The distribution of Pearson residuals in generalized linear models

In general, the distribution of residuals cannot be obtained explicitly. In this paper we give an asymptotic formula for the density of Pearson residuals in continuous generalized linear models corrected to order n⁻¹, where n is the sample size. We define a set of corrected Pearson residuals for these models that, to this order of approximation, have exactly the same distribution of the true Pearson residuals. An application to a real data set and simulation results for a gamma model illustrate the usefulness of our corrected Pearson residuals.     

Size and power of tests for a zero random effect variance or polynomial regression in additive and linear mixed models

Several tests for a zero random effect variance in linear mixed models are compared. This testing problem is non-regular because the tested parameter is on the boundary of the parameter space. Size and power of the different tests are investigated in an extensive simulation study that covers a variety of important settings. These include testing for polynomial regression versus a general smooth alternative using penalized splines. Among the test procedures considered, three are based on the restricted likelihood ratio test statistic (RLRT), while six are different extensions of the linear model F-test to the linear mixed model. Four of the tests with unknown null distributions are based on a parametric bootstrap, the other tests rely on approximate or asymptotic distributions. The parametric bootstrap-based tests all have a similar performance. Tests based on approximate F-distributions are usually the least powerful among the tests under consideration. The chi-square mixture approximation for the RLRT is confirmed to be conservative, with corresponding loss in power. A recently developed approximation to the distribution of the RLRT is identified as a rapid, powerful and reliable alternative to computationally intensive parametric bootstrap procedures. This novel method extends the exact distribution available for models with one random effect to models with several random effects.     

Sample sizes required to detect interactions between two binary fixed-effects in a mixed-effects linear regression model

Mixed-effects linear regression models have become more widely used for analysis of repeatedly measured outcomes in clinical trials over the past decade. There are formulae and tables for estimating sample sizes required to detect the main effects of treatment and the treatment by time interactions for those models. A formula is proposed to estimate the sample size required to detect an interaction between two binary variables in a factorial design with repeated measures of a continuous outcome. The formula is based, in part, on the fact that the variance of an interaction is fourfold that of the main effect. A simulation study examines the statistical power associated with the resulting sample sizes in a mixed-effects linear regression model with a random intercept. The simulation varies the magnitude (Δ) of the standardized main effects and interactions, the intraclass correlation coefficient (ρ), and the number (k) of repeated measures within-subject. The results of the simulation study verify that the sample size required to detect a 2×2 interaction in a mixed-effects linear regression model is fourfold that to detect a main effect of the same magnitude.     

The likelihood ratio test for hidden Markov models in two-sample problems

The asymptotic distribution of the likelihood ratio test statistic in two-sample testing problems for hidden Markov models is derived when allowing for unequal sample sizes as well as for different families of state-dependent distributions. In both cases under regularity conditions the limit distribution is a standard χ²-distribution, and in particular does not depend on the ratio of the distinct sample sizes. In a simulation study, the finite sample properties are investigated, and the methodology is illustrated in an application to modeling the movement of Drosophila larvae.     

Sample size and power calculations based on generalized linear mixed models with correlated binary outcomes

The generalized linear mixed model (GLIMMIX) provides a powerful technique to model correlated outcomes with different types of distributions. The model can now be easily implemented with SAS PROC GLIMMIX in version 9.1. For binary outcomes, linearization methods of penalized quasi-likelihood (PQL) or marginal quasi-likelihood (MQL) provide relatively accurate variance estimates for fixed effects. Using GLIMMIX based on these linearization methods, we derived formulas for power and sample size calculations for longitudinal designs with attrition over time. We found that the power and sample size estimates depend on the within-subject correlation and the size of random effects. In this article, we present tables of minimum sample sizes commonly used to test hypotheses for longitudinal studies. A simulation study was used to compare the results. We also provide a Web link to the SAS macro that we developed to compute power and sample sizes for correlated binary outcomes.     

Effect of imbalance and intracluster correlation coefficient in cluster randomized trials with binary outcomes

Cluster randomization trials are increasingly popular among healthcare researchers. Intact groups (called ‘clusters’) of subjects are randomized to receive different interventions, and all subjects within a cluster receive the same intervention. In cluster randomized trials, a cluster is the unit of randomization, and a subject is the unit of analysis. Variation in cluster sizes can affect the sample size estimate or the power of the study. [Guittet, L., Ravaud, P., Giraudeau, B., 2006. Planning a cluster randomized trial with unequal cluster sizes: Practical issues involving continuous outcomes. BMC Medical Research Methodology 6 (17), 1-15] investigated the impact of an imbalance in cluster size on the power of trials with continuous outcomes through simulations. In this paper, we examine the impact of cluster size variation and intracluster correlation on the power of the study for binary outcomes through simulations. Because the sample size formula for cluster randomization trials is based on a large sample approximation, we evaluate the performance of the sample size formula with small sample sizes through simulation. Simulation study findings show that the sample size formula (m_p) accounting for unequal cluster sizes yields empirical powers closer to the nominal power than the sample size formula (m_a) for the average cluster size method. The differences in sample size estimates and empirical powers between m_a and m_p get smaller as the imbalance in cluster sizes gets smaller.     

Goodness-of-fit tests for modeling longitudinal ordinal data

Longitudinal studies involving categorical responses are extensively applied in many fields of research and are often fitted by the generalized estimating equations (GEE) approach and generalized linear mixed models (GLMMs). The assessment of model fit is an important issue for model inference. The purpose of this article is to extend Pan’s (2002a) goodness-of-fit tests for GEE models with longitudinal binary data to the tests for logistic proportional odds models with longitudinal ordinal data. Two proposed methods based on Pearson chi-squared test and unweighted sum of residual squares are developed, and the approximate expectations and variances of the test statistics are easily computed. Four major variants of working correlation structures, independent, AR(1), exchangeable and unspecified, are considered to estimate the variances of the proposed test statistics. Simulation studies in terms of type I error rate and the power performance of the proposed tests are presented for various sample sizes. Furthermore, the approaches are demonstrated by two real data sets.     

Empirical likelihood based diagnostics for heteroscedasticity in partial linear models

In this paper, we propose a diagnostic technique for checking heteroscedasticity based on empirical likelihood for the partial linear models. We construct an empirical likelihood ratio test for heteroscedasticity. Also, under mild conditions, a nonparametric version of Wilk’s theorem is derived, which says that our proposed test has an asymptotic chi-square distribution. Simulation results reveal that the finite sample performance of our proposed test is satisfactory in both size and power. An empirical likelihood bootstrap simulation is also conducted to overcome the size distortion in small sample sizes.     

Exact optimal inference in regression models under heteroskedasticity and non-normality of unknown form

Simple point-optimal sign-based tests are developed for inference on linear and nonlinear regression models with non-Gaussian heteroskedastic errors. The tests are exact, distribution-free, robust to heteroskedasticity of unknown form, and may be inverted to build confidence regions for the parameters of the regression function. Since point-optimal sign tests depend on the alternative hypothesis considered, an adaptive approach based on a split-sample technique is proposed in order to choose an alternative that brings power close to the power envelope. The performance of the proposed quasi-point-optimal sign tests with respect to size and power is assessed in a Monte Carlo study. The power of quasi-point-optimal sign tests is typically close to the power envelope, when approximately 10% of the sample is used to estimate the alternative and the remaining sample to compute the test statistic. Further, the proposed procedures perform much better than common least-squares-based tests which are supposed to be robust against heteroskedasticity.     

Generalized F-tests for the multivariate normal mean

Based on Läuter’s [Läuter, J., 1996. Exact t and F tests for analyzing studies with multiple endpoints. Biometrics 52, 964-970] exact t test for biometrical studies related to the multivariate normal mean, we develop a generalized F-test for the multivariate normal mean and extend it to multiple comparison. The proposed generalized F-tests have simple approximate null distributions. A Monte Carlo study and two real examples show that the generalized F-test is at least as good as the optional individual Läuter’s test and can improve its performance in some situations where the projection directions for the Läuter’s test may not be suitably chosen. The generalized F-test could be superior to individual Läuter’s tests and the classical Hotelling T²-test for the general purpose of testing the multivariate normal mean. It is shown by Monte Carlo studies that the extended generalized F-test outperforms the commonly-used classical test for multiple comparison of normal means in the case of high dimension with small sample sizes.     

Practical methods for computing power in testing the multivariate general linear hypothesis

Four test statistics are commonly used in multivariate general linear hypothesis tests such as MANOVA: Roy's largest root, Wilks' Lambda, Hotelling-Lawley trace and Pillai-Bartlett trace. Closed form, finite series expressions do not exist for the distribution functions in either the null or non-null case (except for special cases). In practice, asymptotic approximations, based on F or chi square distributions, are used for the null case. It is not widely known that similarly accurate and general approximations have been published for the non-null case, for all except the largest root [15,38].In this paper new approximations, based on noncentral F's, are provided for power for all but the largest root. These generalize existing F approximations for the central case. Much less calculation is needed than for earlier approximations and accuracy appears not to suffer in practice. An upper bound F approximation is provided for the largest root power.Power calculation can be used to help choose a test statistic as well as for experimental design. A particularly convenient method for estimating power, using standard linear models programs [20], is generalized to the multivariate case.     

On a test statistic for testing normality

This paper presents a statistic for testing a complete sample for normality. The test statistic is defined to be a linear combination of the standardized sample order statistics. It is a location and scale invariant statistic and also could be generalized to test the distributional assumptions for the other location and scale family of distributions. Empirical percentage points are provided for samples of size n = 3(1), 40(10), 150. A Monte Carlo study was conducted to compare the power of the proposed test with that of W test for samples of sizes n = 10, 15, 20, 35, 50 and 36 non-normal alternatives. It is concluded that for the skewed alternatives both statistics have similar power.     

Generalized likelihood ratio test for varying-coefficient models with different smoothing variables

Varying-coefficient models are popular multivariate nonparametric fitting techniques. When all coefficient functions in a varying-coefficient model share the same smoothing variable, inference tools available include the F-test, the sieve empirical likelihood ratio test and the generalized likelihood ratio (GLR) test. However, when the coefficient functions have different smoothing variables, these tools cannot be used directly to make inferences on the model because of the differences in the process of estimating the functions. In this paper, the GLR test is extended to models of the latter case by the efficient estimators of these coefficient functions. Under the null hypothesis the new proposed GLR test follows the χ²-distribution asymptotically with scale constant and degree of freedom independent of the nuisance parameters, known as Wilks phenomenon. Further, we have derived its asymptotic power which is shown to achieve the optimal rate of convergence for nonparametric hypothesis testing. A simulation study is conducted to evaluate the test procedure empirically.     

Finite sample multivariate tests of asset pricing models with coskewness

Exact inference methods are proposed for asset pricing models with unobservable risk-free rates and coskewness; specifically, the Quadratic Market Model (QMM) which incorporates the effect of asymmetry of return distribution on asset valuation. In this context, exact tests are appealing given (i) the increasing popularity of such models in finance, (ii) the fact that traditional market models (which assume that asset returns move proportionally to the market) have not fared well in empirical tests, (iii) finite sample QMM tests are unavailable even with Gaussian errors. Empirical models are considered where the procedure to assess the significance of coskewness preference is LR-based, and relates to the statistical and econometric literature on dimensionality tests which are interesting in their own right. Exact versions of these tests are obtained, allowing for non-normality of fundamentals. A simulation study documents the size and power properties of asymptotic and finite sample tests. Empirical results with well-known data sets reveal temporal instabilities over the full sampling period, namely 1961-2000, though tests fail to reject the QMM restrictions over 5-year sub-periods.     

Computing a consistent approximation to a generalized pairwise comparisons matrix

This paper presents an algorithm for computing a consistent approximation to a generalized pairwise comparisons matrix (that is, without the reciprocity property or even 1s on the main diagonal). The algorithm is based on a logarithmic transformation of the generalized pairwise comparisons matrix into a linear space with the Euclidean metric. It uses both the row and (reciprocals of) column geometric means and is thus a generalization of the ordinary geometric means method. The resulting approximation is not only consistent, but also closest to the original matrix, i.e., deviates least from an expert's original judgments. The computational complexity of the algorithm is O(n²).     

Transmission Disequilibrium Test (TDT) for a pair of linked marker loci

A general theory of the Transmission Disequilibrium Test for two linked flanking marker loci used in interval mapping of a disease gene with an arbitrary mode of inheritance based on the genotypic relative risk model is presented from first principles. The expectations of all the cells in a contingency table possible with four marker haplotypes (transmitted vs. not transmitted) are derived. Although algebraic details of the six possible linkage tests are given, only the test involving doubly heterozygous parents has been considered in detail. Based on a test of symmetry of a square contingency table, chi-square tests are proposed for the null hypothesis of no linkage between the markers and the disease gene. The power of the tests is discussed in terms of the corresponding non-centrality parameters for each of the four modes of inheritance viz. additive, recessive, dominant and multiplicative. Sample sizes required for 80% power at the significance level of 0.05 have also been computed in each case. The results have been presented both for the case when the pair of markers is at the disease susceptibility locus as well as for the case when it is not so. In addition to the marker gene frequencies, recombination probabilities, and various association parameters, etc., it is found that the results depend on a composite parameter involving the genotypic relative risk of the homozygous disease genotype and the disease gene frequency instead of its constituents individually. The power increases with the decrease in the recombination probability in general but their magnitudes differ across the modes of inheritance. Additive and multiplicative modes of inheritance, in general, are found to give almost similar sample sizes. The sample sizes are found to be higher when the marker haplotype is not at the disease susceptibility locus than when the markers are there, indicating loss of power of the tests in the former case. But these are lower than the sample sizes required in the single marker case, thereby showing the superiority of the strategy in adopting the two marker loci for the transmission disequilibrium test. The use of linkage information between the markers seems to improve matters when this strategy is adapted for disease gene identification. The computations for sample sizes required for 80% power at the significance level of 5×10⁻⁸ used in TDT for fine mapping and genome-wide association studies indicate that the sample sizes needed could be several times larger than those for the traditional significance level of 0.05.     

Bootstrap based tests for generalized negative binomial distribution

Goodness of fit test statistics based on the empirical distribution function (EDF) are considered for the generalized negative binomial distribution. The small sample levels of the tests are found to be very close to the nominal significance levels. For small sample sizes, the tests are compared with respect to their simulated power of detecting some alternative hypotheses against a null hypothesis of generalized negative binomial distribution. The discrete Anderson—Darling test is the most powerful among the EDF tests. Two numerical examples are used to illustrate the application of the goodness of fit tests. The support received from the Research Professorship Program at Central Michigan University under the grant #22159 is gratefully acknowledged.     

The sizes of the three popular asymptotic tests for testing homogeneity of two binomial proportions

In statistical hypothesis testing it is important to ensure that the type I error rate is preserved under the nominal level. This paper addresses the sizes and the type I errors rates of the three popular asymptotic tests for testing homogeneity of two binomial proportions: the chi-square test with and without continuity correction, the likelihood ratio test. Although it has been recognized that, based on limited simulation studies, the sizes of the tests are inflated in small samples, it has been thought that the sizes are well preserved under the nominal level when the sample size is sufficiently large. But, Loh [1989. Bounds on the size of the χ² test of independence in a contingency table. Ann. Statist. 17, 1709-1722], and Loh and Yu [1993. Bounds on the size of the likelihood ratio test of independence in a contingency table. J. Multivariate Anal. 45, 291-304] showed theoretically that the sizes are always greater than or equal to the nominal level when the sample size is infinite. In this paper, we confirm their results by computing the large-sample lower bounds of the sizes numerically. Applying complete enumeration which does not have any error, we confirm again the results by computing the sizes precisely on computer in moderate sample sizes. When the sample sizes are unbalanced, the peaks of the type I error rates occur at the extremes of the nuisance parameter. But, the type I error rates of the three tests are close to the nominal level in most values of the nuisance parameter except the extremes. We also find that, when the sample sizes are severely unbalanced and the value of the nuisance parameter is very small, the size of the chi-square test with continuity correction can exceed the nominal level excessively (for instance, the size could be at least 0.877 at 5% nominal level in some cases).     

Direct simulation for CAD models undergoing parametric modifications

We propose a novel approach—direct simulation—for interactive simulation with accuracy control, for CAD models undergoing parametric modifications which leave Dirichlet boundary conditions unchanged. This is achieved by computing offline a generic solution as a function of the design modification parameters. Using this parametric expression, each time the model parameters are edited, the associated simulation solution for this model instance can be cheaply and quickly computed online by evaluating the derived parametric solution for these parameter values. The proposed approach furthermore works for models undergoing topological changes, and does not need any mesh regeneration or mesh mapping. These results are achieved by use of the proper generalized decomposition model reduction technique, in combination with R-functions. We believe this is the first approach that can interactively simulate the physical properties of a CAD model, even undergoing topological change, without expensive re-computation. The approach is demonstrated for linear elasticity analysis; numerical results demonstrate its simulation accuracy and efficiency in comparison with the classic FE method.     

Small sample asymptotics: a review with applications to robust statistics

The aim of this paper is to review concepts, theory, and applications of small sample asymptotic techniques. The striking characteristic of these techniques is that they give uniformly very accurate approximation in the tails of the distribution of a statistics based on n observations even for very small sample sizes. These ideas will be applied to various classes of estimators and tests, including L-estimators, rank procedures, maximum likelihood estimators for general models, and multivariate M-estimators. Other applications to robust statistics, density estimation, and to an efficiency's criterion discussed by Rao will also be discussed.