Statistical Methods and Inference

Longitudinal data analysis using generalized linear models

This paper proposes an extension of generalized linear models to the analysis of longitudinal data. We introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence. The estimating equations are derived without specifying the joint distribution of a subject's observations yet they reduce to the score equations for niultivariate Gaussian outcomes. Asymptotic theory is presented for the general class of estimators. Specific cases in which we assume independence, m-dependence and exchangeable correlation structures from each subject are discussed. Efficiency of the pioposecl estimators in two simple situations is considered. The approach is closely related to quasi-likelihood.

Regularization Paths for Generalized Linear Models via Coordinate Descent.

We develop fast algorithms for estimation of generalized linear models with convex penalties. The models include linear regression, two-class logistic regression, and multinomial regression problems while the penalties include ℓ(1) (the lasso), ℓ(2) (ridge regression) and mixtures of the two (the elastic net). The algorithms use cyclical coordinate descent, computed along a regularization path. The methods can handle large problems and can also deal efficiently with sparse features. In comparative timings we find that the new algorithms are considerably faster than competing methods.

Deciding on the Number of Classes in Latent Class Analysis and Growth Mixture Modeling: A Monte Carlo Simulation Study

Mixture modeling is a widely applied data analysis technique used to identify unobserved heterogeneity in a population. Despite mixture models' usefulness in practice, one unresolved issue in the application of mixture models is that there is not one commonly accepted statistical indicator for deciding on the number of classes in a study population. This article presents the results of a simulation study that examines the performance of likelihood-based tests and the traditionally used Information Criterion (ICs) used for determining the number of classes in mixture modeling. We look at the performance of these tests and indexes for 3 types of mixture models: latent class analysis (LCA), a factor mixture model (FMA), and a growth mixture models (GMM). We evaluate the ability of the tests and indexes to correctly identify the number of classes at three different sample sizes (n = 200, 500, 1,000). Whereas the Bayesian Information Criterion performed the best of the ICs, the bootstrap likelihood ratio test proved to be a very consistent indicator of classes across all of the models considered.

MULTIVARIABLE PROGNOSTIC MODELS: ISSUES IN DEVELOPING MODELS, EVALUATING ASSUMPTIONS AND ADEQUACY, AND MEASURING AND REDUCING ERRORS

Multivariable regression models are powerful tools that are used frequently in studies of clinical outcomes. These models can use a mixture of categorical and continuous variables and can handle partially observed (censored) responses. However, uncritical application of modelling techniques can result in models that poorly fit the dataset at hand, or, even more likely, inaccurately predict outcomes on new subjects. One must know how to measure qualities of a model's fit in order to avoid poorly fitted or overfitted models. Measurement of predictive accuracy can be difficult for survival time data in the presence of censoring. We discuss an easily interpretable index of predictive discrimination as well as methods for assessing calibration of predicted survival probabilities. Both types of predictive accuracy should be unbiasedly validated using bootstrapping or cross-validation, before using predictions in a new data series. We discuss some of the hazards of poorly fitted and overfitted regression models and present one modelling strategy that avoids many of the problems discussed. The methods described are applicable to all regression models, but are particularly needed for binary, ordinal, and time-to-event outcomes. Methods are illustrated with a survival analysis in prostate cancer using Cox regression.

Modeling Survival Data: Extending the Cox Model

Model Selection and Estimation in Regression with Grouped Variables

Summary We consider the problem of selecting grouped variables (factors) for accurate prediction in regression. Such a problem arises naturally in many practical situations with the multifactor analysis-of-variance problem as the most important and well-known example. Instead of selecting factors by stepwise backward elimination, we focus on the accuracy of estimation and consider extensions of the lasso, the LARS algorithm and the non-negative garrotte for factor selection. The lasso, the LARS algorithm and the non-negative garrotte are recently proposed regression methods that can be used to select individual variables. We study and propose efficient algorithms for the extensions of these methods for factor selection and show that these extensions give superior performance to the traditional stepwise backward elimination method in factor selection problems. We study the similarities and the differences between these methods. Simulations and real examples are used to illustrate the methods.

Bayesian Model Selection in Social Research

It is argued that P-values and the tests based upon them give unsatisfactory results, especially in large samples. It is shown that, in regression, when there are many candidate independent variables, standard variable selection procedures can give very misleading results. Also, by selecting a single model, they ignore model uncertainty and so underestimate the uncertainty about quantities of interest. The Bayesian approach to hypothesis testing, model selection, and accounting for model uncertainty is presented. Implementing this is straightforward through the use of the simple and accurate BIC approximation, and it can be done using the output from standard software. Specific results are presented for most of the types of model commonly used in sociology. It is shown that this approach overcomes the difficulties with P-values and standard model selection procedures based on them. It also allows easy comparison of nonnested models, and permits the quantification of the evidence for a null hypothesis of interest, such as a convergence theory or a hypothesis about societal norms.

Regression Modeling Strategies

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Regression and time series model selection in small samples

A bias correction to the Akaike information criterion, AIC, is derived for regression and autoregressive time series models. The correction is of particular use when the sample size is small, or when the number of fitted parameters is a moderate to large fraction of the sample size. The corrected method, called AICC, is asymptotically efficient if the true model is infinite dimensional. Furthermore, when the true model is of finite dimension, AICC is found to provide better model order choices than any other asymptotically efficient method. Applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed.

Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses

In this paper, we develop a classical approach to model selection. Using the Kullback-Leibler Information Criterion to measure the closeness of a model to the truth, we propose simple likelihood-ratio based statistics for testing the null hypothesis that the competing models are equally close to the true data generating process against the alternative hypothesis that one model is closer. The tests are directional and are derived successively for the cases where the competing models are non-nested, overlapping, or nested and whether both, one, or neither is misspecified. As a prerequisite, we fully characterize the asymptotic distribution of the likelihood ratio statistic under the most general conditions. We show that it is a weighted sum of chi-square distribution or a normal distribution depending on whether the distributions in the competing models closest to the truth are observationally identical. We also propose a test of this latter condition.

Marginal Structural Models and Causal Inference in Epidemiology

In observational studies with exposures or treatments that vary over time, standard approaches for adjustment of confounding are biased when there exist time-dependent confounders that are also affected by previous treatment. This paper introduces marginal structural models, a new class of causal models that allow for improved adjustment of confounding in those situations. The parameters of a marginal structural model can be consistently estimated using a new class of estimators, the inverse-probability-of-treatment weighted estimators.

Regression Models for Ordinal Data

Summary A general class of regression models for ordinal data is developed and discussed. These models utilize the ordinal nature of the data by describing various modes of stochastic ordering and this eliminates the need for assigning scores or otherwise assuming cardinality instead of ordinality. Two models in particular, the proportional odds and the proportional hazards models are likely to be most useful in practice because of the simplicity of their interpretation. These linear models are shown to be multivariate extensions of generalized linear models. Extensions to non-linear models are discussed and it is shown that even here the method of iteratively reweighted least squares converges to the maximum likelihood estimate, a property which greatly simplifies the necessary computation. Applications are discussed with the aid of examples.