The Estimation of the Segregation Parameter when Ascertainment is not Complete

Procedures to estimate the genetic segregation parameter when ascertainment of families is incomplete, have previously relied on iterative computer algorithms since estimators with closed form are lacking. We now present the Minimum Variance Unbiased Estimator for the segregation parameter under any ascertainment probability. This estimator assumes a simple form when ascertainment is complete. We also present a simple estimator, akin to Li and Mantel's (1968) estimator, but without the restriction that ascertainment be complete. The performance of these estimators is compared with respect to asymptotic efficiency. We also provide tables that define the required number of families of a given size that need to be sampled to achieve a specific power for testing simple hypothesis on the segregation parameter.     

Segregation of HLA A,B haplotypes and the distribution of antigen mismatches between mother and offspring in Hutterite families

Segregation of HLA haplotypes and offspring genotype distributions were analyzed in families from an inbred Caucasoid population, the Dariusleut Hutterite Brethren. Both parents and from one to 12 offspring were typed for HLA-A and -B antigens in 108 families. Segregation of paternal haplotypes was analyzed conditional on sibship size in 95 sibships (a total of 547 offspring), and segregation of maternal haplotypes, in 90 sibships (a total of 515 offspring). The distribution of the number of different genotypes among the offspring was analyzed conditional on sibship size in 90 families (515 offspring) where four equiprobable genotypes were expected. The distribution of the number of antigenic differences or mismatches for broad specificities between mother and offspring was analyzed in pooled family data consisting of a total of 377 offspring comprising 68 families. Compared with the multinomial distribution of segregation classes of haplotypes, there was no significant departure (probability .05 or less) from the expected segregation ratio for either paternal or maternal haplotypes. Compared with the multinomial distribution of the number of genotypes among the offspring, only two of the 11 sibship sizes had configurations that exceeded the 5% level of significance. Given the number of statistical tests performed, it is likely that these results could be explained by chance variation. Finally, there was no relative deficiency of offspring who were less mismatched with their mother for HLA-A and -B broad specificities. Therefore, if HLA-A,B region variation does have a major effect on the differential survival of fetuses in some families, it is an uncommon factor among fertile couples from this inbred population.     

Ascertainment: An Overwiew of the Classical Segregation Analysis Model for Independent Sibships

Several different methodologies for parameter estimation under various ascertainment sampling schemes have been proposed in the past. In this article, some of the methodologies that have been proposed for independent sibships under the classical segregation analysis model are synthesized, and the general likelihoods derived for single, multiple and complete ascertainment. The issue of incorporating the sibship size distribution into the analysis is addressed, and the effect of conditioning the likelihood on the observed sibship sizes is discussed. It is shown that when the number of probands in a sibship is not specified, the corresponding likelihood can be used for a broader class of ascertainment schemes than is subsumed by the classical model.     

Optimized group sequential study designs for tests of genetic linkage and association in complex diseases

The study of genetic linkage or association in complex traits requires large sample sizes, as the expected effect sizes are small and extremely low significance levels need to be adopted. One possible way to reduce the numbers of phenotypings and genotypings is the use of a sequential study design. Here, average sample sizes are decreased by conducting interim analyses with the possibility to stop the investigation early if the result is significant. We applied optimized group sequential study designs to the analysis of genetic linkage (one-sided mean test) and association (two-sided transmission/disequilibrium test). For designs with two and three stages at overall significance levels of.05 and.0001 and a power of.8, we calculated necessary sample sizes, time points, and critical boundaries for interim and final analyses. Monte Carlo simulation analyses were performed to confirm the validity of the asymptotic approximation. Furthermore, we calculated average sample sizes required under the null and alternative hypotheses in the different study designs. It was shown that the application of a group sequential design led to a maximal increase in sample size of 8% under the null hypothesis, compared with the fixed-sample design. This was contrasted by savings of up to 20% in average sample sizes under the alternative hypothesis, depending on the applied design. These savings affect the amounts of genotyping and phenotyping required for a study and therefore lead to a significant decrease in cost and time.     

A general approach for sample size and power calculations based on the Haseman-Elston method

Risch and Zhang (1995; Science 268: 1584-9) reported a simple sample size and power calculation approach for the Haseman-Elston method and based their computations on the null hypothesis of no genetic effect. We argue that the more reasonable null hypothesis is that of no recombination. For this null hypothesis, we provide a general approach for sample size and power calculations within the Haseman-Elston framework. We demonstrate the validity of our approach in a Monte-Carlo simulation study and illustrate the differences using data from published segregation analyses on body weight and heritability estimates on carotid artery artherosclerotic lesions.     

A Discordant-Sibship Test for Disequilibrium and Linkage: No Need for Parental Data

The sibship disequilibrium test (SDT) is designed to detect both linkage in the presence of association and association in the presence of linkage (linkage disequilibrium). The test does not require parental data but requires discordant sibships with at least one affected and one unaffected sibling. The SDT has many desirable properties: it uses all the siblings in the sibship; it remains valid if there are misclassifications of the affectation status; it does not detect spurious associations due to population stratification; asymptotically it has a chi2 distribution under the null hypothesis; and exact P values can be easily computed for a biallelic marker. We show how to extend the SDT to markers with multiple alleles and how to combine families with parents and data from discordant sibships. We discuss the power of the test by presenting sample-size calculations involving a complex disease model, and we present formulas for the asymptotic relative efficiency (which is approximately the ratio of sample sizes) between SDT and the transmission/disequilibrium test (TDT) for special family structures. For sib pairs, we compare the SDT to a test proposed both by Curtis and, independently, by Spielman and Ewens. We show that, for discordant sib pairs, the SDT has good power for testing linkage disequilibrium relative both to Curtis''s tests and to the TDT using trios comprising an affected sib and its parents. With additional sibs, we show that the SDT can be more powerful than the TDT for testing linkage disequilibrium, especially for disease prevalence >.3.     

The effects of conditioning on probands to correct for multiple ascertainment.

Cannings and Thompson suggested conditioning on the phenotypes of the probands to correct for ascertainment in the analysis of pedigree data. The method assumes single ascertainment and can be expected to yield asymptotically biased parameter estimates except in this specific case. However, because the method is easy to apply, we investigated the degree of bias in the more typical situation of multiple ascertainment, in the hope that the bias might be small and that the method could be applied more generally. To explore the utility of conditioning on probands to correct for multiple ascertainment, we calculated the asymptotic value of the segregation ratio for two versions of the simple Mendelian segregation model on sibship data. For both versions, we found that this asymptotic value decreased approximately linearly as the ascertainment probability increased. When ascertainment was complete, the segregation-ratio estimates were zero, not just asymptotically but for finite sample size as well. In some cases, conditioning on probands actually resulted in greater parameter bias than no ascertainment correction at all. These results hold for a variety of sibship-size distributions, several modes of inheritance, and a wide range of population prevalences of affected individuals.     

A Comparison of Tests for Marginal Homogeneity in Square Contingency Tables

Several asymptotic tests were proposed for testing the null hypothesis of marginal homogeneity in square contingency tables with r categories. A simulation study was performed for comparing the power of four finite conservative conditional test procedures and of two asymptotic tests for twelve different contingency schemes for small sample sizes. While an asymptotic test proposed by STUART (1955) showed a rather satisfactory behaviour for moderate sample sizes, an asymptotic test proposed by BHAPKAR (1966) was quite anticonservative. With no a priori information the performance of (r - 1) simultaneous conditional binomial tests with a Bonferroni adjustment proved to be a quite efficient procedure. With assumptions about where to expect the deviations from the null hypothesis, other procedures favouring the larger or smaller conditional sample sizes, respectively, can have a great efficiency. The procedures are illustrated by means of a numerical example from clinical psychology.     

On the power of some binomial modifications of the Bonferroni multiple test

Widely used in testing statistical hypotheses, the Bonferroni multiple test has a rather low power that entails a high risk to accept falsely the overall null hypothesis and therefore to not detect really existing effects. We suggest that when the partial test statistics are statistically independent, it is possible to reduce this risk by using binomial modifications of the Bonferroni test. Instead of rejecting the null hypothesis when at least one of n partial null hypotheses is rejected at a very high level of significance (say, 0.005 in the case of n = 10), as it is prescribed by the Bonferroni test, the binomial tests recommend to reject the null hypothesis when at least k partial null hypotheses (say, k = [n/2]) are rejected at much lower level (up to 30-50%). We show that the power of such binomial tests is essentially higher as compared with the power of the original Bonferroni and some modified Bonferroni tests. In addition, such an approach allows us to combine tests for which the results are known only for a fixed significance level. The paper contains tables and a computer program which allow to determine (retrieve from a table or to compute) the necessary binomial test parameters, i.e. either the partial significance level (when k is fixed) or the value of k (when the partial significance level is fixed).     

Statistical properties of Teng and Risch's sibship type tests for detecting an association between disease and a candidate allele

Risch and Teng [Genome Res 1998;8:1273-1288] and Teng and Risch [Genome Res 1999;9:234-241] proposed a class of transmission/disequilibrium test-like statistical tests based on the difference between the estimated allele frequencies in the affected and control populations. They evaluated the power of a variety of family-based and nonfamily-based designs for detecting an association between a candidate allele and disease. Because they were concerned with diseases with low penetrances, their power calculations assumed that unaffected individuals can be treated as a random sample from the population. They predicted that this assumption rendered their sample size calculations slightly conservative. We generalize their partial ascertainment conditioning by including the status of the unaffected sibs in the calculations of the distribution and power of the statistic used to compare the allele frequency in affected offspring to the estimated frequency in the parents, based on sibships with genotyped affected and unaffected sibs. Sample size formulas for our full ascertainment methods are presented. The sample sizes for our procedure are compared to those of Teng and Risch. The numerical results and simulations indicate that the simplifying assumption used in Teng and Risch can produce both conservative and anticonservative results. The magnitude of the difference between the sample sizes needed by their partial ascertainment approximation and the full ascertainment is small in the circumstances they focused on but can be appreciable in others, especially when the baseline penetrances are moderate. Two other statistics, using different estimators for the variance of the basic statistic comparing the allele frequencies in the affected and unaffected sibs are introduced. One of them incorporates an estimate of the null variance obtained from an auxiliary sample and appears to noticeably decrease the sample sizes required to achieve a prespecified power.     

On the assessment of statistical significance in disease-gene discovery.

One of the major challenges facing genome-scan studies to discover disease genes is the assessment of the genomewide significance. The assessment becomes particularly challenging if the scan involves a large number of markers collected from a relatively small number of meioses. Typically, this assessment has two objectives: to assess genomewide significance under the null hypothesis of no linkage and to evaluate true-positive and false-positive prediction error rates under alternative hypotheses. The distinction between these goals allows one to formulate the problem in the well-established paradigm of statistical hypothesis testing. Within this paradigm, we evaluate the traditional criterion of LOD score 3.0 and a recent suggestion of LOD score 3.6, using the Monte Carlo simulation method. The Monte Carlo experiments show that the type I error varies with the chromosome length, with the number of markers, and also with sample sizes. For a typical setup with 50 informative meioses on 50 markers uniformly distributed on a chromosome of average length (i.e., 150 cM), the use of LOD score 3.0 entails an estimated chromosomewide type I error rate of.00574, leading to a genomewide significance level >.05. In contrast, the corresponding type I error for LOD score 3.6 is.00191, giving a genomewide significance level of slightly <.05. However, with a larger sample size and a shorter chromosome, a LOD score between 3.0 and 3.6 may be preferred, on the basis of proximity to the targeted type I error. In terms of reliability, these two LOD-score criteria appear not to have appreciable differences. These simulation experiments also identified factors that influence power and reliability, shedding light on the design of genome-scan studies.     

The quantitative LOD score: test statistic and sample size for exclusion and linkage of quantitative traits in human sibships.

We present a test statistic, the quantitative LOD (QLOD) score, for the testing of both linkage and exclusion of quantitative-trait loci in randomly selected human sibships. As with the traditional LOD score, the boundary values of 3, for linkage, and -2, for exclusion, can be used for the QLOD score. We investigated the sample sizes required for inferring exclusion and linkage, for various combinations of linked genetic variance, total heritability, recombination distance, and sibship size, using fixed-size sampling. The sample sizes required for both linkage and exclusion were not qualitatively different and depended on the percentage of variance being linked or excluded and on the total genetic variance. Information regarding linkage and exclusion in sibships larger than size 2 increased as approximately all possible pairs n(n-1)/2 up to sibships of size 6. Increasing the recombination (theta) distance between the marker and the trait loci reduced empirically the power for both linkage and exclusion, as a function of approximately (1-2theta)4.     

Statistical tests for taxonomic distinctiveness from observations of monophyly

The observation of monophyly for a specified set of genealogical lineages is often used to place the lineages into a distinctive taxonomic entity. However, it is sometimes possible that monophyly of the lineages can occur by chance as an outcome of the random branching of lineages within a single taxon. Thus, especially for small samples, an observation of monophyly for a set of lineages--even if strongly supported statistically--does not necessarily indicate that the lineages are from a distinctive group. Here I develop a test of the null hypothesis that monophyly is a chance outcome of random branching. I also compute the sample size required so that the probability of chance occurrence of monophyly of a specified set of lineages lies below a prescribed tolerance. Under the null model of random branching, the probability that monophyly of the lineages in an index group occurs by chance is substantial if the sample is highly asymmetric, that is, if only a few of the sampled lineages are from the index group, or if only a few lineages are external to the group. If sample sizes are similar inside and outside the group of interest, however, chance occurrence of monophyly can be rejected at stringent significance levels (P < 10(-5)) even for quite small samples (approximately 20 total lineages). For a fixed total sample size, rejection of the null hypothesis of random branching in a single taxon occurs at the most stringent level if samples of nearly equal size inside and outside the index group--with a slightly greater size within the index group--are used. Similar results apply, with smaller sample sizes needed, when reciprocal monophyly of two groups, rather than monophyly of a single group, is of interest. The results suggest minimal sample sizes required for inferences to be made about taxonomic distinctiveness from observations of monophyly.     

POLYTOMIES AND THE POWER OF PHYLOGENETIC INFERENCE

Although phylogenetic hypotheses can provide insights into mechanisms of evolution, their utility is limited by our inability to differentiate simultaneous speciation events (hard polytomies) from rapid cladogenesis (soft polytomies). In the present paper, we tested the potential for statistical power analysis to differentiate between hard and soft polytomies in molecular phytogenies. Classical power analysis typically is used a priori to determine the sample size required to detect a particular effect size at a particular level of significance (a) with a certain power (1 – β). A posteriori, power analysis is used to infer whether failure to reject a null hypothesis results from lack of an effect or from insufficient data (i.e., low power). We adapted this approach to molecular data to infer whether polytomies result from simultaneous branching events or from insufficient sequence information. We then used this approach to determine the amount of sequence data (sample size) required to detect a positive branch length (effect size). A worked example is provided based on the auklets (Charadriiformes: Alcidae), a group of seabirds among which relationships are represented by a polytomy, despite analyses of over 3000 bp of sequence data. We demonstrate the calculation of effect sizes and sample sizes from sequence data using a normal curve test for difference of a proportion from an expected value and a t-test for a difference of a mean from an expected value. Power analyses indicated that the data for the auklets should be sufficient to differentiate speciation events that occurred at least 100,000 yr apart (the duration of the shortest glacial and interglacial events of the Pleistocene), 2.6 million years ago.     

How often are invasion-induced ecological impacts missed?

Managers and policy makers depend on empirical research to guide and support biosecurity measures that mitigate introduced species’ impacts. Research contributing to this knowledge base generally uses null hypothesis significance testing to determine the significance of data patterns. However, reliance on traditional statistical significance testing methods, combined with small effect and sample size and large variability inherent to many impact studies, may obscure effects on native species, communities or ecosystems. This may result in false certainty of no impact. We investigated potential Type II error rates and effect sizes for 31 non-significant empirical evaluations of impact for introduced algal and crustacean species. We found low power consistently led to acceptance of Type II errors at rates 5.6–19 times greater than Type I errors (despite moderate to large effect sizes). Our results suggest that introduced species for which impact studies have statistically non-significant outcomes (often interpreted as “no impact”) may potentially have large impacts that are missed due to small sample or effect sizes and/or high variation. This alarming willingness to “miss” impacts has severe implications for conservation efforts, including under-managing species’ impacts and discounting the costs of Type II errors.     

Sample size determination for establishing equivalence/noninferiority via ratio of two proportions in matched-pair design

In this article, we propose approximate sample size formulas for establishing equivalence or noninferiority of two treatments in match-pairs design. Using the ratio of two proportions as the equivalence measure, we derive sample size formulas based on a score statistic for two types of analyses: hypothesis testing and confidence interval estimation. Depending on the purpose of a study, these formulas can be used to provide a sample size estimate that guarantees a prespecified power of a hypothesis test at a certain significance level or controls the width of a confidence interval with a certain confidence level. Our empirical results confirm that these score methods are reliable in terms of true size, coverage probability, and skewness. A liver scan detection study is used to illustrate the proposed methods.     

The effect of proband designation on segregation analysis

In many family studies, it is often difficult to know exactly how the families were ascertained. Even if known, the circumstances under which the families came to the attention of the study may violate the assumptions of classical ascertainment bias correction. The purpose of this work was to investigate the effect on segregation analysis of violations of the assumptions of the classical ascertainment model. We simulated family data generated under a simple recessive model of inheritance. We then ascertained families under different "scenarios." These scenarios were designed to simulate actual conditions under which families come to the attention of-and then interact with-a clinic or genetic study. We show that how one designates probands, which one must do under the classical ascertainment model, can influence parameter estimation and hypothesis testing. We demonstrate that, in some cases, there may be no "correct" way to designate probands. Further, we show that interactions within the family, the conditions under which the genetic study must function, and even social influences can have a profound effect on segregation analysis. We also propose a method for dealing with the ascertainment problem that is applicable to almost any study situation.     

Exact sample sizes needed to detect dependence in 2 x 3 tables

Many medical and biological studies entail classifying a number of observations according to two factors, where one has two and the other three possible categories. This is the case of, for example, genetic association studies of complex traits with single-nucleotide polymorphisms (SNPs), where the a priori statistical planning, analysis, and interpretation of results are of critical importance. Here, we present methodology to determine the minimum sample size required to detect dependence in 2 x 3 tables based on Fisher's exact test, assuming that neither of the two margins is fixed and only the grand total N is known in advance. We provide the numerical tools necessary to determine these sample sizes for desired power, significance level, and effect size, where only the computational time can be a limitation for extreme parameter values. These programs can be accessed at . This solution of the sample size problem for an exact test will permit experimentalists to plan efficient sampling designs, determine the extent of statistical support for their hypotheses, and gain insight into the repeatability of their results. We apply this solution to the sample size problem to three empirical studies, and discuss the results with specified power and nominal significance levels.     

Sample-size considerations and strategies for linkage analysis in autosomal recessive disorders.

The opportunity raised by recombinant DNA technology to develop a linkage marker panel that spans the human genome requires cost-efficient strategies for its optimal utilization. Questions arise as to whether it is more cost-effective to convert a dimorphic restriction enzyme marker system into a highly polymorphic system or, instead, to increase the number of families studied, simply using the available marker alleles. The choice is highly dependent on the population available for study, and, therefore, an examination of the informational content of the various family structures is important to obtain the most informative data. To guide such decisions, we have developed tables of the average sample number of families required to detect linkage for autosomal recessive disorders under single backcross and under "fully informative" matings. The latter cross consists of a marker locus with highly polymorphic codominant alleles such that the parental marker genotypes can be uniquely distinguished. The sampling scheme considers families with unaffected parents of known mating types ascertained via affected offspring, for sibship sizes ranging from two to four and various numbers of affected individuals. The sample-size tables, calculated for various values of the recombination fractions and lod scores, may serve as a guide to a more efficient application of the restriction fragment length polymorphism technology to sequential linkage analysis.     

Linkage data on MN and the Hb beta locus.

A sample of 28 informative families was studied for linkage between Hb beta and MN. Values of the neuterized recombination fraction from these and other families from the literature excluded a recombination fraction of less than .30 between these loci. Our results support different recombination values for males and females (theta equals .34 abd .50, respectively). A simple approach to estimate the sample size required as well as a study of the relationship between sibship size and sample size under conditions of loose linkage are also presented.