Comparison of population-averaged and subject-specific approaches for analyzing repeated binary outcomes

Several approaches have been proposed to model binary outcomes that arise from longitudinal studies. Most of the approaches can be grouped into two classes: the population-averaged and subject-specific approaches. The generalized estimating equations (GEE) method is commonly used to estimate population-averaged effects, while random-effects logistic models can be used to estimate subject-specific effects. However, it is not clear to many epidemiologists how these two methods relate to one another or how these methods relate to more traditional stratified analysis and standard logistic models. The authors address these issues in the context of a longitudinal smoking prevention trial, the Midwestern Prevention Project. In particular, the authors compare results from stratified analysis, standard logistic models, conditional logistic models, the GEE models, and random-effects models by analyzing a binary outcome from two and seven repeated measurements, respectively. In the comparison, the authors focus on the interpretation of both time-varying and time-invariant covariates under different models. Implications of these methods for epidemiologic research are discussed.     

Between- and within-cluster covariate effects in the analysis of clustered data

Standard methods for the regression analysis of clustered data postulate models relating covariates to the response without regard to between- and within-cluster covariate effects. Implicit in these analyses is the assumption that these effects are identical. Example data show that this is frequently not the case and that analyses that ignore differential between- and within-cluster covariate effects can be misleading. Consideration of between- and within-cluster effects also helps to explain observed and theoretical differences between mixture model analyses and those based on conditional likelihood methods. In particular, we show that conditional likelihood methods estimate purely within-cluster covariate effects, whereas mixture model approaches estimate a weighted average of between- and within-cluster covariate effects.     

Refined crystal structure of methylamine dehydrogenase from Paracoccus denitrificans at 1.75 A resolution

We extend the random intercept logistic model to accommodate negative intracluster correlations for bivariate binary response data. This approach assumes a single random effect per cluster, but entails separate affine transformations of this random effect for the two responses of the pair. We show this approach works for two data sets and a simulation, whereas other mixed effects approaches fail. The two data sets are from a crossover trial and a developmental toxicity study of the effects of chemical exposure on malformation risk among rat pups. Comparisons are made with the conditional likelihood approach and with generalized estimating equations estimation of the population-averaged logit model. Simulations show the conditional likelihood approach does not perform well for moderate to strong negative correlations, as a positive intracluster correlation is assumed. The proposed mixed effects approach appears to be slightly more conservative than the population-averaged approach with respect to coverage of confidence intervals. Nonetheless, the statistical literature suggests that mixed effects models provide information in addition to that provided by population-averaged models under scientific contexts such as crossover trials. Extensions to trivariate and higher-dimensional responses also are addressed. However, such extensions require certain constraints on the correlation structure.     

Random effects probit and logistic regression models for three-level data

In analysis of binary data from clustered and longitudinal studies, random effect models have been recently developed to accommodate two-level problems such as subjects nested within clusters or repeated classifications within subjects. Unfortunately, these models cannot be applied to three-level problems that occur frequently in practice. For example, multicenter longitudinal clinical trials involve repeated assessments within individuals and individuals are nested within study centers. This combination of clustered and longitudinal data represents the classic three-level problem in biometry. Similarly, in prevention studies, various educational programs designed to minimize risk taking behavior (e.g., smoking prevention and cessation) may be compared where randomization to various design conditions is at the level of the school and the intervention is performed at the level of the classroom. Previous statistical approaches to the three-level problem for binary response data have either ignored one level of nesting, treated it as a fixed effect, or used first- and second-order Taylor series expansions of the logarithm of the conditional likelihood to linearize these models and estimate model parameters using more conventional procedures for measurement data. Recent studies indicate that these approximate solutions exhibit considerable bias and provide little advantage over use of traditional logistic regression analysis ignoring the hierarchical structure. In this paper, we generalize earlier results for two-level random effects probit and logistic regression models to the three-level case. Parameter estimation is based on full-information maximum marginal likelihood estimation (MMLE) using numerical quadrature to approximate the multiple random effects. The model is illustrated using data from 135 classrooms from 28 schools on the effects of two smoking cessation interventions.     

An application of maximum likelihood and generalized estimating equations to the analysis of ordinal data from a longitudinal study with cases missing at random

Data are analysed from a longitudinal psychiatric study in which there are no dropouts that do not occur completely at random. A marginal proportional odds model is fitted that relates the response (severity of side effects) to various covariates. Two methods of estimation are used: generalized estimating equations (GEE) and maximum likelihood (ML). Both the complete set of data and the data from only those subjects completing the study are analysed. For the completers-only data, the GEE and ML analyses produce very similar results. These results differ considerably from those obtained from the analyses of the full data set. There are also marked differences between the results obtained from the GEE and ML analysis of the full data set. The occurrence of such differences is consistent with the presence of a non-completely-random dropout process and it can be concluded in this example that both the analyses of the completers only and the GEE analysis of the full data set produce misleading conclusions about the relationships between the response and covariates.     

Probabilistic models for sequential taste effects in triadic choice.

Sequential effects and positional response bias are accounted for in new models for triadic choice. These models were applied to data on distilled water and dilute NaCl solutions by use of the triangular and 3-alternative forced-choice methods with 4 participants. The concept of a "conditional stimulus" is introduced to describe stimuli that are created partially by prior oral environmental effects. The effects of 1 or 2 prior stimuli on triadic choice were evaluated. The triad models used were based on a Thurstonian variant of M. W. Richardson's (1938) method of triads and a Thurstonian model for 1st choice among 3 possibly different stimuli. Maximum likelihood estimates of the scale values for conditional stimuli and bias parameters showed that it was necessary only to consider 1 prior stimulus. It was also shown that salt concentration differences are not the physical analog of the mental representations for the conditional stimuli. The results strongly suggest a water taste to salt taste continuum. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Mixed effects logistic regression models for longitudinal binary response data with informative drop-out

A shared parameter model with logistic link is presented for longitudinal binary response data to accommodate informative drop-out. The model consists of observed longitudinal and missing response components that share random effects parameters. To our knowledge, this is the first presentation of such a model for longitudinal binary response data. Comparisons are made to an approximate conditional logit model in terms of a clinical trial dataset and simulations. The naive mixed effects logit model that does not account for informative drop-out is also compared. The simulation-based differences among the models with respect to coverage of confidence intervals, bias, and mean squared error (MSE) depend on at least two factors: whether an effect is a between- or within-subject effect and the amount of between-subject variation as exhibited by variance components of the random effects distributions. When the shared parameter model holds, the approximate conditional model provides confidence intervals with good coverage for within-cluster factors but not for between-cluster factors. The converse is true for the naive model. Under a different drop-out mechanism, when the probability of drop-out is dependent only on the current unobserved observation, all three models behave similarly by providing between-subject confidence intervals with good coverage and comparable MSE and bias but poor within-subject confidence intervals, MSE, and bias. The naive model does more poorly with respect to the within-subject effects than do the shared parameter and approximate conditional models. The data analysis, which entails a comparison of two pain relievers and a placebo with respect to pain relief, conforms to the simulation results based on the shared parameter model but not on the simulation based on the outcome-driven drop-out process. This comparison between the data analysis and simulation results may provide evidence that the shared parameter model holds for the pain data.     

Analysis of prevention program effectiveness with clustered data using generalized estimating equations.

Experimental studies of prevention programs often randomize clusters of individuals rather than individuals to treatment conditions. When the correlation among individuals within clusters is not accounted for in statistical analysis, the standard errors are biased, potentially resulting in misleading conclusions about the significance of treatment effects. This study demonstrates the generalized estimating equations (GEE) method, focusing specifically on the GEE-independent method, to control for within-cluster correlation in regression models with either continuous or binary outcomes. The GEE-independent method yields consistent and robust variance estimates. Data from Project DARE, a youth substance abuse prevention program, are used for illustration. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A generalized estimating equations approach for testing ordered group effects with repeated measurements

In repeated measures studies, we are often interested in comparing group effects in which groups are associated with a certain order relation. We propose testing procedures for ordered group effects using the generalized estimating equations (GEE) approach of Liang and Zeger (1986, Biometrika 73, 13-22). The order-constrained GEE estimators of group effects are approximated by the isotonic regression of the unconstrained GEE estimators. Based on these constrained estimators, we construct test statistics for detecting ordered group effects. The limiting distributions of the test statistics are mixtures of chi-square distributions. A Monte Carlo experiment shows improved performances of the proposed tests over the usual chi-square tests in detecting ordered group effects. The proposed test procedures are illustrated by familial polyposis supplementation trial data.     

Generalized full-information item bifactor analysis.

Full-information item bifactor analysis is an important statistical method in psychological and educational measurement. Current methods are limited to single-group analysis and inflexible in the types of item response models supported. We propose a flexible multiple-group item bifactor analysis framework that supports a variety of multidimensional item response theory models for an arbitrary mixing of dichotomous, ordinal, and nominal items. The extended item bifactor model also enables the estimation of latent variable means and variances when data from more than 1 group are present. Generalized user-defined parameter restrictions are permitted within or across groups. We derive an efficient full-information maximum marginal likelihood estimator. Our estimation method achieves substantial computational savings by extending Gibbons and Hedeker's (1992) bifactor dimension reduction method so that the optimization of the marginal log-likelihood requires only 2-dimensional integration regardless of the dimensionality of the latent variables. We use simulation studies to demonstrate the flexibility and accuracy of the proposed methods. We apply the model to study cross-country differences, including differential item functioning, using data from a large international education survey on mathematics literacy. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Regression models for multiple outcomes in large epidemiologic studies

In situations in which one cannot specify a single primary outcome, epidemiologic analyses often examine multiple associations between outcomes and explanatory covariates or risk factors. To compare alternative approaches to the analysis of multiple outcomes in regression models, I used generalized estimating equations (GEE) models, a multivariate extension of generalized linear models, to incorporate the dependence among the outcomes from the same subject and to provide robust variance estimates of the regression coefficients. I applied the methods in a hospital-population-based study of complications of surgical anaesthesia, using GEE model fitting and quasi-likelihood score and Wald tests. In one GEE model specification, I allowed the associations between each of the outcomes and a covariate to differ, yielding a regression coefficient for each of the outcome and covariate combinations; I obtained the covariances among the set of outcome-specific regression coefficients for each covariate from the robust 'sandwich' variance estimator. To address the problem of multiple inference, I used simultaneous methods that make adjustments to the test statistic p-values and the confidence interval widths, to control type I error and simultaneous coverage, respectively. In a second model specification, for each of the covariates I assumed a common association between the outcomes and the covariate, which eliminates the problem of multiplicity by use of a global test of association. In an alternative approach to multiplicity, I used empirical Bayes methods to shrink the outcome-specific coefficients toward a pooled mean that is similar to the common effect coefficient. GEE regression models can provide a flexible framework for estimation and testing of multiple outcomes.     

Sequence Effects in the Categorization of Tones Varying in Frequency.

In contrast to exemplar and decision-bound categorization models, the memory and contrast models described here do not assume that long-term representations of stimulus magnitudes are available. Instead, stimuli are assumed to be categorized using only their differences from a few recent stimuli. To test this alternative, the authors examined sequential effects in a binary categorization of 10 tones varying in frequency. Stimuli up to 2 trials back in the sequence had a significant effect on the response to the current stimulus. The effects of previous stimuli interacted with one another. A memory and contrast model, according to which only ordinal information about the differences between the current stimulus and recent preceding stimuli is used, best accounted for these data (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Effects of stimulus intensity on the lateralized readiness potential.

Information-processing models differ about whether stimulus intensity affects the speed of motor processes involved in response activation and execution. Previous studies of intensity are reviewed, but they are not decisive on this point because they have used indirect approaches requiring strong assumptions. Two experiments examined the effects of stimulus intensity on the lateralized readiness potential (LRP), a measure of hand-specific response activation. In Experiment 1, visual stimulus intensity influenced the time from stimulus onset to LRP onset but not the time from LRP onset to the keypress response. In Experiment 2, auditory stimulus intensity did not influence either of these time intervals, although it did influence the time from stimulus onset to the N100 and P300 components of the evoked potential. The results indicate that stimulus intensity does not influence the duration of motor processes in choice reaction time tasks. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Conditional and unconditional automaticity: A dual-process model of effects of spatial stimulus-response correspondence.

Distributional analyses and event-related brain potentials were used to show that effects of irrelevant spatial stimulus–response correspondence consist of 2 qualitatively different automatic components that can be distinguished on the basis of their dependencies on relative response speed and on computational requirements of the primary task. One component reflects priming of the spatially corresponding response by an abrupt stimulus onset that does not depend on the nature of the primary task. This unconditional component exhibits a biphasic pattern, with initial facilitation later turning into inhibition, analogous to that found for spatial cuing in visual detection tasks. The 2nd component reflects automatic generalization of task-defined transformations of relevant stimulus information to spatial codes; this conditional component does not depend on relative response speed. Possible connectionist implementations of the conditional mechanism are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Posterior likelihood methods for multivariate survival data

This article deals with the semiparametric analysis of multivariate survival data with random block (group) effects. Survival times within the same group are correlated as a consequence of a frailty random block effect. The standard approaches assume either a parametric or a completely unknown baseline hazard function. This paper considers an intermediate solution, that is, a nonparametric function that is reasonably smooth. This is accomplished by a Bayesian model in which the conditional proportional hazards model is used with a correlated prior process for the baseline hazard. The posterior likelihood based on data, as well as the prior process, is similar to the discretized penalized likelihood for the frailty model. The methodology is exemplified with the recurrent kidney infections data of McGilchrist and Aisbett (1991, Biometrics 47, 461-466), in which the times to infections within the same patients are expected to be correlated. The reanalysis of the data has shown that the estimates of the parameters of interest and the associated standard errors depend on the prior knowledge about the smoothness of the baseline hazard.     

A general framework for random effects survival analysis in the Cox proportional hazards setting

The use of random effects modeling in statistics has increased greatly in recent years. The introduction of such modeling into event-time analysis has proceeded more slowly, however. Previously, random effects models for survival data have either required assumptions regarding the form of the baseline hazard function or restrictions on the classes of models that can be fit. In this paper, we develop a method of random effect analysis of survival data, the hierarchical Cox model, that is an extension of Cox's original formulation in that the baseline hazard function remains unspecified. This method also allows an arbitrary distribution for the random effects. We accomplish this using Markov chain Monte Carlo methods in a Bayesian setting. The method is illustrated with three models for a dataset with times to multiple occurrences of mammory tumors for 48 rats treated with a carcinogen and then randomized to either treatment or control. This analysis is more satisfying than standard approaches, such as studying the first event for each subject, which does not fully use the data, or assuming independence, which in this case would overestimate the precision.     

Cerebral gangliogliomas. Morphological and immunohistochemical study

Recent studies have shown that experimental tumors could be treated more efficiently with ionizing radiation if genetic material was transfered into tumor cells. Several approaches have been reported, and among them, the first one consisted of increasing the apoptotic response to radiation by modulating genes involved in the regulation of the apoptotic pathway. Indeed the modulation of p53 and bcl-2 gene expression has recently been used successfully in several experimental models to increase the apoptotic death after radiation. A second approach consisted of taking advantage of the conditional expression of some genes after exposure to ionizing radiation. Indeed, some genes exhibit a radio-inducible promoter which can be combined to a gene, able to enhance or decrease the biological effect of radiation. The irradiation of such a transgene under the control of a radio-inducible promoter can lead to a second biological effect, concomitant to the irradiation, as reported for the TNF alpha under the control of the EGR (early growth response) promoter. A third approach consisted of enhancing the effect of radiation induced tumor cell death by the expression of a suicide gene in these cells, as suggested recently for the HSV-tk (herpes virus thymidine kinase gene). These preliminary results obtained in experimental models appear to be very promising and might improve the efficacy and specificity of radiation therapy in a not too distant future.     

Modeling hazard functions in families

A genetic frailty model is presented for censored age of onset data in nuclear families where individuals carrying a genetic susceptibility gene have an increased risk of becoming affected. We use maximum likelihood via the EM algorithm to estimate the genetic relative risk and the allele frequency under a dominant susceptibility type and a proportional hazards model. When sampling is from a disease registry, likelihood corrections are necessary for reducing bias in the parameter estimates. In these biased samples, the full conditional likelihood is approximated by a likelihood conditional on the proband's age of onset. For unbiased samples, simulations show the distributions of the estimates are similar under both a semiparametric and the correctly specified parametric likelihoods. For biased samples, simulations under the approximate conditional likelihood show the median estimates of the allele frequency and genetic relative risk tend to under- and overestimate, respectively, the true values; however, the approximation is better for rarer allele frequencies (0.0033 vs. 0.01). In practice, large samples or more complex ascertainment corrections are recommended. Using the approximate conditional likelihood on familial breast cancer onset data collected as part of a case-control study at the Fred Hutchinson Cancer Research Center in Seattle, Washington, we estimate an allele frequency of 0.0009 (approximate 95% CI 0.0006-0.002) and a genetic relative risk of 104 (approximate 95% CI 55-181).     

Ordinal invariant measures for individual and group changes in ordered categorical data

Subjective judgements of complex variables are commonly recorded as ordered categorical data. The rank-invariant properties of such data are well known, and there are various statistical approaches to the analysis and modelling of ordinal data. This paper focuses on the non-additive property of ordered categorical data in the analysis of change. A rank-invariant non-parametric method of analysis is presented that is valid regardless of the number of response categories. The unique feature of this method is the augmented ranking approach that is related to the joint distribution of paired observations. This approach makes it possible to measure separately the individual order-preserved categorical changes, which are attributable to the group change, and the individual categorical changes that are not consistent with the pattern of group change. The method is applied to analysis of change in a three-point scale and in a visual analogue scale of continuous ordinal responses.