Robust prediction and extrapolation designs for misspecified generalized linear regression models

We study minimax robust designs for response prediction and extrapolation in biased linear regression models. We extend previous work of others by considering a nonlinear fitted regression response, by taking a rather general extrapolation space and, most significantly, by dropping all restrictions on the structure of the regressors. Several examples are discussed.     

A superiority problem in misspecified restricted linear models

Abstract

In this article we study the relationship between principal component analysis and a multivariate dependency measure. It is shown, via simulated examples and real data, that the information provided by principal components is compatible with that obtained via the dependency measure δ. Furthermore, we show that in some instances in which principal component analysis fails to give reasonable results due to nonlinearity among the random variables, the dependency statistic δ still provides good results. Finally, we give some ideas about using the statistic δ in order to reduce the dimensionality of a given data set.     

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.     

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     

Computationally efficient familywise error rate control in genome-wide association studies using score tests for generalized linear models

In genetic association studies, detecting phenotype–genotype association is a primary goal. We assume that the relationship between the data—phenotype, genetic markers and environmental covariates—can be modeled by a generalized linear model. The number of markers is allowed to be far greater than the number of individuals of the study. A multivariate score statistic is used to test each marker for association with a phenotype. We assume that the test statistics asymptotically follow a multivariate normal distribution under the complete null hypothesis of no phenotype–genotype association. We present the familywise error rate order k approximation method to find a local significance level (alternatively, an adjusted p-value) for each test such that the familywise error rate is controlled. The special case k=1 gives the Šidák method. As a by-product, an effective number of independent tests can be defined. Furthermore, if environmental covariates and genetic markers are uncorrelated, or no environmental covariates are present, we show that covariances between score statistics depend on genetic markers alone. This not only leads to more efficient calculations but also to a local significance level that is determined only by the collection of markers used, independent of the phenotypes and environmental covariates of the experiment at hand.     

On a superiority problem in misspecified restricted linear models

Razzaghi (1987) conjectures that a wrong choice of covariance matrix in a restricted linear model results in loss of efficiency, This conjecture is proved to be correct.     

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.     

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.     

Collinearity in generalized linear models

This paper defines collinearity for generalized linear models (GLMs), investigates its consequences and proposes diagnostic criteria. The relationship between collinearity in GLMs and standard linear models (SLMs) is explored and bounds which relate the degree of collinearity in these two models are given. Estimation based on ridge methods is discussed.     

Transformed generalized linear models

The estimation of data transformation is very useful to yield response variables satisfying closely a normal linear model. Generalized linear models enable the fitting of models to a wide range of data types. These models are based on exponential dispersion models. We propose a new class of transformed generalized linear models to extend the Box and Cox models and the generalized linear models. We use the generalized linear model framework to fit these models and discuss maximum likelihood estimation and inference. We give a simple formula to estimate the parameter that index the transformation of the response variable for a subclass of models. We also give a simple formula to estimate the r$r$th moment of the original dependent variable. We explore the possibility of using these models to time series data to extend the generalized autoregressive moving average models discussed by Benjamin et al. [Generalized autoregressive moving average models. J. Amer. Statist. Assoc. 98, 214–223]. The usefulness of these models is illustrated in a simulation study and in applications to three real data sets.     

On power and sample size calculations for Wald tests in generalized linear models

A Wald test-based approach for power and sample size calculations has been presented recently for logistic and Poisson regression models using the asymptotic normal distribution of the maximum likelihood estimator, which is applicable to tests of a single parameter. Unlike the previous procedures involving the use of score and likelihood ratio statistics, there is no simple and direct extension of this approach for tests of more than a single parameter. In this article, we present a method for computing sample size and statistical power employing the discrepancy between the noncentral and central chi-square approximations to the distribution of the Wald statistic with unrestricted and restricted parameter estimates, respectively. The distinguishing features of the proposed approach are the accommodation of tests about multiple parameters, the flexibility of covariate configurations and the generality of overall response levels within the framework of generalized linear models. The general procedure is illustrated with some special situations that have motivated this research. Monte Carlo simulation studies are conducted to assess and compare its accuracy with existing approaches under several model specifications and covariate distributions.     

Analysis of accelerated life tests with weibull failure distribution via generalized linear models

Efficient industrial experiments for reliability analysis of manufactured goods may consist in subjecting the units to higher stress levels than those of the usual working conditions. This results in the so called "accelerated life tests" where, for each pre-fixed stress level, the experiment ends after the failure of a certain pre-fixed proportion of units or a certain test time is reached. The aim of this paper is to determine estimates of the mean lifetime of the units under usual working conditions from censored failure data obtained under stress conditions. This problem is approached through generalized linear modelling and related inferential techniques, considering a Weibull failure distribution and a log-linear stress-response relationship. The general framework considered has as particular cases, the Inverse Power Law model, the Eyring model, the Arrhenius model and the generalized Eyring model. In order to illustrate the proposed methodology, a numerical example is provided.     

Performance of the principal component two-parameter estimator in misspecified linear regression model

In this article, we consider the performance of the principal component two-parameter estimator in situation of multicollinearity for misspecified linear regression model where misspecification is due to omission of some relevant explanatory variables. The conditions of superiority of the principal component two-parameter estimator over some estimators under the Mahalanobis loss function by the average loss criterion are derived. Furthermore, a real data example and a Monte Carlo simulation study are provided to illustrate some of the theoretical results.     

for misspecified regression models

The authors propose minimax robust designs for regression models whose response function is possibly misspecified. These designs, which minimize the maximum of the mean squared error matrix, can control the bias caused by model misspecification and provide the desired efficiency through one parameter. The authors call on a nonsmooth optimization technique to derive these designs analytically. Their results extend those of Heo, Schmuland & Wiens (2001). The authors also discuss several examples for approximately polynomial regression.     

Prediction in multilevel generalized linear models

Summary.  We discuss prediction of random effects and of expected responses in multilevel generalized linear models. Prediction of random effects is useful for instance in small area estimation and disease mapping, effectiveness studies and model diagnostics. Prediction of expected responses is useful for planning, model interpretation and diagnostics. For prediction of random effects, we concentrate on empirical Bayes prediction and discuss three different kinds of standard errors; the posterior standard deviation and the marginal prediction error standard deviation (comparative standard errors) and the marginal sampling standard deviation (diagnostic standard error). Analytical expressions are available only for linear models and are provided in an appendix . For other multilevel generalized linear models we present approximations and suggest using parametric bootstrapping to obtain standard errors. We also discuss prediction of expectations of responses or probabilities for a new unit in a hypothetical cluster, or in a new (randomly sampled) cluster or in an existing cluster. The methods are implemented in gllamm and illustrated by applying them to survey data on reading proficiency of children nested in schools. Simulations are used to assess the performance of various predictions and associated standard errors for logistic random-intercept models under a range of conditions.     

Residual components in generalized linear models

The adequacy of a postulated generalized linear model can often be improved by transforming predictors and/or including additional explanatory variables. To assess the fit relative to a given predictor, we define its corresponding residual component. Asymptotic bias and variance of the residual component are considered, paying particular attention to the case that the presumed model is valid.     

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.     

Incomplete covariates data in generalized linear models

We consider regression analysis when part of covariates are incomplete in generalized linear models. The incomplete covariates could be due to measurement error or missing for some study subjects. We assume there exists a validation sample in which the data is complete and is a simple random subsample from the whole sample. Based on the idea of projection-solution method in Heyde (1997, Quasi-Likelihood and its Applications: A General Approach to Optimal Parameter Estimation. Springer, New York), a class of estimating functions is proposed to estimate the regression coefficients through the whole data. This method does not need to specify a correct parametric model for the incomplete covariates to yield a consistent estimate, and avoids the ‘curse of dimensionality’ encountered in the existing semiparametric method. Simulation results shows that the finite sample performance and efficiency property of the proposed estimates are satisfactory. Also this approach is computationally convenient hence can be applied to daily data analysis.     

Hidden Markov chains in generalized linear models

We show how the concept of hidden Markov model may be accommodated in a setting involving multiple sequences of observations. The resulting class of models allows for both interrelationships between different sequences and serial dependence within sequences. Missing values in the observation sequences may be handled in a straightforward manner. We also examine a group of methods, based upon the observed Fisher Information matrix, for estimating the covariance matrix of the parameter estimates. We illustrate the methods with both real and simulated data sets.