On two tests of fit for HLA data with no double blanks.

Stevens suggests a test of fit, based on Bernstein's estimators, of the Hardy-Weinberg law for the ABO system. Nam and Gart extend this test to the generalized ABO-like system and apply it to HLA data. When the recessive gene is rare, Huether and Murphy recall Haldane's point that its Bernstein's estimator is negatively biased and go on to suggest novel corrected versions of it. With the identification of more HLA antigens, it is not uncommon to find, in certain populations, that the sample data contain no double blanks; that is, every individual reacts to at least one antigen for a given locus. Gart and Nam give a simple score test of a zero true recessive-gene frequency for such situations. Here we examine the extended test of Stevens as a test of this hypothesis. We find that it is fully efficient for two codominant alleles but that when the number exceeds two its efficiency may be 50% or lower or as high as 100%, depending on the number of alleles and the pattern of gene frequencies. The tests are applied to a set of HLA data.     

The ML estimation and testing of generalized ABO-like data with no observed double recessives

The general method of the discrepancy or heterogeneity chi-square is applied to ABO-like data in which there are no observed double blanks in either the disease or the control group. When the recessive gene frequency is assumed zero, this method leads to an approximate chi-square test identical to that suggested by Smouse and Williams (1982, Biometrics 38, 757-768). When this assumption is relaxed, there arise two cases which are determined by whether the maximum likelihood estimate of this frequency is zero or not. It is shown that the value of the simple score statistic of Gart and Nam (1984, Biometrics 40, 887-894) discriminates between the two cases. The various omnibus test statistics for comparing groups are shown to differ little in several practical examples. However, under the more general assumption the appropriate degrees of freedom is one more than the number previously suggested.     

Approximate interval estimation of the difference in binomial parameters: correction for skewness and extension to multiple tables

Recently, Beal (1987, Biometrics 43, 941-950) found Mee's modification of Anbar's approximate interval estimation for the difference in binomial parameters to be a good choice in small sample sizes. As this method can be derived from the score theory of Bartlett, it is easily corrected for skewness. Exact numerical evaluation shows that this correction is not as important for this case as for the ratio of binomial parameters (Gart and Nam, 1988, Biometrics 44, 323-338). The score theory is also used to extend this method to the stratified or multiple-table case. Thus, good approximate interval estimates for differences, ratios, and odds ratios of binomial parameters can all be derived from the same general theory.     

Testing the Hardy-Weinberg equilibrium in the HLA system

A new method of testing the Hardy-Weinberg equilibrium in the human leukocyte antigen (HLA) system is proposed and applied to real data. The derivation is based on the maximum likelihood method and closely related to standard regression theory. The test statistic has a closed representation of residual sum of squares by a projection mapping of data onto the estimated regression plane. Under the Hardy-Weinberg law the noniterative estimates for the gene frequencies are suggested by the use of the projection mapping. The test statistic and gene frequency estimates are shown to be asymptotically equivalent to the maximum likelihood method and to be more efficient than the other suggested test statistic when there are more than two identified alleles.     

A score test for the possible presence of recessive alleles in generalized ABO-like genetic systems

In generalized ABO-like genetic systems, particularly for certain human leukocyte antigen (HLA) loci, it is quite likely, even for large sample sizes, that no double recessives (or 'double blanks') may be found. It has been suggested that all such data should be analyzed as if the true recessive gene frequency were 0. We derive an easily computed score [or C(alpha)] statistic which tests for the possibility that, in such instances, this frequency may exceed 0. An associated point estimator is suggested. The methods are applied to the data analyzed by Smouse and Williams (1982, Biometrics 38, 757-768).     

Some goodness-of-fit methods for the Poisson plus added zeros distribution

Methods for making inferences about the Poisson plus added zeros distribution and the truncated Poisson distribution are presented and illustrated with bacteriological data. Some of the methods are designed for testing the compatibility of the zero frequency with the Poisson distribution, whereas others are given for testing the goodness of fit for the truncated Poisson. In particular, a modified form of the Fisher index of dispersion is presented which is suitable for the truncated case. It is shown that the use of the usual expression of the index of dispersion for testing the adequacy of the truncated Poisson is not correct and leads to accepting inadequate fits more frequently than expected on the basis of test of significance. Furthermore, three test statistics are presented for testing the compatability of the zero frequency with the Poisson distribution. The results of the simulation show that two test statistics, one due to Cochran (W. G. Cochran, Biometrics 10:417-451, 1954) and the other to Rao and Chakravarti (C. R. Rao and I. M. Chakravarti, Biometrics 12:264-282, 1956), are preferable to those from the likelihood ratio test.     

The variance for the disequilibrium coefficient in the individual Hardy-Weinberg test

It is of interest to detect the deviation from Hardy-Weinberg proportion (HWP) for a particular heterozygote. Hernández and Weir (1989, Biometrics 45, 53-70) suggested a disequilibrium coefficient approach and proposed a 1-d.f. chi2 test. This note derives the appropriate variance under the null hypothesis for performing this individual HWP test. Examples of applying the test to human genetic disease data are presented.     

Some goodness-of-fit methods for the Poisson plus added zeros distribution.

Methods for making inferences about the Poisson plus added zeros distribution and the truncated Poisson distribution are presented and illustrated with bacteriological data. Some of the methods are designed for testing the compatibility of the zero frequency with the Poisson distribution, whereas others are given for testing the goodness of fit for the truncated Poisson. In particular, a modified form of the Fisher index of dispersion is presented which is suitable for the truncated case. It is shown that the use of the usual expression of the index of dispersion for testing the adequacy of the truncated Poisson is not correct and leads to accepting inadequate fits more frequently than expected on the basis of test of significance. Furthermore, three test statistics are presented for testing the compatability of the zero frequency with the Poisson distribution. The results of the simulation show that two test statistics, one due to Cochran (W. G. Cochran, Biometrics 10:417-451, 1954) and the other to Rao and Chakravarti (C. R. Rao and I. M. Chakravarti, Biometrics 12:264-282, 1956), are preferable to those from the likelihood ratio test.     

Genotype relative risks: methods for design and analysis of candidate-gene association studies.

Design and analysis methods are presented for studying the association of a candidate gene with a disease by using parental data in place of nonrelated controls. This alternative design eliminates spurious differences in allele frequencies between cases and nonrelated controls resulting from different ethnic origins and population stratification for these two groups. We present analysis methods which are based on two genetic relative risks: (1) the relative risk of disease for homozygotes with two copies of the candidate gene versus homozygotes without the candidate gene and (2) the relative risk for heterozygotes with one copy of the candidate gene versus homozygotes without the candidate gene. In addition to estimating the magnitude of these relative risks, likelihood methods allow specific hypotheses to be tested, namely, a test for overall association of the candidate gene with disease, as well as specific genetic hypotheses, such as dominant or recessive inheritance. Two likelihood methods are presented: (1) a likelihood method appropriate when Hardy-Weinberg equilibrium holds and (2) a likelihood method in which we condition on parental genotype data when Hardy-Weinberg equilibrium does not hold. The results for the relative efficiency of these two methods suggest that the conditional approach may at times be preferable, even when equilibrium holds. Sample-size and power calculations are presented for a multitiered design. The purpose of tier 1 is to detect the presence of an abnormal sequence for a postulated candidate gene among a small group of cases. The purpose of tier 2 is to test for association of the abnormal variant with disease, such as by the likelihood methods presented. The purpose of tier 3 is to confirm positive results from tier 2. Results indicate that required sample sizes are smaller when expression of disease is recessive, rather than dominant, and that, for recessive disease and large relative risks, necessary sample sizes may be feasible, even if only a small percentage of the disease can be attributed to the candidate gene.     

Testing Heterozygote Excess and Deficiency

Currently used tests of Hardy-Weinberg proportions do not take into account the nature of the alternative hypothesis, which is generally a heterozygote deficiency. Different exact tests, appropriate for small sample size and large number of alleles, are proposed in this perspective, and their properties are evaluated by power comparisons. Some tests are found to be close to optimal for the detection of inbreeding or heterozygote excess, one of which is a score test closely related to Robertson and Hill's estimator of the inbreeding coefficient. This test is also easily applied to multiple samples. Such tests are not always the most appropriate if alternative hypotheses differ from those considered here.     

随机漂移亚群体的两种模拟结果的比较

本文研究影响H-W平衡的亚群体结构，采用两种方式模拟随机漂移亚群体，比较两种模拟结果，分别依矩估计、极大似然估计计算近交系数，得到两种模拟结果是基本一致的     

Approximate interval estimation of the ratio of binomial parameters: a review and corrections for skewness

Various methods for finding confidence intervals for the ratio of binomial parameters are reviewed and evaluated numerically. It is found that the method based on likelihood scores (Koopman, 1984, Biometrics 40, 513-517; Miettinen and Nurminen, 1985, Statistics in Medicine 4, 213-226) performs best in achieving the nominal confidence coefficient, but it may distribute the tail probabilities quite disparately. Using general theory of Bartlett (1953, Biometrika 40, 306-317; 1955, Biometrika 42, 201-203), we correct this method for asymptotic skewness. Following Gart (1985, Biometrika 72, 673-677), we extend this correction to the case of estimating the common ratio in a series of two-by-two tables. Computing algorithms are given and applied to numerical examples. Parallel methods for the odds ratio and the ratio of Poisson parameters are noted.     

On logit confidence intervals for the odds ratio with small samples

Unless the true association is very strong, simple large-sample confidence intervals for the odds ratio based on the delta method perform well even for small samples. Such intervals include the Woolf logit interval and the related Gart interval based on adding .5 before computing the log odds ratio estimate and its standard error. The Gart interval smooths the observed counts toward the model of equiprobability, but one obtains better coverage probabilities by smoothing toward the independence model and by extending the interval in the appropriate direction when a cell count is zero.     

INBREEDING AND THE RATE OF SELF-FERTILIZATION IN A GRAPE FERN,BOTRYCHIUM DISSECTUM

Botrychium dissectum is a homosporous fern with bisexual, subterranean gametophytes. Because of these features, B. dissectum would be suspected of displaying a very high frequency of self-fertilization. Sporophytes collected from three populations of this species were assayed for heterozygosity by determining the electrophoretic mobility patterns displayed by two polymorphic enzymes. Extreme deviations from Hardy-Weinberg expectations were observed in each population and analyzed by means of F-statistics. The average inbreeding coefficient was found to be 0.951. A population genetic model is derived that demonstrates that the rate of intragametophytic self-fertilization in homosporous ferns is equal to the inbreeding coefficient calculated from deviations from Hardy-Weinberg expectations. It is therefore concluded that B. dissectum outcrosses about 5% of the time.     

Homogeneity score test for the intraclass version of the kappa statistics and sample-size determination in multiple or stratified studies

When the intraclass correlation coefficient or the equivalent version of the kappa agreement coefficient have been estimated from several independent studies or from a stratified study, we have the problem of comparing the kappa statistics and combining the information regarding the kappa statistics in a common kappa when the assumption of homogeneity of kappa coefficients holds. In this article, using the likelihood score theory extended to nuisance parameters (Tarone, 1988, Communications in Statistics-Theory and Methods 17(5), 1549-1556) we present an efficient homogeneity test for comparing several independent kappa statistics and, also, give a modified homogeneity score method using a noniterative and consistent estimator as an alternative. We provide the sample size using the modified homogeneity score method and compare it with that using the goodness-of-fit method (GOF) (Donner, Eliasziw, and Klar, 1996, Biometrics 52, 176-183). A simulation study for small and moderate sample sizes showed that the actual level of the homogeneity score test using the maximum likelihood estimators (MLEs) of parameters is satisfactorily close to the nominal and it is smaller than those of the modified homogeneity score and the goodness-of-fit tests. We investigated statistical properties of several noniterative estimators of a common kappa. The estimator (Donner et al., 1996) is essentially efficient and can be used as an alternative to the iterative MLE. An efficient interval estimation of a common kappa using the likelihood score method is presented.     

Powerful Haplotype-Based Hardy-Weinberg Equilibrium Tests for Tightly Linked Loci

Recently, there have been many case-control studies proposed to test for association between haplotypes and disease, which require the Hardy-Weinberg equilibrium (HWE) assumption of haplotype frequencies. As such, haplotype inference of unphased genotypes and development of haplotype-based HWE tests are crucial prior to fine mapping. The goodness-of-fit test is a frequently-used method to test for HWE for multiple tightly-linked loci. However, its degrees of freedom dramatically increase with the increase of the number of loci, which may lack the test power. Therefore, in this paper, to improve the test power for haplotype-based HWE, we first write out two likelihood functions of the observed data based on the Niu''s model (NM) and inbreeding model (IM), respectively, which can cause the departure from HWE. Then, we use two expectation-maximization algorithms and one expectation-conditional-maximization algorithm to estimate the model parameters under the HWE, IM and NM models, respectively. Finally, we propose the likelihood ratio tests LRT and LRT for haplotype-based HWE under the NM and IM models, respectively. We simulate the HWE, Niu''s, inbreeding and population stratification models to assess the validity and compare the performance of these two LRT tests. The simulation results show that both of the tests control the type I error rates well in testing for haplotype-based HWE. If the NM model is true, then LRT is more powerful. While, if the true model is the IM model, then LRT has better performance in power. Under the population stratification model, LRT is still more powerful. To this end, LRT is generally recommended. Application of the proposed methods to a rheumatoid arthritis data set further illustrates their utility for real data analysis.     

Inbreeding Depression and Inferred Deleterious-Mutation Parameters in Daphnia

DENG and LYNCH recently proposed a method for estimating deleterious genomic mutation parameters from changes in the mean and genetic variance of fitness traits upon inbreeding in outcrossing populations. Such observations are readily acquired in cyclical parthenogens. Selfing and life-table experiments were performed for two such Daphnia populations. We observed a significant inbreeding depression and an increase of genetic variance for all traits analyzed. DENG and LYNCH's original procedures were extended to estimate genomic mutation rate (U), mean dominance coefficient (h), mean selection coefficient (s), and scaled genomic mutational variance (V(m)/V(e)). On average, U, h, s and V(m)/V(e) (^ indicates an estimate) are 0.74, 0.30, 0.14 and 4.6E-4, respectively. For the true values, the U and h are lower bounds, and s and V(m)/V(e) upper bounds. The present U, h and V(m)/V(e) are in general concordance with earlier results. The discrepancy between the present s and that from mutation-accumulation experiments in Drosophila (~0.04) is discussed. It is shown that different reproductive modes do not affect gene frequency at mutation-selection equilibrium if mutational effects on fitness are multiplicative and not completely recessive.     

Inferring purging from pedigree data

The harmful effects of inbreeding can be reduced if deleterious recessive alleles were removed (purged) by selection against homozygotes in earlier generations. If only a few generations are involved, purging is due almost entirely to recessive alleles that reduce fitness to near zero. In this case the amount of purging and allele frequency change can be inferred approximately from pedigree data alone and are independent of the allele frequency. We examined pedigrees of 59,778 U.S. Jersey cows. Most of the pedigrees were for six generations, but a few went back slightly farther. Assuming recessive homozygotes have fitness 0, the reduction of total genetic load due to purging is estimated at 17%, but most of this is not expressed, being concealed by dominant alleles. Considering those alleles that are currently expressed due to inbreeding, the estimated amount of purging is such as to reduce the expressed load (inbreeding depression) in the current generation by 12.6%. That the reduction is not greater is due mainly to (1) generally low inbreeding levels because breeders in the past have tended to avoid consanguineous matings, and (2) there is essentially no information more than six generations back. The methods used here should be applicable to other populations for which there is pedigree information.     

Genetic diversity and breeding system in a group of neotropical epiphytic ferns (Pleopeltis; Polypodiaceae)

Epiphytes are ecologically important components of tropical forests worldwide and yet they have been underrepresented in studies of reproductive biology. Given the presumed ephemeral nature of their substrates, and the importance of dispersal and colonization, epiphytes might be expected to undergo substantial inbreeding to ensure reproductive success, as in weedy terrestrial plants. While there is some evidence for inbreeding in epiphytic angiosperms, the only previous studies of fern epiphytes indicate that they are predominantly outcrossing. The present study reports on the genetic diversity and breeding system of six members of the Neotropical epiphytic fern genus Pleopeltis (Polypodiaceae). A survey of isozyme variability using starch gel electrophoresis revealed high population levels of polymorphism (P = 0.62), allelic diversity (A = 2.3), and individual heterozygosity (Ho = 0.181), but little differentiation among conspecific populations (I 3 0.98; Gst = 0.048), and high interpopulational gene flow rates (Nm > 1). In addition, there was no indication of homozygote excess within populations that might indicate a history of selfing in these ferns: populations generally conformed to Hardy-Weinberg expected genotype frequencies, and both Wright's inbreeding coefficient (Fis) and Holsinger's intragametophytic selfing rates approached zero. Possible mechanisms limiting inbreeding in these ferns include antheridiogen activity and high levels of genetic load that would lead to inbreeding depression upon selfing.     

Detection of inbreeding effects by the chi-square test on genotypic and phenotypic frequencies.

The problem of detection of inbreeding effects by the chi-square goodness-of-fit test is investigated. The noncentrality parameter of the limiting chi-square distribution is considered as a function of the inbreeding coefficient in two models: (1) a gene locus with m codominant alleles and (2) the generalized ABO model. Previous results by Ward and Sing and by Chakraborty and Rao were based on inadequate statistical reasoning. In the first model, the resulting noncentrality parameter is identical with that found by Ward and Sing. For the second model, the limiting noncentrality parameter is found to be zero.