Testing the non-unity of rate ratio under inverse sampling

Inverse sampling is considered to be a more appropriate sampling scheme than the usual binomial sampling scheme when subjects arrive sequentially, when the underlying response of interest is acute, and when maximum likelihood estimators of some epidemiologic indices are undefined. In this article, we study various statistics for testing non-unity rate ratios in case-control studies under inverse sampling. These include the Wald, unconditional score, likelihood ratio and conditional score statistics. Three methods (the asymptotic, conditional exact, and Mid-P methods) are adopted for P-value calculation. We evaluate the performance of different combinations of test statistics and P-value calculation methods in terms of their empirical sizes and powers via Monte Carlo simulation. In general, asymptotic score and conditional score tests are preferable for their actual type I error rates are well controlled around the pre-chosen nominal level, and their powers are comparatively the largest. The exact version of Wald test is recommended if one wants to control the actual type I error rate at or below the pre-chosen nominal level. If larger power is expected and fluctuation of sizes around the pre-chosen nominal level are allowed, then the Mid-P version of Wald test is a desirable alternative. We illustrate the methodologies with a real example from a heart disease study.     

Permutation tests for random effects in linear mixed models

Inference regarding the inclusion or exclusion of random effects in linear mixed models is challenging because the variance components are located on the boundary of their parameter space under the usual null hypothesis. As a result, the asymptotic null distribution of the Wald, score, and likelihood ratio tests will not have the typical χ(2) distribution. Although it has been proved that the correct asymptotic distribution is a mixture of χ(2) distributions, the appropriate mixture distribution is rather cumbersome and nonintuitive when the null and alternative hypotheses differ by more than one random effect. As alternatives, we present two permutation tests, one that is based on the best linear unbiased predictors and one that is based on the restricted likelihood ratio test statistic. Both methods involve weighted residuals, with the weights determined by the among- and within-subject variance components. The null permutation distributions of our statistics are computed by permuting the residuals both within and among subjects and are valid both asymptotically and in small samples. We examine the size and power of our tests via simulation under a variety of settings and apply our test to a published data set of chronic myelogenous leukemia patients.     

Testing inflated zeros in binomial regression models

Binomial regression models are commonly applied to proportion data such as those relating to the mortality and infection rates of diseases. However, it is often the case that the responses may exhibit excessive zeros; in such cases a zero‐inflated binomial (ZIB) regression model can be applied instead. In practice, it is essential to test if there are excessive zeros in the outcome to help choose an appropriate model. The binomial models can yield biased inference if there are excessive zeros, while ZIB models may be unnecessarily complex and hard to interpret, and even face convergence issues, if there are no excessive zeros. In this paper, we develop a new test for testing zero inflation in binomial regression models by directly comparing the amount of observed zeros with what would be expected under the binomial regression model. A closed form of the test statistic, as well as the asymptotic properties of the test, is derived based on estimating equations. Our systematic simulation studies show that the new test performs very well in most cases, and outperforms the classical Wald, likelihood ratio, and score tests, especially in controlling type I errors. Two real data examples are also included for illustrative purpose.     

Inference of haplotype effects in case-control studies using unphased genotype and environmental data

A retrospective likelihood-based approach was proposed to test and estimate the effect of haplotype on disease risk using unphased genotype data with adjustment for environmental covariates. The proposed method was also extended to handle the data in which the haplotype and environmental covariates are not independent. Likelihood ratio tests were constructed to test the effects of haplotype and gene-environment interaction. The model parameters such as haplotype effect size was estimated using an Expectation Conditional-Maximization (ECM) algorithm developed by Meng and Rubin (1993). Model-based variance estimates were derived using the observed information matrix. Simulation studies were conducted for three different genetic effect models, including dominant effect, recessive effect, and additive effect. The results showed that the proposed method generated unbiased parameter estimates, proper type I error, and true beta coverage probabilities. The model performed well with small or large sample sizes, as well as short or long haplotypes.     

Testing for group effect in a 2 x k heteroscedastic ANOVA model

It is natural to want to relax the assumption of homoscedasticity and Gaussian error in ANOVA models. For a two-way ANOVA model with 2 x k cells, one can derive tests of main effect for the factor with two levels (referred to as group) without assuming homoscedasticity or Gaussian error. Empirical likelihood can be used to derive testing procedures. An approximate empirical likelihood ratio test (AELRT) is derived for the test of group main effect. To approximate the distributions of the test statistics under the null hypothesis, simulation from the approximate empirical maximum likelihood estimate (AEMLE) restricted by the null hypothesis is used. The homoscedastic ANOVA F -test and a Box-type approximation to the distribution of the heteroscedastic ANOVA F -test are compared to the AELRT in level and power. The AELRT procedure is shown by simulation to have appropriate type I error control (although possibly conservative) when the distribution of the test statistics are approximated by simulation from the constrained AEMLE. The methodology is motivated and illustrated by an analysis of folate levels in the blood among two alcohol intake groups while accounting for gender.     

A note on permutation tests for variance components in multilevel generalized linear mixed models

In many applications of generalized linear mixed models to multilevel data, it is of interest to test whether a random effects variance component is zero. It is well known that the usual asymptotic chi-square distribution of the likelihood ratio and score statistics under the null does not necessarily hold. In this note we propose a permutation test, based on randomly permuting the indices associated with a given level of the model, that has the correct Type I error rate under the null. Results from a simulation study suggest that it is more powerful than tests based on mixtures of chi-square distributions. The proposed test is illustrated using data on the familial aggregation of sleep disturbance.     

Trend Test for Overdispersed Proportions

The Cochran-Armitage test has commonly been used for a trend test in binomial proportions. The quasi-likelihood method provides a simple approach to model extra-binomial proportions. Two versions of the score and Wald tests using different parameterizations for the extra-binomial variance were investigated: one in terms of intercluster correlation, and another in terms of variance. The Monte Carlo simulation was used to evaluate the performance of the each version of the score test and the Wald test, and the Cochran-Armitage test. The simulation shows that the Cochran-Armitage test has the proper size only for the binomial sample data, and the test is no longer valid when applied to the extra-binomial data. The Wald test is more likely to exceed the nominal level than the score test under either intercluster correlation model or variance model. Both score tests performed very well even with the binomial data; the tests control the type I error and in the meantime maintain the power of detecting the dose effects. Based on the design considered in this paper, the two scores test are comparable. The score test based on the intercluster correlations model seems better controlling the Type I error but appears less powerful than that based on the variance model. An example from a developmental toxicity experiment is given.     

A hybrid omnibus test for generalized semiparametric single‐index models with high‐dimensional covariate sets

Numerous statistical methods have been developed for analyzing high‐dimensional data. These methods often focus on variable selection approaches but are limited for the purpose of testing with high‐dimensional data. They are often required to have explicit‐likelihood functions. In this article, we propose a “hybrid omnibus test” for high‐dicmensional data testing purpose with much weaker requirements. Our hybrid omnibus test is developed under a semiparametric framework where a likelihood function is no longer necessary. Our test is a version of a frequentist‐Bayesian hybrid score‐type test for a generalized partially linear single‐index model, which has a link function being a function of a set of variables through a generalized partially linear single index. We propose an efficient score based on estimating equations, define local tests, and then construct our hybrid omnibus test using local tests. We compare our approach with an empirical‐likelihood ratio test and Bayesian inference based on Bayes factors, using simulation studies. Our simulation results suggest that our approach outperforms the others, in terms of type I error, power, and computational cost in both the low‐ and high‐dimensional cases. The advantage of our approach is demonstrated by applying it to genetic pathway data for type II diabetes mellitus.     

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.     

Rejoinder to ``Asymptotic Distribution of Score Statistics for Spatial Cluster Detection with Censored Data"

Summary. This short note evaluates the assumptions required for a permutation test to approximate the null distribution of the spatial scan statistic for censored outcomes proposed in Cook et al. (2007). In particular, we study the exchangeability conditions required for such a test under survival models. A simulation study is further performed to assess the impact on the type I error when the global exchangeability assumption is violated and to determine whether the permutation test still well approximates the null distribution.     

Exact tests for the analysis of case-control studies of genetic markers

OBJECTIVES: The association of a candidate gene with disease can be evaluated by a case-control study in which the genotype distribution is compared for diseased cases and unaffected controls. Usually, the data are analyzed with Armitage's test using the asymptotic null distribution of the test statistic. Since this test does not generally guarantee a type I error rate less than or equal to the significance level alpha, tests based on exact null distributions have been investigated. METHODS: An algorithm to generate the exact null distribution for both Armitage's test statistic and a recently proposed modification of the Baumgartner-Weiss-Schindler statistic is presented. I have compared the tests in a simulation study. RESULTS: The asymptotic Armitage test is slightly anticonservative whereas the exact tests control the type I error rate. The exact Armitage test is very conservative, but the exact test based on the modification of the Baumgartner-Weiss-Schindler statistic has a type I error rate close to alpha. The exact Armitage test is the least powerful test; the difference in power between the other two tests is often small and the comparison does not show a clear winner. CONCLUSION: Simulation results indicate that an exact test based on the modification of the Baumgartner-Weiss-Schindler statistic is preferable for the analysis of case-control studies of genetic markers.     

Statistical tests of genetic association for case-control study designs

The central theme in case-control genetic association studies is to efficiently identify genetic markers associated with trait status. Powerful statistical methods are critical to accomplishing this goal. A popular method is the omnibus Pearson's chi-square test applied to genotype counts. To achieve increased power, tests based on an assumed trait model have been proposed. However, they are not robust to model misspecification. Much research has been carried out on enhancing robustness of such model-based tests. An analysis framework that tests the equality of allele frequency while allowing for different deviation from Hardy-Weinberg equilibrium (HWE) between cases and controls is proposed. The proposed method does not require specification of trait models nor HWE. It involves only 1 degree of freedom. The likelihood ratio statistic, score statistic, and Wald statistic associated with this framework are introduced. Their performance is evaluated by extensive computer simulation in comparison with existing methods.     

Independent contrasts and regression through the origin

Following the pioneering work of Felsenstein and Garland, phylogeneticists have been using regression through the origin to analyze comparative data using independent contrasts. The reason why regression through the origin must be used with such data was revisited. The demonstration led to the formulation of a permutation test for the coefficient of determination and the regression coefficient estimates in regression through the origin. Simulations were carried out to measure type I error and power of the parametric and permutation tests under two models of data generation: regression models I and II (correlation model). Although regression through the origin assumes model I data, in independent contrast data error is present in the explanatory as well as the response variables. Two forms of permutations were investigated to test the regression coefficients: permutation of the values of the response variable y, and permutation of the residuals of the regression model. The simulations showed that the parametric tests or any of the permutation tests can be used when the error is normal, which is the usual assumption in independent contrast studies; only the test by permutation of y should be used when the error is highly asymmetric; and the parametric tests should be used when extreme values are present in covariables. Two examples are presented. The first one concerns non-specificity in fish parasites of the genus Lamellodiscus, the second the richness in parasites in 78 species of mammals.     

Testing approaches for overdispersion in poisson regression versus the generalized poisson model

Overdispersion is a common phenomenon in Poisson modeling, and the negative binomial (NB) model is frequently used to account for overdispersion. Testing approaches (Wald test, likelihood ratio test (LRT), and score test) for overdispersion in the Poisson regression versus the NB model are available. Because the generalized Poisson (GP) model is similar to the NB model, we consider the former as an alternate model for overdispersed count data. The score test has an advantage over the LRT and the Wald test in that the score test only requires that the parameter of interest be estimated under the null hypothesis. This paper proposes a score test for overdispersion based on the GP model and compares the power of the test with the LRT and Wald tests. A simulation study indicates the score test based on asymptotic standard Normal distribution is more appropriate in practical application for higher empirical power, however, it underestimates the nominal significance level, especially in small sample situations, and examples illustrate the results of comparing the candidate tests between the Poisson and GP models. A bootstrap test is also proposed to adjust the underestimation of nominal level in the score statistic when the sample size is small. The simulation study indicates the bootstrap test has significance level closer to nominal size and has uniformly greater power than the score test based on asymptotic standard Normal distribution. From a practical perspective, we suggest that, if the score test gives even a weak indication that the Poisson model is inappropriate, say at the 0.10 significance level, we advise the more accurate bootstrap procedure as a better test for comparing whether the GP model is more appropriate than Poisson model. Finally, the Vuong test is illustrated to choose between GP and NB2 models for the same dataset.     

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.     

Fitting Mixed Poisson Regression Models Using Quasi-Likelihood Methods

This paper is to investigate the use of the quasi-likelihood, extended quasi-likelihood, and pseudo-likelihood approach to estimating and testing the mean parameters with respect to two variance models, M1: φ μ^θ(1+μphis;) and M2: φ μ^θ(1+τ). Simulation was conducted to compare the bias and standard deviation, and type I error of the Wald tests, based on the model-based and robust variance estimates, using the three semi-parametric approaches under four mixed Poisson models, two variance structures, and two sample sizes. All methods perform reasonably well in terms of bias. Type I error of the Wald test, based on either the model-based or robust estimate, tends to be larger than the nominal level when over-dispersion is moderate. The extended quasi-likelihood method with the variance model M1 performs more consistently in terms of the efficiency and controlling the type I error than with the model M2, and better than the pseudo-likelihood approach with either the M1 or M2 model. The model-based estimate seems to perform better than the robust estimate when the sample size is small.     

TYPE I ERROR RATES FOR TESTING GENETIC DRIFT WITH PHENOTYPIC COVARIANCE MATRICES: A SIMULATION STUDY

Studies of evolutionary divergence using quantitative genetic methods are centered on the additive genetic variance–covariance matrix ( G ) of correlated traits. However, estimating G properly requires large samples and complicated experimental designs. Multivariate tests for neutral evolution commonly replace average G by the pooled phenotypic within‐group variance–covariance matrix ( W ) for evolutionary inferences, but this approach has been criticized due to the lack of exact proportionality between genetic and phenotypic matrices. In this study, we examined the consequence, in terms of type I error rates, of replacing average G by W in a test of neutral evolution that measures the regression slope between among‐population variances and within‐population eigenvalues (the Ackermann and Cheverud [AC] test) using a simulation approach to generate random observations under genetic drift. Our results indicate that the type I error rates for the genetic drift test are acceptable when using W instead of average G when the matrix correlation between the ancestral G and P is higher than 0.6, the average character heritability is above 0.7, and the matrices share principal components. For less‐similar G and P matrices, the type I error rates would still be acceptable if the ratio between the number of generations since divergence and the effective population size (t/N_e) is smaller than 0.01 (large populations that diverged recently). When G is not known in real data, a simulation approach to estimate expected slopes for the AC test under genetic drift is discussed.     

Testing Hardy-Weinberg proportions in a frequency-matched case-control genetic association study

In case-control genetic association studies, cases are subjects with the disease and controls are subjects without the disease. At the time of case-control data collection, information about secondary phenotypes is also collected. In addition to studies of primary diseases, there has been some interest in studying genetic variants associated with secondary phenotypes. In genetic association studies, the deviation from Hardy-Weinberg proportion (HWP) of each genetic marker is assessed as an initial quality check to identify questionable genotypes. Generally, HWP tests are performed based on the controls for the primary disease or secondary phenotype. However, when the disease or phenotype of interest is common, the controls do not represent the general population. Therefore, using only controls for testing HWP can result in a highly inflated type I error rate for the disease- and/or phenotype-associated variants. Recently, two approaches, the likelihood ratio test (LRT) approach and the mixture HWP (mHWP) exact test were proposed for testing HWP in samples from case-control studies. Here, we show that these two approaches result in inflated type I error rates and could lead to the removal from further analysis of potential causal genetic variants associated with the primary disease and/or secondary phenotype when the study of primary disease is frequency-matched on the secondary phenotype. Therefore, we proposed alternative approaches, which extend the LRT and mHWP approaches, for assessing HWP that account for frequency matching. The goal was to maintain more (possible causative) single-nucleotide polymorphisms in the sample for further analysis. Our simulation results showed that both extended approaches could control type I error probabilities. We also applied the proposed approaches to test HWP for SNPs from a genome-wide association study of lung cancer that was frequency-matched on smoking status and found that the proposed approaches can keep more genetic variants for association studies.     

Non-inferiority of new procedure to standard procedure in stratified matched-pair design

We consider the statistical testing for non-inferiority of a new treatment compared with the standard one under matched-pair setting in a stratified study or in several trials. A non-inferiority test based on the efficient scores and a Mantel-Haenszel (M-H) like procedure with restricted maximum likelihood estimators (RMLEs) of nuisance parameters and their corresponding sample size formulae are presented. We evaluate the above tests and the M-H type Wald test in level and power. The stratified score test is conservative and provides the best power. The M-H like procedure with RMLEs gives an accurate level. However, the Wald test is anti-conservative and we suggest caution when it is used. The unstratified score test is not biased but it is less powerful than the stratified score test when base-line probabilities related to strata are not the same. This investigation shows that the stratified score test possesses optimum statistical properties in testing non-inferiority. A common difference between two proportions across strata is the basic assumption of the stratified tests, we present appropriate tests to validate the assumption and related remarks.     

Haplotype sharing analysis using mantel statistics

OBJECTIVE: The potential value of haplotypes has attracted widespread interest in the mapping of complex traits. Haplotype sharing methods take the linkage disequilibrium information between multiple markers into account, and may have good power to detect predisposing genes. We present a new approach based on Mantel statistics for spacetime clustering, which is developed in order to improve the power of haplotype sharing analysis for gene mapping in complex disease. METHODS: The new statistic correlates genetic similarity and phenotypic similarity across pairs of haplotypes for case-only and case-control studies. The genetic similarity is measured as the shared length between haplotypes around a putative disease locus. The phenotypic similarity is measured as the mean-corrected cross-product based on the respective phenotypes. We analyzed two tests for statistical significance with respect to type I error: (1) assuming asymptotic normality, and (2) using a Monte Carlo permutation procedure. The results were compared to the chi(2) test for association based on 3-marker haplotypes. RESULTS: The results of the type I error rates for the Mantel statistics using the permutational procedure yielded pointwise valid tests. The approach based on the assumption of asymptotic normality was seriously liberal. CONCLUSION: Power comparisons showed that the Mantel statistics were better than or equal to the chi(2) test for all simulated disease models.