Penalized Quasi-likelihood Research Articles

Can biological tests assist prediction of suicide in mood disorders?

Predicting suicide is difficult due to its low base-rate and the limited specificity of clinical predictors. Prospective biological studies suggest that dysfunctions in the serotonergic system and hypothalamic-pituitary-adrenal axis have some predictive power for completed suicide in mood disorders. A prediction model that incorporates biological testing to increase specificity and sensitivity of prediction of suicide is of potential clinical value. Meta-analyses of prospective biological studies of suicide and cerebrospinal fluid 5-hydroxyindoleacetic acid (CSF 5-HIAA) and suicide and the dexamethasone suppression test (DST) in mood disorders using the penalized quasi-likelihood (PQL) and bootstrap method yield odds ratios for prediction of suicide of 4.48 and 4.65 respectively. Two combinatory prediction models, the first requiring positive results on more than one test, and the second requiring a positive result on either one of two tests, were tested to assess their sensitivity, specificity, and predictive power using biological data from published and unpublished studies. The prediction model that requires both DST and CSF 5-HIAA tests to be positive results in 37.5% sensitivity, 88% specificity, and has a positive predictive value of 23%. The prediction model that requires either DST or CSF 5-HIAA tests to be positive results in 87.5% sensitivity, 28% specificity, and has a positive predictive value of 10%. Thus, models attempting to predict a lethal outcome that is uncommon perform very differently making model choice of major importance. Further work on refining biological predictors and integration with clinical predictors is needed to optimize a model to predict suicide in the clinic.

Optimal Experimental Designs for Multilevel Logistic Models with Two Binary Predictors

ABSTRACT Many experiments aim at populations with persons nested within clusters. In the design stage of such experiments one has to decide whether to randomize complete clusters or persons within clusters to treatment conditions. Furthermore, the optimal sample sizes have to be calculated. In this article these two design issues will be dealt with for logistic models with a binary treatment condition and a binary covariate. The multilevel model is used to relate treatment condition and the covariate to the binary outcome. The optimal design is analytically derived for first order Marginal Quasi Likelihood (MQL) by linearizing the model using a Taylor series expansion. A simulation study shows results for second order Penalized Quasi Likelihood, (PQL), which is known to produce less biased estimates. The results show that person level randomization is preferred, especially when the cluster size and variance component at the cluster level are large. Two examples show how optimal sample sizes can be calculated in practice.

Fitting Generalized Linear Mixed Models For Point-Referenced Spatial Data

Non-Gaussian point-referenced spatial data are frequently modeled using generalized linear mixed models (GLMM) with location-specific random effects. Spatial dependence can be introduced in the covariance matrix of the random effects. Maximum likelihood-based or Bayesian estimation implemented via Markov chain Monte Carlo (MCMC) for such models is computationally demanding especially for large sample sizes because of the large number of random effects and the inversion of the covariance matrix involved in the likelihood. We review three fitting procedures, the Penalized Quasi Likelihood method, the MCMC, and the Sampling-Importance-Resampling method. They are assessed in terms of estimation accuracy, ease of implementation, and computational efficiency using a spatially structured dataset on infant mortality from Mali.

A Comparison of Estimation Methods for Multilevel Logistic Models

In this paper a comparison between Penalized Quasi Likelihood (PQL) and estimation by numerical integration is made for the analysis of two-level experimental binary data with two treatment conditions. The comparison between the estimation methods is made for three situations: randomization to treatment conditions at the cluster level, randomization at the person level with treatment by cluster interaction, and without such interaction. Criteria for comparison are convergence of the estimation process and improper estimates (i.e. unrealistic high point estimates), criteria concerning the point estimation (bias, variance, and mean squared error) and testing (bias of the point estimates and of the variances as reported by the software) of the treatment effect. The results show that non-convergence occurs more often when estimation is done by numerical integration. This method may also lead to improper estimates. First order PQL performs best in terms of point estimation of the treatment effect, but should not be used for testing. For the latter purpose second order PQL is more applicable.

Penalized quasi-likelihood with spatially correlated data

This article discusses and evaluates penalized quasi-likelihood (PQL) estimation techniques for the situation where random effects are correlated, as is typical in mapping studies. This is an approximate fitting technique which uses a Laplace approximation to the integrated mixed model likelihood. It is much easier to implement than usual maximum likelihood estimation. Our results show that the PQL estimates are reasonably unbiased for analysis of mixed Poisson models when there is correlation in the random effects, except when the means are sufficiently small to yield sparse data. However, although the normal approximation to the distribution of the parameter estimates works fairly well for the parameters in the mean it does not perform as well for the variance components. In addition, when the mean mortality counts are small, the estimated standard errors of the variance components tend to become more biased than those for the mean. We illustrate our approaches by applying PQL for mapping mortality in British Columbia, Canada, over the five-year period 1985–1989.

Optimal Permutation Tests for the Analysis of Group Randomized Trials

Two facts complicate the comparison of interventions in group randomized trials (GRT's), a family of clinical trials in which each member of a particular group receives the same treatment assignment. First, individual outcomes within each group are often correlated. Second, the number of groups in a GRT is often not sufficient to make asymptotic approximations possible. Therefore, the tests used with methods, such as generalized estimating equations (GEE) and penalized quasi likelihood (PQL), originally developed for longitudinal studies, may not be valid. As an alternative, a class of permutation tests is derived to maximize the power of testing for an intervention effect in a GRT while maintaining a nominal test size. The test uses a statistic that is a weighted sum of residuals, with the weights based on the group sizes and the variability of each individual outcome. Through simulation, we demonstrate the importance of weights to a permutation test's power and compare the power of permutation tests, GEE, and PQL. Last, we apply our methods to an actual GRT to study smoking cessation, discuss the findings based on permutation tests, and compare those findings with traditional methods.

Multilevel Modeling of Cognitive Function in Schizophrenic Patients and Their First Degree Relatives

We describe multilevel modeling of cognitive function in subjects with schizophrenia, their healthy first degree relatives and controls. The purpose of the study was to compare mean cognitive performance between the three groups after adjusting for various covariates, as well as to investigate differences in the variances. Multilevel models were required because subjects were nested within families and some of the measures were repeated several times on the same subject. The following four methodological issues that arose during the analysis of the data are discussed. First, when the random effects distribution was not normal, non-parametric maximum likelihood (NPML) was employed, leading to a different conclusion than the conventional multilevel model regarding one of the main study hypotheses. Second, the between-subject (within-family) variance was allowed to differ between the three groups. This corresponded to the variance at level 1 or level 2 depending on whether repeated measures were analyzed. Third, a positively skewed response was analyzed using a number of different generalized linear mixed models. Finally, penalized quasilikelihood (PQL) estimates for a binomial response were compared with estimates obtained using Gaussian quadrature. A small simulation study was carried out to assess the accuracy of the latter.

On Marginal Quasi-Likelihood Inference in Generalized Linear Mixed Models

In view of the cumbersome and often intractable numerical integrations required for a full likelihood analysis, several suggestions have been made recently for approximate inference in generalized linear mixed models (GLMMs). Two closely related approximate methods are the penalized quasi-likelihood (PQL) method and the marginal quasi-likelihood (MQL) method. The PQL approach generally produces biased estimates for the regression effects and the variance component of the random effects. Recently, some corrections have been proposed to remove these biases. But the corrections appear to be satisfactory only when the variance component of the random effects is small. The MQL approach has also been used for inference in the GLMMs. This approach requires the computations of the joint moments of the clustered observations, up to order four. But the derivation of these moments are not easy. Consequently, different “working” formulas have been used, especially for the mean and covariance matrix of the observations, which may not lead to desirable estimates. In this paper, we use a small variance component (of the random effects) approach and develop the MQL estimating equations for the parameters based on the joint moments of order up to four. The proposed approach thus avoids the use of the so-called “working” covariance and higher order moment matrices, leading to better estimates for the regression and the overdispersion parameters, in the sense of efficiency in particular.

Modelling spatial disease rates using maximum likelihood

This paper concerns maximum likelihood estimation for a generalized linear mixed model (GLMM) useful for modelling spatial disease rates. The model allows for log-linear covariate adjustment and local smoothing of rates through estimation of spatially correlated random effects. The covariance structure of the random effects is based on a recently proposed model which parameterizes spatial dependence through the inverse covariance matrix. A Markov chain Monte Carlo algorithm for performing maximum likelihood estimation for this model is described. Results of a computer simulation study that compared maximum likelihood (ML) and penalized quasi-likelihood (PQL) estimators are presented. Compared with PQL, ML produced less biased estimates of the intercept but the ML estimates were slightly more variable. Estimates of the other regression coefficients were unbiased and nearly identical for the two methods. ML estimators of the random effects standard deviation and spatial correlation were more biased than the corresponding PQL estimators. The conclusion is that ML estimators for GLMMs cannot be expected to perform better than PQL for small samples.

Parametric bootstrap and penalized quasi-likelihood inference in conditional autoregressive models.

This paper discusses a variety of conditional autoregressive (CAR) models for mapping disease rates, beyond the usual first-order intrinsic CAR model. We illustrate the utility and scope of such models for handling different types of data structures. To encourage their routine use for map production at statistical and health agencies, a simple algorithm for fitting such models is presented. This is derived from penalized quasi-likelihood (PQL) inference which uses an analogue of best-linear unbiased estimation for the regional risk ratios and restricted maximum likelihood for the variance components. We offer the practitioner here the use of the parametric bootstrap for inference. It is more reliable than standard maximum likelihood asymptotics for inference purposes since relevant hypotheses for the mapping of rates lie on the boundary of the parameter space. We illustrate the parametric bootstrap test of the practically relevant and important simplifying hypothesis that there is no spatial autocorrelation. Although the parametric bootstrap requires computational effort, it is straightforward to implement and offers a wealth of information relating to the estimators and their properties. The proposed methodology is illustrated by analysing infant mortality in the province of British Columbia in Canada.

On estimating equations for parameters in generalized linear mixed models with application to binary data

In view of the cumbersome and often intractable numerical integrations required for a full likelihood analysis, several suggestions have been made recently for approximate inference in generalized linear mixed models and other nonlinear variance component models. For example, we refer to the penalized quasi-likelihood (PQL) approach of Breslow and Clayton (1993), the Stein-type estimating function based approach of Waclawiw and Liang (1993), and the corrected PQL approach of Breslow and Lin (1995). Recently, Sutradhar and Godambe (1998) provided a semiparametric solution to the estimation problem dealt with by Waclawiw and Liang. In the present article, we provide a semiparametric solution to the estimation problem investigated by Breslow and Lin, and Breslow and Clayton. More specifically, we propose a two-step joint estimating equations approach to estimate the model parameters. In the first step, we use an estimating function based approach to obtain the estimates of the random effects. In the second step, following Prentice and Zhao (1991), we construct the first two moment based joint estimating equations for the regression parameters and the variance component of the random effects. As the exact first and second order moments of the generalized linear mixed models are not available, these moments are obtained first by expanding the conditional moments for given random effects about their estimating function based estimates and then taking the expectation of the conditional moments over the true distribution of the random effects. Since binary mixed models arise in many biomedical application areas, for example, computations are provided in detail for this special case. We also discuss a second approach, namely, a three-step ad hoc estimating equations approach, in the context of the special binary mixed models. The performance of the proposed estimators are examined through a simulation study. The estimation procedure is also illustrated through an analysis of the child health and development study (CHDS) data from Yerushalmy (1970).

Generalized linear mixed models for meta-analysis

We examine two strategies for meta-analysis of a series of 2 x 2 tables with the odds ratio modelled as a linear combination of study level covariates and random effects representing between-study variation. Penalized quasi-likelihood (PQL), an approximate inference technique for generalized linear mixed models, and a linear model fitted by weighted least squares to the observed log-odds ratios are used to estimate regression coefficients and dispersion parameters. Simulation results demonstrate that both methods perform adequate approximate inference under many conditions, but that neither method works well in the presence of highly sparse data. Under certain conditions with small cell frequencies the PQL method provides better inference.

Bias Correction in Generalized Linear Mixed Models With Multiple Components of Dispersion

Abstract General formulas are derived for the asymptotic bias in regression coefficients and variance components estimated by penalized quasi-likelihood (PQL) in generalized linear mixed models with canonical link function and multiple sets of independent random effects. Easily computed correction matrices result in variance component estimates that have satisfactory asymptotic behavior for small values of the variance components and significantly reduce bias for larger values. Both first-order and second-order correction procedures are developed for regression coefficients estimated by PQL. The methods are illustrated through an analysis of an experiment on salamander matings involving crossed male and female random effects, and their properties are evaluated in a simulation study.

Analysis of correlated binomial data in logistic-normal models

Estimation in logistic-normal models for correlated and overdispersed binomial data is complicated by the numerical evaluation of often intractable likelihood functions. Penalized quasilikelihood (PQL) estimators of fixed effects and variance components are known to be seriously biased for binary data. A simple correction procedure has been proposed to improve the performance of the PQL estimators. The proposed method is illustrated by analyzing infectious disease data. Its performance is compared, by means of simulations, with that of the Bayes approach using the Gibbs sampler.

Bias Correction in Generalized Linear Mixed Models with Multiple Components of Dispersion

Abstract General formulas are derived for the asymptotic bias in regression coefficients and variance components estimated by penalized quasi-likelihood (PQL) in generalized linear mixed models with canonical link function and multiple sets of independent random effects. Easily computed correction matrices result in variance component estimates that have satisfactory asymptotic behavior for small values of the variance components and significantly reduce bias for larger values. Both first-order and second-order correction procedures are developed for regression coefficients estimated by PQL. The methods are illustrated through an analysis of an experiment on salamander matings involving crossed male and female random effects, and their properties are evaluated in a simulation study.