X-linked extension of the revised Haseman-Elston algorithm for linkage analysis in sib pairs

Haseman and Elston (H-E) proposed a regression-based robust test of linkage between a marker and an autosomal quantitative trait locus, using the squared sib pair trait difference as a dependent variable and the proportion of alleles shared identical by descent by the sib pair as an independent variable. Several authors have proposed improvement of the original H-E's seminal work by using an optimal linear combination of squared sum and squared difference as the dependent variable. In this paper, we extend Haseman and Elston's sib pair method to an X-linked locus. We give a general formulation of the complete regression model and details of the regression coefficients in terms of variance components. Simulation results are presented to describe the power of this technique for a theoretical best case scenario.     

A unified Haseman-Elston method for testing linkage with quantitative traits

The Haseman and Elston (H-E) method uses a simple linear regression to model the squared trait difference of sib pairs with the shared allele identical by descent (IBD) at marker locus for linkage testing. Under this setting, the squared mean-corrected trait sum is also linearly related to the IBD sharing. However, the resulting slope estimate for either model is not efficient. In this report, we propose a simple linkage test that optimally uses information from the estimates of both models. We also demonstrate that the new test is more powerful than both the traditional one and the recently revisited H-E methods.     

The effect of linkage disequilibrium on linkage analysis of incomplete pedigrees

Dense SNP maps can be highly informative for linkage studies. But when parental genotypes are missing, multipoint linkage scores can be inflated in regions with substantial marker-marker linkage disequilibrium (LD). Such regions were observed in the Affymetrix SNP genotypes for the Genetic Analysis Workshop 14 (GAW14) Collaborative Study on the Genetics of Alcoholism (COGA) dataset, providing an opportunity to test a novel simulation strategy for studying this problem. First, an inheritance vector (with or without linkage present) is simulated for each replicate, i.e., locations of recombinations and transmission of parental chromosomes are determined for each meiosis. Then, two sets of founder haplotypes are superimposed onto the inheritance vector: one set that is inferred from the actual data and which contains the pattern of LD; and one set created by randomly selecting parental alleles based on the known allele frequencies, with no correlation (LD) between markers. Applying this strategy to a map of 176 SNPs (66 Mb of chromosome 7) for 100 replicates of 116 sibling pairs, significant inflation of multipoint linkage scores was observed in regions of high LD when parental genotypes were set to missing, with no linkage present. Similar inflation was observed in analyses of the COGA data for these affected sib pairs with parental genotypes set to missing, but not after reducing the marker map until r2 between any pair of markers was 相似文献   

Adding further power to the Haseman and Elston method for detecting linkage in larger sibships: weighting sums and differences

Haseman and Elston (H-E) proposed a robust test to detect linkage between a quantitative trait and a genetic marker. In their method the squared sib-pair trait difference is regressed on the estimated proportion of alleles at a locus shared identical by descent by sib pairs. This method has recently been improved by changing the dependent variable from the squared difference to the mean-corrected product of the sib-pair trait values, a significantly positive regression indicating linkage. Because situations arise in which the original test is more powerful, a further improvement of the H-E method occurs when the dependent variable is changed to a weighted average of the squared sib-pair trait difference and the squared sib-pair mean-corrected trait sum. Here we propose an optimal method of performing this weighting for larger sibships, allowing for the correlation between pairs within a sibship. The optimal weights are inversely proportional to the residual variances obtained from the two different regressions based on the squared sib-pair trait differences and the squared sib-pair mean-corrected trait sums, respectively, allowing for correlations among sib pairs. The proposed method is compared with the existing extension of the H-E approach for larger sibships. Control of the type I error probabilities for sibships of any size can be improved by using a generalized estimating equation approach and the robust sandwich estimate of the variance, or a Monte-Carlo permutation test.     

Effect of Box-Cox transformation on power of Haseman-Elston and maximum-likelihood variance components tests to detect quantitative trait Loci

Non-normality of the phenotypic distribution can affect power to detect quantitative trait loci in sib pair studies. Previously, we observed that Winsorizing the sib pair phenotypes increased the power of quantitative trait locus (QTL) detection for both Haseman-Elston (HE) least-squares tests [Hum Hered 2002;53:59-67] and maximum likelihood-based variance components (MLVC) analysis [Behav Genet (in press)]. Winsorizing the phenotypes led to a slight increase in type 1 error in H-E tests and a slight decrease in type I error for MLVC analysis. Herein, we considered transforming the sib pair phenotypes using the Box-Cox family of transformations. Data were simulated for normal and non-normal (skewed and kurtic) distributions. Phenotypic values were replaced by Box-Cox transformed values. Twenty thousand replications were performed for three H-E tests of linkage and the likelihood ratio test (LRT), the Wald test and other robust versions based on the MLVC method. We calculated the relative nominal inflation rate as the ratio of observed empirical type 1 error divided by the set alpha level (5, 1 and 0.1% alpha levels). MLVC tests applied to non-normal data had inflated type I errors (rate ratio greater than 1.0), which were controlled best by Box-Cox transformation and to a lesser degree by Winsorizing. For example, for non-transformed, skewed phenotypes (derived from a chi2 distribution with 2 degrees of freedom), the rates of empirical type 1 error with respect to set alpha level=0.01 were 0.80, 4.35 and 7.33 for the original H-E test, LRT and Wald test, respectively. For the same alpha level=0.01, these rates were 1.12, 3.095 and 4.088 after Winsorizing and 0.723, 1.195 and 1.905 after Box-Cox transformation. Winsorizing reduced inflated error rates for the leptokurtic distribution (derived from a Laplace distribution with mean 0 and variance 8). Further, power (adjusted for empirical type 1 error) at the 0.01 alpha level ranged from 4.7 to 17.3% across all tests using the non-transformed, skewed phenotypes, from 7.5 to 20.1% after Winsorizing and from 12.6 to 33.2% after Box-Cox transformation. Likewise, power (adjusted for empirical type 1 error) using leptokurtic phenotypes at the 0.01 alpha level ranged from 4.4 to 12.5% across all tests with no transformation, from 7 to 19.2% after Winsorizing and from 4.5 to 13.8% after Box-Cox transformation. Thus the Box-Cox transformation apparently provided the best type 1 error control and maximal power among the procedures we considered for analyzing a non-normal, skewed distribution (chi2) while Winzorizing worked best for the non-normal, kurtic distribution (Laplace). We repeated the same simulations using a larger sample size (200 sib pairs) and found similar results.     

The bias introduced by population stratification in IBD based linkage analysis

The lack of replication of model-free linkage analyses performed on complex diseases raises questions about the robustness of these methods to various biases. The confounding effect of population stratification on a genetic association study has long been recognized in the genetic epidemiology community. Because the estimation of the number of alleles shared identical by descent (IBD) does not depend on the marker allele frequency when founders of families are observed, model-free linkage analysis is usually thought to be robust to population stratification. However, for common complex diseases, the genotypes of founders are often unobserved and therefore population stratification has the potential to impair model-free linkage analysis. Here, we demonstrate that, when some or all of the founder genotypes are missing, population stratification can introduce deleterious effects on various model-free linkage methods or designs. For an affected sib pair design, it can cause excess false-positive discoveries even when the trait distribution is homogeneous among subpopulations. After incorporating a control group of discordant sib pairs or for a quantitative trait, two circumstances must be met for population stratification to be a confounder: the distributions for both the marker and the trait must be heterogeneous among subpopulations. When this occurs, the bias can result in either a liberal, and hence invalid, test or a conservative test. Bias can be eliminated or alleviated by inclusion of founders' or other family members' genotype data. When this is not possible, new methods need to be developed to be robust to population stratification.     

Comparison of Haseman-Elston regression analyses using single, summary, and longitudinal measures of systolic blood pressure

To compare different strategies for linkage analyses of longitudinal quantitative trait measures, we applied the "revisited" Haseman-Elston (RHE) regression model (the cross product of centered sib-pair trait values is regressed on expected identical-by-descent allele sharing) to cross-sectional, summary, and repeated measurements of systolic blood pressure (SBP) values in replicate 34, randomly selected from the Genetic Analysis Workshop 13 simulated data. RHE linkage scans were performed without knowledge of the generating model using the following phenotypes derived from untreated SBP measurements: the first, the last, the mean, the ratio of the change between the first and last over time, and the estimated linear regression slope coefficient. Estimates of allele sharing in sibling pairs were obtained from the complete genotype data of Cohorts 1 and 2, but linkage analyses were restricted to the five visits of Cohort 2 siblings. Evidence for linkage was suggestive (p < 0.001) at markers neighboring SBP genes Gb35, Gs10, and Gs12, but weaker signals (p < 0.01) were obtained at markers mapping close to Gb34 and Gs11. Linkage to baseline genes Gb34 and Gb35 was best detected using the first SBP measurement, whereas linkage to slope genes Gs10-12 was best detected using the last or mean SBP value. At markers on chromosomes 13 and 21 displaying strongest linkage signals, marginal RHE-type models including repeated SBP measures were fit to test for overall and time-dependent genetic effects. These analyses assumed independent sib pairs and employed generalized estimating equations (GEE) with a first-order autoregressive working correlation structure to adjust for serial correlation present among repeated observations from the same sibling pair.     

Dissecting the correlation structure of a bivariate phenotype: Common genes or shared environment?

High correlations between two quantitative traits may be either due to common genetic factors or common environmental factors or a combination of both. In this study, we develop statistical methods to extract the genetic contribution to the total correlation between the components of a bivariate phenotype. Using data on bivariate phenotypes and marker genotypes for sib-pairs, we propose a test for linkage between a common QTL and a marker locus based on the conditional cross-sib trait correlations (trait 1 of sib 1—trait 2 of sib 2 and conversely) given the identity-by-descent (i.b.d.) sharing at the marker locus. We use Monte-Carlo simulations to evaluate the performance of the proposed test under different trait parameters and quantitative trait distributions. An application of the method is illustrated using data on two alcohol-related phenotypes from a project on the collaborative study on the genetics of alcoholism.     

Matrix representation of the Haseman-Elston method.

The Hasemann-Elston method of linkage detection is based on the probabilities of a sib pair having 0, 1, or 2 alleles identical by descent (IBD) at a marker and a trait locus. These probabilities form a 3x3 matrix. Here, the characteristic values and characteristic vectors of this matrix were used to clarify the structure of the equations and to simplify calculations. As examples, the regression coefficients were derived for three genetic systems: a trait and a marker, two epistatic traits and two markers, and one trait locus and two markers. The last model was studied under the assumption of no crossover interference, the expression for allele IBD sharing at a trait locus was derived as a function of allele IBD sharing at two marker loci, and the regression is shown to be non-linear.     

Selection and subsequent analysis of sib pair data for QTL detection.

Haseman and Elston (1972) developed a robust regression method for the detection of linkage between a marker and a quantitative trait locus (QTL) using sib pair data. The principle underlying this method is that the difference in phenotypes between pairs of sibs becomes larger as they share a decreasing number of alleles at a particular QTL identical by descent (IBD) from their parents. In this case, phenotypically very different sibs will also on average share a proportion of alleles IBD at any marker linked to the QTL that is lower than the expected value of 0.5. Thus, the deviation of the proportion of marker alleles IBD from the expected value in pairs of sibs selected to be phenotypically different (i.e. discordant) can provide a test for the presence of a QTL. A simple regression method for QTL detection in sib pairs selected for high phenotypic differences is presented here. The power of the analytical method was found to be greater than the power obtained using the standard analysis when samples of sib pairs with high phenotypic differences were used. However, the use of discordant sib pairs was found to be less powerful for QTL detection than alternative selective genotyping schemes based on the phenotypic values of the sibs except with intense selection, when its advantage was only marginal. The most effective selection scheme overall was the use of sib pairs from entire families selected on the basis of high within-family variance for the trait in question. There is little effect of selection on QTL position estimates, which are in good agreement with the simulated values. However, QTL variance estimates are biased to a greater or lesser degree, depending on the selection method.     

Consanguinity and the sib-pair method: an approach using identity by descent between and within individuals.

To test for linkage between a trait and a marker, one can consider identical marker alleles in related individuals, for instance, sibs. For recessive diseases, it has been shown that some information may be gained from the identity by descent (IBD) of the two alleles of an affected inbred individual at the marker locus. The aim of this paper is to extend the sib-pair method of linkage analysis to the situation of sib pairs sampled from consanguineous populations. This extension takes maximum advantage of the information provided by both the IBD pattern between sibs and allelic identity within each sib of the pair. This is possible through the use of the condensed identity coefficients. Here, we propose a new test of linkage based on a chi2. We compare the performance of this test with that of the classical chi2 test based on the distribution of sib pairs sharing 0, 1, or 2 alleles IBD. For sib pairs from first-cousin matings, the proposed test can better detect the role of a disease-susceptibility (DS) locus. Its power is shown to be greater than that of the classical test, especially for models where the DS allele may be common and incompletely penetrant; that is to say for situations that may be encountered in multifactorial diseases. A study of the impact of inbreeding on the expected proportions of sib pairs sharing 0, 1, or 2 alleles IBD is also performed here. Ignoring inbreeding, when in fact inbreeding exists, increases the rate of type I errors in tests of linkage.     

Robustness and power of the maximum-likelihood-binomial and maximum-likelihood-score methods,in multipoint linkage analysis of affected-sibship data.

The maximum-likelihood-binomial (MLB) method, based on the binomial distribution of parental marker alleles among affected offspring, recently was shown to provide promising results by two-point linkage analysis of affected-sibship data. In this article, we extend the MLB method to multipoint linkage analysis, using the general framework of hidden Markov models. Furthermore, we perform a large simulation study to investigate the robustness and power of the MLB method, compared with those of the maximum-likelihood-score (MLS) method as implemented in MAPMAKER/SIBS, in the multipoint analysis of different affected-sibship samples. Analyses of multiple-affected sibships by means of the MLS were conducted by consideration of all possible sib pairs, with (weighted MLS [MLSw]) or without (unweighted MLS [MLSu]) application of a classic weighting procedure. In simulations under the null hypothesis, the MLB provided very consistent type I errors regardless of the type of family sample (sib pairs or multiple-affected sibships), as did the MLS for samples with sib pairs only. When samples included multiple-affected sibships, the MLSu led to inflation of low type I errors, whereas the MLSw yielded very conservative tests. Power comparisons showed that the MLB generally was more powerful than the MLS, except in recessive models with allele frequencies <.3. Missing parental marker data did not strongly influence type I error and power results in these multipoint analyses. The MLB approach, which in a natural way accounts for multiple-affected sibships and which provides a simple likelihood-ratio test for linkage, is an interesting alternative for multipoint analysis of sibships.     

Comparison of a multipoint identity-by-descent method with parametric multipoint linkage analysis for mapping quantitative traits.

We previously developed a method of partitioning genetic variance of a quantitative trait to loci in specific chromosomal regions. In this paper, we compare this method--multipoint IBD (identical by descent) method (MIM)--with parametric multipoint linkage analysis (MLINK). A simulation study was performed comparing the methods for the major-locus, mixed, and two-locus models. The criterion for comparisons between MIM and MLINK was the average lod score from multiple replicates of simulated data sets. The effect of gene frequency, dominance, model misspecification, marker spacing, and informativeness are also considered in a smaller set of simulations. Within the context of the models examined, the MIM approach was found to be comparable in power with parametric multipoint linkage analysis when (a) parental data are unknown, (b) the effect of the major locus is small and there is additional genetic variation, or (c) the parameters of the major-locus model are misspecified. The performance of the MIM method relative to MLINK was markedly lower when the allele frequency at the trait locus was .2 versus .5, particularly for the case when parental data were assumed to be known. Dominance at the trait major locus, as well as marker spacing and heterozygosity, did not appear to have a large effect on the ELOD comparisons.     

Accounting for linkage in family-based tests of association with missing parental genotypes

In studies of complex diseases, a common paradigm is to conduct association analysis at markers in regions identified by linkage analysis, to attempt to narrow the region of interest. Family-based tests for association based on parental transmissions to affected offspring are often used in fine-mapping studies. However, for diseases with late onset, parental genotypes are often missing. Without parental genotypes, family-based tests either compare allele frequencies in affected individuals with those in their unaffected siblings or use siblings to infer missing parental genotypes. An example of the latter approach is the score test implemented in the computer program TRANSMIT. The inference of missing parental genotypes in TRANSMIT assumes that transmissions from parents to affected siblings are independent, which is appropriate when there is no linkage. However, using computer simulations, we show that, when the marker and disease locus are linked and the data set consists of families with multiple affected siblings, this assumption leads to a bias in the score statistic under the null hypothesis of no association between the marker and disease alleles. This bias leads to an inflated type I error rate for the score test in regions of linkage. We present a novel test for association in the presence of linkage (APL) that correctly infers missing parental genotypes in regions of linkage by estimating identity-by-descent parameters, to adjust for correlation between parental transmissions to affected siblings. In simulated data, we demonstrate the validity of the APL test under the null hypothesis of no association and show that the test can be more powerful than the pedigree disequilibrium test and family-based association test. As an example, we compare the performance of the tests in a candidate-gene study in families with Parkinson disease.     

X-APL: an improved family-based test of association in the presence of linkage for the X chromosome

Family-based association methods have been developed primarily for autosomal markers. The X-linked sibling transmission/disequilibrium test (XS-TDT) and the reconstruction-combined TDT for X-chromosome markers (XRC-TDT) are the first association-based methods for testing markers on the X chromosome in family data sets. These are valid tests of association in family triads or discordant sib pairs but are not theoretically valid in multiplex families when linkage is present. Recently, XPDT and XMCPDT, modified versions of the pedigree disequilibrium test (PDT), were proposed. Like the PDT, XPDT compares genotype transmissions from parents to affected offspring or genotypes of discordant siblings; however, the XPDT can have low power if there are many missing parental genotypes. XMCPDT uses a Monte Carlo sampling approach to infer missing parental genotypes on the basis of true or estimated population allele frequencies. Although the XMCPDT was shown to be more powerful than the XPDT, variability in the statistic due to the use of an estimate of allele frequency is not properly accounted for. Here, we present a novel family-based test of association, X-APL, a modification of the test for association in the presence of linkage (APL) test. Like the APL, X-APL can use singleton or multiplex families and properly infers missing parental genotypes in linkage regions by considering identity-by-descent parameters for affected siblings. Sampling variability of parameter estimates is accounted for through a bootstrap procedure. X-APL can test individual marker loci or X-chromosome haplotypes. To allow for different penetrances in males and females, separate sex-specific tests are provided. Using simulated data, we demonstrated validity and showed that the X-APL is more powerful than alternative tests. To show its utility and to discuss interpretation in real-data analysis, we also applied the X-APL to candidate-gene data in a sample of families with Parkinson disease.     

The power to detect linkage disequilibrium with quantitative traits in selected samples

Results from power studies for linkage detection have led to many ongoing and planned collections of phenotypically extreme nuclear families. Given the great expense of collecting these families and the imminent availability of a dense diallelic marker map, the families are likely to be used in allelic-association as well as linkage studies. However, optimal selection strategies for linkage may not be equally powerful for association. We examine the power to detect linkage disequilibrium for quantitative traits after phenotypic selection. The results encompass six selection strategies that are in widespread use, including single selection (two designs), affected sib pairs, concordant and discordant pairs, and the extreme-concordant and -discordant design. Selection of sibships on the basis of one extreme proband with high or low trait scores provides as much power as discordant sib pairs but requires the screening and phenotyping of substantially fewer initial families from which to select. Analysis of the role of allele frequencies within each selection design indicates that common trait alleles generally offer the most power, but similarities between the marker- and trait-allele frequencies are much more important than the trait-locus frequency alone. Some of the most widespread selection designs, such as single selection, yield power gains only when both the marker and quantitative trait loci (QTL) are relatively rare in the population. In contrast, discordant pairs and the extreme-proband design provide power for the broadest range of QTL-marker-allele frequency differences. Overall, proband selection from either tail provides the best balance of power, robustness, and simplicity of ascertainment for family-based association analysis.     

Comparative informativeness for linkage of multiple SNPs and single microsatellites

Single nucleotide polymorphisms (SNPs), or biallelic markers, are popular in genetic linkage studies due to their abundance in the genome, stability, and ease of scoring. We determined the 'information ratio' (IR) of closely spaced SNPs in simulated nuclear families and affected sib pairs (ASPs). (The IR is the ratio of actual average maximum lod score to the maximum lod score attainable if the marker were fully informative.) The nuclear families included parental information, whereas the ASPs did not. We analyzed these SNPs in two ways: (1) using multipoint analysis, and (2) treating the SNPs as 'composite markers' (i.e., haplotypes, as assigned by GENEHUNTER). (3) We also calculated the IR of a single microsatellite marker with multiple alleles and compared with the IR from the SNPs. For each set of input conditions, we simulated 1000 nuclear families, of 2, 3, 4, or 5 children each, as well as 1000 ASPs. We generated SNP marker data for strings of k = 1, 2, 3, 5, 7, and 10 SNP loci, with no recombination (theta = 0) and no linkage disequilibrium among the SNPs. The MAF (minor allele frequency) was either 0.5 or 0.25, and allele frequencies were the same for all k loci in any analysis. We also generated marker data for one single-locus microsatellite marker, with m = 3, 4, 5, 6, 7, and 9 equally frequent alleles. In all simulations, the disease was fully penetrant dominant, and there was no recombination or linkage disequilibrium among markers or between marker and disease. When multipoint analysis was used, we found that 5-7 closely spaced SNPs were usually enough to yield an IR of approximately 100%, for nuclear families of any size. However, for the ASPs, even 7-10 SNPs yielded an IR of only 70-80%. A microsatellite with 9 equally frequent alleles yielded about the same IR (86-88%) as a string of 4-5 SNPs, in nuclear families. SNPs analyzed as 'composite markers' analyses performed worse, due to the inherent ambiguity of SNP haplotyping.     

Analysis of quantitative trait loci using hybrid pedigrees: quantitative traits of animals

A method is proposed for analysis of quantitative traits in animal hybrid pedigrees formed by crosses between outbred lines differing in allele frequencies of the genes controlling the trait studied. The method is based on the decomposition of trait variances into components and uses maximization of the likelihood function for estimating model parameters, which allows the estimation of additive and dominance effects of the gene involved in trait determination and its allele frequencies, as well as determination of the chromosomal position of this gene relative to genotyped markers. To test the linkage of this gene with markers, a statistic with the noncentral chi(2) distribution has been chosen. Analytical expressions for the power of this method have been derived. The method has been tested on small model hybrid pedigrees. Phenotypic values of the trait and information on marker genotypes for each individual in hybrid pedigrees are original data for the analysis of a quantitative trait.     

Association studies of QTL for multi-allele markers by mixed models

In this paper, we extend association study methods of both Fan et al. [Hum Hered 2002;53:130-145], in which a quantitative trait locus (QTL) and a multi-allele marker are considered for trio families, and Fan and Xiong [Biostatistics 2003, in press], in which a QTL and a bi-allelic marker are considered for nuclear families. The objective is to build mixed models for association study between a QTL and a multi-allelic marker for nuclear families with any number of offspring. Two types of nuclear family data are considered: the first is genetic data of offspring from at least one heterozygous parents, and the second is genetic data of offspring of nuclear family. (1) For the data of offspring from at least one heterozygous parents, we assume that at least one parent is heterozygous at the marker locus, and we may infer clearly the transmission of parental marker alleles to the offspring. We show that it can be used in association study in the presence of linkage. The theoretical basis is the difference between the conditional mean of trait value given an allele is transmitted and the conditional mean of trait value given the allele is not transmitted from a heterozygous parent. To build valid models, we calculate the variance covariance structure of trait values of offspring. Besides, the reduction of the number of parameters is discussed under an assumption of tight linkage between the trait locus and the marker. (2) For the data of offspring of nuclear family, we show that it can be used in general association study. In this case, the theoretical basis is the difference between the conditional mean of trait values given an allele is transmitted from a parent and the population mean. Then, we calculate variance-covariance structure of trait values of offspring. (3) Based on the theoretical analysis, mixed models are built for each type of the data, and related test statistics are proposed for association study. By power calculation and comparison, we show that, in some instances, the proposed test statistics have higher power than that by collapsing alleles to be new ones. The proposed models are used to analyze chromosomes 4 and chromosome 16 data of the Oxford asthma data, Genetic Analysis Workshop 12.     

Standardization and conversion of marker polymorphism measures

Large scale gene mapping efforts in domestic animals have generated and mapped a large number of genetic markers that are useful for mapping quantitative trait and disease loci and for DNA diagnostic purposes such as parentage testing. Marker polymorphism is an important criterion for selecting genetic markers in planning experiment for mapping quantitative trait loci or for DNA diagnostic purposes. Current formulations of marker polymorphism measures are functions of marker allele frequencies. In this study, two measures of marker polymorphism that are available from gene mapping studies and do not require allele frequencies were proposed and analyzed: the observed polymorphic information content (PIC) and the observed family information content (FIC). The observed FIC was more stable than the observed PIC because the observed FIC is unaffected by the variation in the frequency of heterozygous parents. However, both FIC and PIC are dependent on the gene mapping design. The effective number of alleles is recommended as a tool to standardize marker polymorphism measures so that polymorphism of different markers can be compared on an equal basis, and to obtain a new polymorphism measure (such an exclusion probability) from an existing measure (such as FIC). The usage of the effective number of alleles to standardize FIC, PIC and exclusion probabilities is illustrated using genetic markers in a published linkage map.