Modified Wald statistics for generalized linear models

Summary: Wald statistics in generalized linear models are asymptotically ² distributed. The asymptotic chi–squared law of the corresponding quadratic form shows disadvantages with respect to the approximation of the finite–sample distribution. It is shown by means of a comprehensive simulation study that improvements can be achieved by applying simple finite–sample size approximations to the distribution of the quadratic form in generalized linear models. These approximations are based on a ² distribution with an estimated degree of freedom that generalizes an approach by Patnaik and Pearson. Simulation studies confirm that nominal level is maintained with higher accuracy compared to the Wald statistics.     

A comparison of two approaches for power and sample size calculations in logistic regression models

Whittemore (1981) proposed an approach for calculating the sample size needed to test hypotheses with specified significance and power against a given alternative for logistic regression with small response probability. Based on the distribution of covariate, which could be either discrete or continuous, this approach first provides a simple closed-form approximation to the asymptotic covariance matrix of the maximum likelihood estimates, and then uses it to calculate the sample size needed to test a hypothesis about the parameter. Self et al. (1992) described a general approach for power and sample size calculations within the framework of generalized linear models, which include logistic regression as a special case. Their approach is based on an approximation to the distribution of the likelihood ratio statistic. Unlike the Whittemore approach, their approach is not limited to situations of small response probability. However, it is restricted to models with a finite number of covariate configurations. This study compares these two approaches to see how accurate they would be for the calculations of power and sample size in logistic regression models with various response probabilities and covariate distributions. The results indicate that the Whittemore approach has a slight advantage in achieving the nominal power only for one case with small response probability. It is outperformed for all other cases with larger response probabilities. In general, the approach proposed in Self et al. (1992) is recommended for all values of the response probability. However, its extension for logistic regression models with an infinite number of covariate configurations involves an arbitrary decision for categorization and leads to a discrete approximation. As shown in this paper, the examined discrete approximations appear to be sufficiently accurate for practical purpose.     

Score tests for zero inflation in generalized linear models

The authors develop score tests of goodness of fit for discrete generalized linear models against zero inflation. The binomial and Poisson models are treated as examples, and in the latter case the proposed test reduces to that of Broek (1995). Some simulation results and an illustrative example are presented.     

Local power and size properties of the LR,Wald, score and gradient tests in dispersion models

We derive asymptotic expansions for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of dispersion models, under a sequence of Pitman alternatives. The asymptotic distributions of these statistics are obtained for testing a subset of regression parameters and for testing the precision parameter. Based on these nonnull asymptotic expansions, the power of all four tests, which are equivalent to first order, are compared. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, Monte Carlo simulations are presented. An empirical application to a real data set is considered for illustrative purposes.     

Bootstrap tests for variance components in generalized linear mixed models

In many applications of generalized linear mixed models to clustered correlated or longitudinal data, often we are interested in testing whether a random effects variance component is zero. The usual asymptotic mixture of chi‐square distributions of the score statistic for testing constrained variance components does not necessarily hold. In this article, the author proposes and explores a parametric bootstrap test that appears to be valid based on its estimated level of significance under the null hypothesis. Results from a simulation study indicate that the bootstrap test has a level much closer to the nominal one while the asymptotic test is conservative, and is more powerful than the usual asymptotic score test based on a mixture of chi‐squares. The proposed bootstrap test is illustrated using two sets of real‐life data obtained from clinical trials. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

The equivalence of two approaches to incorporating variance uncertainty in sample size calculations for linear statistical models

Sample size determination is one of the most commonly encountered tasks in the design of every applied research. The general guideline suggests that a pilot study can offer plausible planning values for the vital model characteristics. This article examines two viable approaches to taking into account the imprecision of a variance estimate in sample size calculations for linear statistical models. The multiplier procedure employs an adjusted sample variance in the form of a multiple of the observed sample variance. The Bayesian method accommodates the uncertainty of a sample variance through a prior distribution. It is shown that the two seemingly distinct techniques are equivalent for sample size determination under the designated assurance requirements that the actual power exceeds the planned threshold with a given tolerance probability, or the expected power attains the desired level. The selection of optimum pilot sample size for minimizing the expected total cost is also considered.     

Testing bioequivalence for multiple formulations with power and sample size calculations

Bioequivalence (BE) trials play an important role in drug development for demonstrating the BE between test and reference formulations. The key statistical analysis for BE trials is the use of two one‐sided tests (TOST), which is equivalent to showing that the 90% confidence interval of the relative bioavailability is within a given range. Power and sample size calculations for the comparison between one test formulation and the reference formulation has been intensively investigated, and tables and software are available for practical use. From a statistical and logistical perspective, it might be more efficient to test more than one formulation in a single trial. However, approaches for controlling the overall type I error may be required. We propose a method called multiplicity‐adjusted TOST (MATOST) combining multiple comparison adjustment approaches, such as Hochberg's or Dunnett's method, with TOST. Because power and sample size calculations become more complex and are difficult to solve analytically, efficient simulation‐based procedures for this purpose have been developed and implemented in an R package. Some numerical results for a range of scenarios are presented in the paper. We show that given the same overall type I error and power, a BE crossover trial designed to test multiple formulations simultaneously only requires a small increase in the total sample size compared with a simple 2 × 2 crossover design evaluating only one test formulation. Hence, we conclude that testing multiple formulations in a single study is generally an efficient approach. The R package MATOST is available at https://sites.google.com/site/matostbe/ . Copyright © 2012 John Wiley & Sons, Ltd.     

Permutation tests for between-unit fixed effects in multivariate generalized linear mixed models

A permutation testing approach in multivariate mixed models is presented. The solutions proposed allow for testing between-unit effect; they are exact under some assumptions, while approximated in the more general case. The classes of models comprised by this approach include generalized linear models, vector generalized additive models and other nonparametric models based on smoothing. Moreover it does not assume observations of different units to have the same distribution. The extensions to a multivariate framework are presented and discussed. The proposed multivariate tests exploit the dependence among variables, hence increasing the power with respect to other standard solutions (e.g. Bonferroni correction) which combine many univariate tests in an overall one. Examples are given of two applications to real data from psychological and ecological studies; a simulation study provides some insight into the unbiasedness of the tests and their power. The methods were implemented in the R package flip, freely available on CRAN.     

Goodness-of-fit tests of generalized linear mixed models for repeated ordinal responses

Categorical longitudinal data are frequently applied in a variety of fields, and are commonly fitted by generalized linear mixed models (GLMMs) and generalized estimating equations models. The cumulative logit is one of the useful link functions to deal with the problem involving repeated ordinal responses. To check the adequacy of the GLMMs with cumulative logit link function, two goodness-of-fit tests constructed by the unweighted sum of squared model residuals using numerical integration and bootstrap resampling technique are proposed. The empirical type I error rates and powers of the proposed tests are examined by simulation studies. The ordinal longitudinal studies are utilized to illustrate the application of the two proposed tests.     

Performance of tests of association in misspecified generalized linear models

We examine the effects of modelling errors, such as underfitting and overfitting, on the asymptotic power of tests of association between an explanatory variable x and an outcome in the setting of generalized linear models. The regression function for x is approximated by a polynomial or another simple function, and a chi-square statistic is used to test whether the coefficients of the approximation are simultaneously equal to zero. Adding terms to the approximation increases asymptotic power if and only if the fit of the model increases by a certain quantifiable amount. Although a high degree of freedom approximation offers robustness to the shape of the unknown regression function, a low degree of freedom approximation can yield much higher asymptotic power even when the approximation is very poor. In practice, it is useful to compute the power of competing test statistics across the range of alternatives that are plausible a priori. This approach is illustrated through an application in epidemiology.     

On the sample size of curtailed tests

Formulas are given for the asymptotic distribution, mean, and variance of m^-1N_m,where NN_m is the random sample size of the curtailed version of a fixed-sample most powerful test based on sample size m. The adequacy of the formulas is numerically investigated in some important applications where exact formulas can also be derived     

On the conditioning problem in generalized linear models

When weights are assigned to a data matrix, as in the iterative least squares estimator of a generalized linear model, the condition of the data matrix is changed. In this paper a geometrical approach to studying the mechanisms which determine the changed condition is introduced. Specifically, it is found that in some cases strong multicollinearities can be weakened or eliminated by the weights while in other cases the weights can induce an ill-conditioning.     

Bootstrap for generalized linear models

We consider the distribution of the (standardized) ML-estimator of the unknown parameter vector in a Generalized Linear Model with canonical link function. It will be shown that its (parametric) Bootstrap estimator is consistent under the same assumptions needed by Fahrmeir & Kaufmann (1985, 1986) to show its asymptotic normality.     

Prediction for generalized linear models

Regression models are often used to make predictions. All the information needed is contained in the predictive distribution. However, this cannot be evaluated explicitly for most generalized linear models. We construct two approximations to this distribution and demonstrate their use on two sets of survival data, corresponding to the outcome of patients admitted to intensive care units and the survival times of leukaemia patients.Regression models are often used to make predictions. All the information needed is contained in the predictive distribution. However, this cannot be evaluated explicitly for most generalized linear models. We construct two approximations to this distribution and demonstrate their use on two sets of survival data, corresponding to the outcome of patients admitted to intensive care units and the survival times of leukaemia patients.Regression models are often used to make predictions. All the information needed is contained in the predictive distribution. However, this cannot be evaluated explicitly for most generalized linear models. We construct two approximations to this distribution and demonstrate their use on two sets of survival data, corresponding to the outcome of patients admitted to intensive care units and the survival times of leukaemia patients.Regression models are often used to make predictions. All the information needed is contained in the predictive distribution. However, this cannot be evaluated explicitly for most generalized linear models. We construct two approximations to this distribution and demonstrate their use on two sets of survival data, corresponding to the outcome of patients admitted to intensive care units and the survival times of leukaemia patients.     

On small sample properties of the generalized signed rank and generalized sign tests

The generalized signed rank (GSR) and generalized sign (GS) tests were recently proposed for matched pair studies with censored observations (Woolson and Lechenbruch, 1980). The results provided in that paper were asymptotic, and no indicatin of small sample behavior was given. In this paper we report on simulation studied of these statistics for a variety of distributions. We find that the GSR is more powerful than the GS, and that censoring does not affect power greatly. In the original paper, we assumed each member of the pair has the same censoring time. We consider a variant of this in which each member of the pair has a censoring time chosen from a uniform distribution, and the minimum of these times is selected as the censoring time for the pair. It is found that the power of the test is slightly reduced because the number of doubly censored pairs is increased.     

Bayesian sample size calculations for binomial experiments

This paper proposes new methodology for calculating the optimal sample size when a hypothesis test between two binomial proportions is conducted. The problem is addressed from the Bayesian point of view. Following the formulation by DasGupta and Vidakovic (1997, J. Statist. Plann. Inference 65, 335–347), the posterior risk is determined and set not to exceed a prespecified bound. A second constraint deals with the likelihood of data not satisfying the bound on the risk. The cases when the two proportions are equal to a fixed or to a random value are examined.     

Bias in linear model power and sample size due to estimating variance

Planning a study using the General Linear Univariate Model often involves sample size calculation based on a variance estimated in an earlier study. Noncentrality, power, and sample size inherit the randomness. Additional complexity arises if the estimate has been censored. Left censoring occurs when only significant tests lead to a power calculation, while right censoring occurs when only non-significant tests lead to a power calculation. We provide simple expressions for straightforward computation of the distribution function, moments, and quantiles of the censored variance estimate, estimated noncentrality, power, and sample size. We also provide convenient approximations and evaluate their accuracy. The results allow demonstrating that ignoring right censoring falsely widens confidence intervals for noncentrality and power, while ignoring left censoring falsely narrows the confidence intervals. The new results allow assessing and avoiding the potentially substantial bias that censoring may create.     

A note on the sample size required in sequential tests for the generalized binomial distribution

In this paper, we discuss the sample size needed to perform Wald's sequential statistical test for the proportion of non-conforming items generated by a process when the results of the inspections are correlated and the generalized binomial distribution proposed by Madsen (1993) is used. It will be shown that, in the presence of correlation, the sample size increases as the value of the coefficient of correlation increases--being much higher for processes with small failure rates.     

On variable selection in generalized linear and related regression models

This paper is concerned with selection of explanatory variables in generalized linear models (GLM). The class of GLM's is quite large and contains e.g. the ordinary linear regression, the binary logistic regression, the probit model and Poisson regression with linear or log-linear parameter structure. We show that, through an approximation of the log likelihood and a certain data transformation, the variable selection problem in a GLM can be converted into variable selection in an ordinary (unweighted) linear regression model. As a consequence no specific computer software for variable selection in GLM's is needed. Instead, some suitable variable selection program for linear regression can be used. We also present a simulation study which shows that the log likelihood approximation is very good in many practical situations. Finally, we mention briefly possible extensions to regression models outside the class of GLM's.