecteds only was not driven by spurioussegregation distortion.

Although there was good evidence of a QTL onchromosome 13 using full genotypes, we detected noevidence of a QTL on chromosome 13 using affectedsonly (maximum LOD score of 1.25, P . 0.99). Theresults on chromosome 13 demonstrate that the reverseapproach, LODR,seg, provides reliable results only whenthere is little segregation distortion in the overall set ofgenotypes. In this particular data set, there was strongsegregation distortion among the pooled set of affec-teds and unaffecteds, while affected individuals alonehad genotype proportions close to their null values.

Equal numbers of affecteds and unaffecteds geno-typed: By genotyping both affecteds and unaffecteds, weno longer require the use of a segregation assumptionand so can avoid the problem of distorted evidence thatwe encountered in our affecteds only analysis. Here, weconsider the same intercross as above, but now we havegenotyped 35 unaffected mice (selected at random), inaddition to the 35 affected individuals. In this case, weperform analysis using the reverse approach, LODR,and use permutation to get a 5% significance thresholdof 3.65.

The profile of our observed LOD curves (Figure 1) isvery similar to that obtained by standard analysis withcomplete genotyping. The overall strength of the signalobtained by partial genotyping is somewhat attenuated,but we still have reasonable evidence of QTL onchromosome 5 and 13, with P-values 0.085 and 0.026,respectively. After the initial genome scan using thisportion of the cross, we recommend following up withgenotypes on the full cross in genomic regions ofinterest. For example, on chromosome 5 we find thatthe remaining unaffected individuals show D5M357genotype proportions of 5:21:20, compared to 7:18:10among the first 35 genotyped unaffecteds. These resultsconfirm that the unaffecteds overall have a relativelylarger proportion of homozygote BB individuals andsmaller proportion of CC individuals. Further, the peakpositions obtained by our partial genotyping strategyare identical to those obtained by analysis of the fullcross. Thus, genotyping an equal number of affectedand unaffected individuals combined with follow-upusing the full cross provides an effective way to locateQTL while vastly reducing the amount of genotypingrequired.

To examine sensitivity of our results to randomness inthe set of unaffected individuals selected for genotyp-ing, we repeated the analysis with 100 different sets of 35

Figure 1.—Analysis of intercross data from Boyartchuk

et al. (2001) with significant results on chromosomes 5 and13, and chromosome 2 shown for comparison. LOD curvesare generated using four different methods and normalizedby their corresponding 5% significance thresholds for com-parison: (i) The LOD with full genotypes is calculated by stan-dard interval mapping according to the method of Xu andAtchley (1996) (shaded line), with a 5% permutationthreshold of 3.57; (ii) with genotypes on affecteds only, theLOD curve is calculated by the reverse approach using an in-tercross segregation assumption, LODR,seg (dashed shadedline), and the appropriate 5% simulation threshold is 3.57;(iii) using genotypes on 35 unaffecteds and all 35 affecteds,the LOD curve is calculated by the reverse approach, LODR

(solid line), with a corresponding 5% permutation thresholdof 3.65; and (iv) using genotypes on 35 unaffecteds and all 35affecteds, the LOD curve is calculated by the full likelihoodwith the segregation assumption, LODN

Full;seg(dashed solidline), using constrained maximum likelihood. The corre-sponding 5% permutation threshold is 3.56.

868 A. Manichaikul and K. W. Broman

randomly selected unaffected mice. We saw qualitativelysimilar results across this set of replicates, with reason-able evidence of QTL on chromosomes 5 and 13 in themajority of samples (results not shown). This investiga-tion suggests genotyping all affecteds and an equalnumber of unaffecteds is an effective way to captureevidence of QTL in the full cross, while genotyping onlya fraction of the individuals.

Some unaffecteds genotyped using constrainedmaximum likelihood: Incorporating a segregation as-sumption is most useful for affecteds only analysis,where we have zero power to map QTL without suchan assumption. Here, we consider incorporating thisassumption in the more moderate case of selectivegenotyping with equal numbers of affecteds and un-affecteds genotyping.

We examine the same set of genotyped individualsas above, with genotypes on 35 unaffecteds and all35 affected individuals, and perform analysis with theinclusion of a segregation assumption. Toward this end,both the reverse approach and the full likelihood arepossible options to accommodate the assumption. How-ever, as shown in results above, LODR,seg is equivalentto a standard analysis, plus the LOD for segregationdistortion. In mapping QTL, we are generally not in-terested in overall segregation distortion. We performanalysis by LODN

Full;seg to eliminate the possibility ofspurious evidence. Since we have eliminated evidence forsegregation distortion, we use permutation, rather thansimulation, to obtain a 5% significance threshold of 3.56.

The overall shape of the LOD curve produced by thisapproach agrees quite closely with the full and partialgenotyping results shown in Figure 1. Still, we do notedifferences that reflect properties of the methods. Thepeak on chromosome 5 shows a LOD score of 4.06 (P¼0.018), greater than the value of 3.37 using the reverseapproach. On the other hand, the peak evidence onchromosome 13 using LODN

Full;seg was only 2.42 (P ¼0.449), which is considerably less than the reverse LODof 3.98. These results show that constrained maximumlikelihood can improve the strength of our signal,particularly when the segregation assumption matchesthe observed data well. At the same time, using asegregation assumption can also attenuate the strengthof evidence when there is deviation from this assump-tion, as seen on chromosome 13.

POWER STUDIES

Simulations were performed varying cross type, her-itability, and expected proportion of affecteds, to in-vestigate the impact of these factors on power to detect aQTL across four approaches to binary trait mapping.Data were generated for backcrosses of 250 individualsand intercrosses of 500 individuals, using a marker mapbased on the mouse genome with markers about every10 cM [the full map is included with the R/qtl package

(Broman et al. 2003)]. In all simulations, a single QTLwas placed between the sixth and seventh markers onchromosome 1 with heritability of the continuous liabilityphenotype (Xu and Atchley 1996) set at either 5 or10%. Binary traits were generated from the continuousliability values, with thresholds set such that the ex-pected proportion of affecteds was either 10 or 25%.

The four mapping strategies assessed in the simula-tions are the same as those presented in the applica-

tion: (1) full genotyping of the cross using the standardapproach of Xu and Atchley (1996), (2) affecteds onlyanalysis using LODR,seg, (3) genotypes on all affectedsand an equal number of unaffecteds using the reverseapproach, LODR, without the segregation assumption,and (4) genotypes on all affecteds and an equal numberof unaffecteds using constrained full-likelihood analysis,LODN

Full;seg. For each set of parameter values and each ofthe four mapping strategies, we obtained a 5% signifi-cance threshold as the 95% quantile of the distributionof genomewide maximum LOD scores under the null,as estimated by 10,000 simulation replicates.

Power for all combinations of parameters based on10,000 simulation replicates is shown in Table 3. Foreach of the scenarios considered, the highest power wasobtained by full genotyping with standard analysis.Affecteds only analysis and constrained full-likelihoodanalysis with partial genotyping had comparable powerto detect a QTL, with slightly lower power in theaffecteds only analysis under all investigated scenarios.Finally, reverse analysis with partial genotyping showednotably lower power to detect a QTL compared to theother three approaches.

When the affected phenotype is rare, an affectedsonly analysis can provide power comparable to anal-ysis of the full cross and requires only a small fractionof the genotyping. To check for spurious evidence dueto segregation distortion, further genotyping of un-affecteds can serve as a useful supplement to affectedsonly analysis, particularly when incorporated by con-strained full-likelihood analysis, and with greater im-provements seen when the affected phenotype is morecommon. Although analysis of affecteds and unaffec-teds using the reverse approach, LODR, has lowerpower than other approaches, we should keep in mindthat this approach is more robust to segregation dis-tortion than the constrained full-likelihood analysis,which can suffer from reduced power in the presenceof segregation distortion. The relatively weaker perfor-mance of the reverse approach, LODR, suggests thisrobust strategy is more suitable as a follow-up check forsegregation distortion, rather than as a genomewideQTL mapping strategy.

DISCUSSION

We have presented methods for linkage analysis ofbinary phenotypes in the presence of selective genotyp-

Binary Trait Mapping 869

ing. As alternatives to standard interval mapping, wepresented a reverse approach of modeling genotypesconditional on phenotypes and also a full-likelihoodapproach. Our suggested modifications to the standardapproach of Xu and Atchley (1996) are developed interms of fundamental likelihood modeling strategies.Accordingly, a key contribution here is our presentationof approaches to binary trait analysis using a cohesivelikelihood framework, elucidating fundamental rela-tionships among the methods. Our formal develop-ment of allele sharing methods presented as the reverseapproach, LODR, led to the use of hidden Markovmodels to allow interval mapping in the reverse and full-likelihood approaches (appendixes a and c). Throughanalytical comparisons, we found that our reverseapproach, LODR, is identical to a full-likelihood ap-proach at both marker and nonmarker locations.

We also proposed another version of the reverseapproach, LODR,seg, which incorporates a segregationassumption of the expected genotype proportions basedon the type of cross that was performed. This approachformalizes a natural method of analysis for dealing withgenotypes on affecteds only and presents it in a moregeneral form that can be applied with genotypes onboth affecteds and unaffecteds. We found the log10-likelihood ratio, LODR,seg, could be decomposed as thesum of two types of evidence: (1) deviation of genotypeproportions by phenotype group and (2) segregationdistortion. For the case of genotypes on affecteds only,incorporating the segregation assumption was espe-cially crucial, as it provided a view of evidence for aQTL where none was available by LODR or LODF.

The inclusion of evidence for segregation distortionwas deemed inappropriate in dealing with data havinggenotypes available on both affecteds and unaffecteds.For this case, we proposed incorporating the segrega-tion assumption using a constrained full-likelihood

approach. In this way, the segregation assumption wasimposed under both the null and alternative hypothe-ses, so that the resulting test statistic, LODN

Full;seg, did notcontain evidence for segregation distortion. Eliminat-ing evidence for segregation distortion helps ensurethat a large LOD score indicates evidence of a QTL,as segregation distortion can arise simply by randomchance, systematically as a result of genotyping error, oras a result of embryonic lethal alleles that are unrelatedto the trait of interest.

An understanding of these approaches as they relateto one another helps us to decide which method touse on the basis of the existing pattern of selectivegenotyping. In the case that we have genotyped every-body in our sample, we are not interested in overallevidence for segregation distortion and would choosea standard approach LODF. On the other hand, if wehave genotyped affecteds only, the standard approachdoes not allow us to detect association. In this case, allevidence of association will be captured as evidence forsegregation distortion, which shows up only with use ofLODR,seg as LODseg.dist..

Since full genotyping of a cross can be costly, whileaffecteds only analysis can be prone to spurious evi-dence, a reasonable balance is to genotype someaffecteds and some unaffecteds. Specifically, we recom-mend an initial screen with genotypes on all affectedindividuals and an equal number of unaffecteds, fol-lowed by analysis of the full cross in genomic regions ofinterest. As demonstrated in the application, thiseconomical strategy can be an effective way to charac-terize QTL from the full cross and requires only afraction of the genotyping. The ideal selective genotyp-ing approach for any particular study may of course varyfrom this recommendation and could be studied as afunction of animal rearing and phenotyping costsrelative to genotyping cost.

TABLE 3

Power to detect QTL using four different strategies for binary trait mapping

Power (%) of methods

Expected proportionof affecteds (%) Cross type

Heritability(%)

Fullgenotypes

Affectedsonly

Both withreverse

Both with fulllikelihood

10 Backcross 5 9.5 7.7 4.2 7.710 28.6 24.2 12.9 24.8

Intercross 5 19.9 16.5 7.1 16.710 59.6 52.0 25.5 53.0

25 Backcross 5 20.2 12.5 10.4 14.810 58.3 39.9 34.6 46.4

Intercross 5 41.7 26.0 22.1 31.310 89.5 72.4 64.2 79.3

Estimates are based on 10,000 simulation replicates for each combination of parameter values, in backcrossesof 250 individuals and intercrosses of 500 individuals. The segregation assumption is incorporated for affectedsonly analysis, as well as for full-likelihood analysis of affecteds and some unaffecteds.

870 A. Manichaikul and K. W. Broman

The reverse approach, LODR, with no segregationassumption is a natural method of analysis for data withgenotypes on affecteds and some unaffecteds. Althoughthis approach was shown to be quite similar to the standardapproach LODF at marker locations, we prefer LODR as itis not susceptible to biased parameter estimates and soproduces more reliable results in between markers.

A full-likelihood analysis with constrained maximiza-tion under both the null and alternative hypotheses,presented as LODN

Full;seg, is another reasonable way toapproach selective genotyping data with both affectedsand unaffecteds. Incorporating the segregation assump-tion in this setting is a practical compromise to preservethe evidence reported in an affecteds only analysis whilebringing in unaffected individuals. The drawback, asseen in the Listeria example (Boyartchuk et al. 2001),is that evidence can be attenuated when there is overallsegregation distortion in the data. Still, our computersimulation studies indicate constrained full-likelihoodanalysis offers notably higher power than the reverseapproach, LODR, across a variety of parameter val-ues. Thus, our power studies suggest constrained full-likelihood analysis is preferable as long as there is nopervasive segregation distortion in the cross.

A further limitation of the reverse approach lies in thetreatment of multiple-QTL models. While single-QTLmodels may be set up quite naturally by conditioninggenotypes at a single locus on the observed phenotypes,modeling genotypes at multiple loci can be much morecumbersome. Instead, when exploring multiple-QTLmodels with data on both affecteds and unaffectedsavailable, the standard approach of conditioning phe-notypes on genotypes is more natural. The close re-lationship between the forward and reverse approachesin a single-QTL scan makes it quite reasonable to goahead with the forward approach for the considerationof multiple-QTL models in the presence of selectivegenotyping. When using the forward approach formultiple-QTL mapping under selective genotyping,inferences at nonmarker positions may still be some-what unreliable, while entirely valid results will beproduced for models involving marker positions only.

After performing an analysis of a cross experimentunder selective genotyping, we may always follow up bygenotyping all individuals in genomic regions of interestidentified from the initial scan. Such follow-up will beespecially important if the initial scan is performed withaffecteds only, since this strategy is most sensitive tospurious evidence due to segregation distortion.

We thank William Pu at the Children’s Hospital in Boston forpresenting us with data motivating this research. This work wassupported in part by National Institutes of Health grant GM074244(to K.W.B.) and by a National Science Foundation Graduate ResearchFellowship (to A.M.).

LITERATURE CITED

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Communicating editor: R. W. Doerge

Binary Trait Mapping 871

APPENDIX A: THE REVERSE APPROACH AT NONMARKER LOCATIONS

We describe the algorithm for obtaining f̂� here. Analogous methods for obtaining f̂D and f̂D̃ follow directly byapplying the same algorithm within each of the two phenotype groups.

Given the full set of observed genotype data O�i ¼ (O1i, . . . , Opi) for a single individual at the p putativeQTL positions to be considered, let Gi denote the underlying genotype at the mth putative site of interest. Wecan expand the probability of the observed marker data given the underlying genotype at the putative QTL ofinterest as

qig ¼ PrðO1i ; . . . ;Oðm�1Þi jGi ¼ g Þ3 PrðOmi jGi ¼ g Þ3 PrðOðm11Þi ; . . . ;Omi jGi ¼ g Þ¼ bl

miðg Þ3 eðg ; OmiÞ3 brmiðg Þ;

where the conditional probabilities of observed marker data, blmi(g) and br

mi(g), to the left and right of putative QTLmay be obtained inductively using the backward equations in the context of hidden Markov models (HMMs) (Lander

et al. 1987). Here, e(g, Omi) is the corresponding emission probability at the mth genetic position of interest forindividual i, which can also be interpreted as the genotyping error rate.

The likelihood function for the parameter f� based on the observed genotype data O�i on individuals i2 {1, . . . , nobs}is

likðf�Þ ¼Ynobs

i¼1

PrðO�i ; f�Þ

¼Ynobs

i¼1

Xg2fAA;ABg

½qig � PrðGi ¼ g ; f�Þ�:

At iteration s 1 1, we have the parameter estimate, f̂ðsÞ� . In the E-step, we calculate the expected number of

individuals with genotype AA at the putative QTL of interest as

n̂ðs11ÞAA ¼

Xnobs

i¼1

PrðGi ¼ AA;O�i ; f̂ðsÞ� Þ

PrðO�i ; f̂ðsÞ� Þ

¼Xnobs

i¼1

f̂ðsÞ� � qi;AA

f̂ðsÞ� � qi;AA 1 ð1� f̂

ðsÞ� Þ � qi;AB

: ðA1Þ

In the M-step, the updated parameter estimate is simply f̂ðs11Þ� ¼ n̂ðs11Þ

AA =nobs. A reasonable initial estimate of f� is thesample average of conditional genotype probabilities, f̂

ð0Þ� ¼ ð1=nobsÞ

Pnobs

i¼1 pi;AA.

APPENDIX B: ALTERNATE REPRESENTATIONS OF THE FULL LIKELIHOOD

We may expand our presentation of the full likelihood from Equation 1 as follows:

likðfD ;fD̃;p�Þ ¼YNi¼1

PrðDi ;O�i ; fD ;fD̃;p�Þ

¼�Ynobs

i¼1

PrðO�i jDi ; fD ;fD̃Þ���YN

i¼1

PrðDi ; p�Þ�

ðB1Þ

¼ likðfD ; fD̃ jDÞ � likðp�Þ: ðB2Þ

Here, the pattern of missing genotypes generated by selective genotyping depends only on the observed phenotypes,D, and is conditionally independent of the underlying genotypes, G, given D. Hence, the model implicitly condi-tions on the pattern of selective genotyping, with PrðO�i j Di ; fD ; fD̃Þ ¼ 1 for all ungenotyped individuals,i 2 fnobs 1 1; . . . ;N g.

We may reparameterize the full likelihood in terms of parameters pAA, pAB, and f� as

872 A. Manichaikul and K. W. Broman

likðpAA;pAB ;f�Þ ¼YNi¼1

PrðDi ;O�i ; pAA;pAB ;f�Þ

¼YNi¼1

PrðDi jO�i ; pAA;pABÞ � PrðO�i ; f�Þ; ðB3Þ

where ungenotyped individuals, i¼ nobs 1 1, . . . , N, are incorporated by applying the marginal genotype probabilitiesas mixing proportions in modeling disease status using a binomial model with p� ¼pAA �f�1 pAB � (1�f�). To find theappropriate MLEs, note that the likelihood in (B3) is a reparameterization of (B1). Specifically, the parameters ofinterest pAA, pAB, and f� can be written in terms of fD, fD̃, and p�:

f� ¼ fD � p�1 fD̃ � ð1� p�Þ

pAA ¼fD � p�

f�

pAB ¼ð1� fDÞ � p�

1� f�: ðB4Þ

Then, the appropriate MLEs for pAA, pAB, and f� in (B3) can be obtained as plug-in estimates using the relationships in(B4). We have provided these MLEs for completeness, but if our primary aim is the appropriate likelihood-ratiostatistic, then calculating these MLEs is unnecessary.

APPENDIX C: CONSTRAINED FULL-LIKELIHOOD ANALYSIS AT NONMARKER LOCATIONS

To start, we incorporate the constraint by transforming the expression in (2) to yield fD̃ ¼ ðf� � fD � p�Þ=ð1� p�Þ.Hence, our constrained likelihood is equivalent to a two-parameter model in which each individual has the followingcontribution to the likelihood,

likðfD ;p�; O�i ;DiÞ ¼ PrðO�i jDi ; fD ;p�Þ � PrðDi ; p�Þ;

with observed genotypes modeled according to disease status as

PrðO�i jDi ; fD ;p�Þ ¼

Pg2fAA;ABg PrðGi ¼ g ; fDÞ � qig ; Di ¼ 1Pg2fAA;ABg PrðGi ¼ g ; fD̃ ¼

f��fD �p�1�p�

Þ � qig ; Di ¼ 0

(ðC1Þ

for genotyped individuals and identically equal to one for ungenotyped individuals.At iteration s 1 1, we have the parameter estimate, f̂

ðsÞD . In the E-step, we calculate the expected number of affected

and unaffected individuals with genotype AA at the putative QTL as shown in (A1).In the M-step, the updated parameter estimates are obtained by maximizing the likelihood function in (C1), using a

numerical optimization approach such as that of Nelder and Mead (1965). Similar to the EM for the reverseapproach above, we use the sample average of conditional genotype probabilities, pig, among diseased individuals as aninitial guess for fD and take the sample average nD=N as the initial estimate for p�.

APPENDIX D: MORE RELATIONSHIPS BETWEEN THE FULL AND REVERSE APPROACHES

Full likelihood vs. reverse approach: Let LODnobs

Full be the LOD score based on full likelihood for genotypedindividuals i ¼ 1, . . . , nobs only and LODN

Full be the LOD score from full-likelihood analysis using all individualsi ¼ 1, . . . , N, whether genotyped or not. We see the following analytic results comparing LODnobs

Full to LODR from thereverse approach:

LODnobsFull ¼ log10

PrðO�� jD; f̂D ; f̂D̃Þ � PrðD; p̂�ÞPrðO��; f̂�Þ � PrðD; p̂�Þ

� �

¼ log10

PrðO�� jD; f̂D ; f̂D̃ÞPrðO��; f̂�Þ

� �¼ LODR:

Likewise, we can compare LODNFull from a full-likelihood analysis with all N individuals to LODR, which uses

genotyped individuals only. Recall that our full-likelihood analysis conditions on the pattern of missing genotype data,

Binary Trait Mapping 873

with the conditional multipoint marker probabilities identically equal to one for all ungenotyped individuals. UsingEquation B2 above, we see that

LODNFull ¼ log10

maxfD ;fD̃

Qnobsi¼1 PrðO�i jDi ; fD ;fD̃Þ

maxf�

Qnobsi¼1 PrðO�i ; f�Þ

� �3

maxp�

QNi¼1 PrðDi ; p�Þ

maxp�

QNi¼1 PrðDi ; p�Þ

� �

¼ log10

PrðO�� jD; f̂D ; f̂D̃ÞPrðO��; f̂�Þ

� �¼ LODR;

where the estimated parameters f̂D and f̂D̃ are the MLEs specified in the methods section.Constrained full likelihood vs. modified reverse approach: It is difficult to work with the constrained full likelihood

analytically. However, we can derive the following inequality to put an upper bound on LODNFull;seg. First note that

constrained likelihood must be bounded above by the unconstrained likelihood so

maxfD ;fD̃;p� jf�¼1=2

likðfD ;fD̃;p�Þ# maxfD ;fD̃;p�

likðfD ;fD̃;p�Þ

¼ likðf̂D ; f̂D̃Þ � likðp̂�Þ;

where f̂D , f̂D̃, and p̂� are the unconstrained MLEs.Plugging into LODN

Full;seg, we find

LODNFull;seg ¼ log10

maxfD ;fD̃;p� jf�¼1=2 likðfD ;fD̃;p�Þmaxp� likðf� ¼ 1=2;p�Þ

� �

# log10

likðf̂D ; f̂D̃Þ � likðp̂�Þlikðf� ¼ 1=2Þ � likðp̂�Þ

� �¼ LODR;seg:

Hence, the LOD score obtained by constrained full likelihood is bounded above by the LOD score from the modifiedreverse approach.

874 A. Manichaikul and K. W. Broman