Parametric simultaneous robust inferences for regression coefficient under generalized linear models

In this article, the parametric robust regression approaches are proposed for making inferences about regression parameters in the setting of generalized linear models (GLMs). The proposed methods are able to test hypotheses on the regression coefficients in the misspecified GLMs. More specifically, it is demonstrated that with large samples, the normal and gamma regression models can be properly adjusted to become asymptotically valid for inferences about regression parameters under model misspecification. These adjusted regression models can provide the correct type I and II error probabilities and the correct coverage probability for continuous data, as long as the true underlying distributions have finite second moments.     

Choice of Parametric Accelerated Life and Proportional Hazards Models for Survival Data: Asymptotic Results

We discuss the impact of misspecifying fully parametric proportional hazards and accelerated life models. For the uncensored case, misspecified accelerated life models give asymptotically unbiased estimates of covariate effect, but the shape and scale parameters depend on the misspecification. The covariate, shape and scale parameters differ in the censored case. Parametric proportional hazards models do not have a sound justification for general use: estimates from misspecified models can be very biased, and misleading results for the shape of the hazard function can arise. Misspecified survival functions are more biased at the extremes than the centre. Asymptotic and first order results are compared. If a model is misspecified, the size of Wald tests will be underestimated. Use of the sandwich estimator of standard error gives tests of the correct size, but misspecification leads to a loss of power. Accelerated life models are more robust to misspecification because of their log-linear form. In preliminary data analysis, practitioners should investigate proportional hazards and accelerated life models; software is readily available for several such models.     

Nonparametric causal effects based on marginal structural models

A parametric marginal structural model (PMSM) approach to Causal Inference has been favored since the introduction of MSMs by Robins [1998a. Marginal structural models. In: 1997 Proceedings of the American Statistical Association. American Statistical Association, Alexandria, VA, pp. 1–10]. We propose an alternative, nonparametric MSM (NPMSM) approach that extends the definition of causal parameters of interest and causal effects. This approach is appealing in practice as it does not require correct specification of a parametric model but instead relies on a working model which can be willingly misspecified. We propose a methodology for longitudinal data to generate and estimate so-called NPMSM parameters describing so-called nonparametric causal effects and provide insight on how to interpret these parameters causally in practice. Results are illustrated with a point treatment simulation study. The proposed NPMSM approach to Causal Inference is compared to the more typical PMSM approach and we contribute to the general understanding of PMSM estimation by addressing the issue of PMSM misspecification.     

A generalized ordered Probit model

Abstract

The ordered probit and logit models, based on the normal and logistic distributions, can yield biased and inconsistent estimators when the distributions are misspecified. A generalized ordered response model is introduced which can reduce the impact of distributional misspecification. An empirical exploration of various determinants of life satisfaction suggests possible benefits of allowing for diverse distributional characteristics. These improvements are confirmed using a Monte Carlo study to contrast the performance of the flexible parametric specifications to the probit and logit specifications.     

Parameter Orthogonality in Mixed Regression Models for Survival Data

The implications of parameter orthogonality for the robustness of survival regression models are considered. The question of which of the proportional hazards or the accelerated life families of models would be more appropriate for analysis is usually ignored, and the proportional hazards family is applied, particularly in medicine, for convenience. Accelerated life models have conventionally been used in reliability applications. We propose a one-parameter family mixture survival model which includes both the accelerated life and the proportional hazards models. By orthogonalizing relative to the mixture parameter, we can show that, for small effects of the covariates, the regression parameters under the alternative families agree to within a constant. This recovers a known misspecification result. We use notions of parameter orthogonality to explore robustness to other types of misspecification including misspecified base-line hazards. The results hold in the presence of censoring. We also study the important question of when proportionality matters.     

Stationary persistent time series misspecified as nonstationary arima

Long-memory processes, such as Autoregressive Fractionally Integrated Moving-Average processes—ARFIMA—are likely to lead the observer to make serious misspecification errors. Nonstationary ARFIMA processes can easily be misspecified as ARIMA models, thus confusing a fractional degree of integration with an integer one. Stationary persistent ARFIMA processes can be misspecified as nonstationary ARIMA models, thus leading to a serious increase of out-of-sample forecast errors. In this paper, we discuss three prototypical misspecification cases and derive the corresponding increase in mean square forecasting error for different lead times.     

Influences of misspecification on asymptotic relative efficiency of coefficients of determination: Application to agriculture

Violation of correct specification may cause some undesirable results such as biased logistic regression coefficients and less efficient test statistics. In this paper, asymptotic relative efficiency (ARE) of various coefficients of determination in misspecified binary logistic regression models is investigated. Seven types of misspecification have been included. ARE of test statistics for exponential and Weibull distributions as a method of calculating optimal cutpoints is derived to demonstrate misspecification. Theoretical relationships between coefficients of determination have also been analyzed. Extensive simulations using bootstrap method and a real data application reveal more efficient one under various modeling scenarios.     

Partial Linear Models for Longitudinal Data Based on Quadratic Inference Functions

In this paper, we consider improved estimating equations for semiparametric partial linear models (PLM) for longitudinal data, or clustered data in general. We approximate the non‐parametric function in the PLM by a regression spline, and utilize quadratic inference functions (QIF) in the estimating equations to achieve a more efficient estimation of the parametric part in the model, even when the correlation structure is misspecified. Moreover, we construct a test which is an analogue to the likelihood ratio inference function for inferring the parametric component in the model. The proposed methods perform well in simulation studies and real data analysis conducted in this paper.     

Adaptive estimators for parameters of the autoregression function of a Markov chain

Suppose we observe an ergodic Markov chain on the real line, with a parametric model for the autoregression function, i.e. the conditional mean of the transition distribution. If one specifies, in addition, a parametric model for the conditional variance, one can define a simple estimator for the parameter, the maximum quasi-likelihood estimator. It is robust against misspecification of the conditional variance, but not efficient. We construct an estimator which is adaptive in the sense that it is efficient if the conditional variance is misspecified, and asymptotically as good as the maximum quasi-likelihood estimator if the conditional variance is correctly specified. The adaptive estimator is a weighted nonlinear least-squares estimator, with weights given by predictors for the conditional variance.     

Propensity score model specification for estimation of average treatment effects

Treatment effect estimators that utilize the propensity score as a balancing score, e.g., matching and blocking estimators are robust to misspecifications of the propensity score model when the misspecification is a balancing score. Such misspecifications arise from using the balancing property of the propensity score in the specification procedure. Here, we study misspecifications of a parametric propensity score model written as a linear predictor in a strictly monotonic function, e.g. a generalized linear model representation. Under mild assumptions we show that for misspecifications, such as not adding enough higher order terms or choosing the wrong link function, the true propensity score is a function of the misspecified model. Hence, the latter does not bring bias to the treatment effect estimator. It is also shown that a misspecification of the propensity score does not necessarily lead to less efficient estimation of the treatment effect. The results of the paper are highlighted in simulations where different misspecifications are studied.     

Restricted minimax robust designs for misspecified regression models

The authors propose and explore new regression designs. Within a particular parametric class, these designs are minimax robust against bias caused by model misspecification while attaining reasonable levels of efficiency as well. The introduction of this restricted class of designs is motivated by a desire to avoid the mathematical and numerical intractability found in the unrestricted minimax theory. Robustness is provided against a family of model departures sufficiently broad that the minimax design measures are necessarily absolutely continuous. Examples of implementation involve approximate polynomial and second order multiple regression.     

Aspects of smoothing and model inadequacy in generalized regression

We present a Bayesian semiparametric approach to exponential family regression that extends the class of generalized linear regression models. Further, flexibility in the process of modelling is achieved by explicitly accounting for the discrepancy between the ‘true’ response-covariate regression surface and an assumed parametric functional relationship. An approximate full Bayesian analysis is provided, based upon the Gibbs sampling algorithm.     

Robust Designs in Generalized Linear Models: A Quantile Dispersion Graphs Approach

This article studies design selection for generalized linear models (GLMs) using the quantile dispersion graphs (QDGs) approach in the presence of misspecification in the link and/or linear predictor. The uncertainty in the linear predictor is represented by a unknown function and estimated using kriging. For addressing misspecified link functions, a generalized family of link functions is used. Numerical examples are shown to illustrate the proposed methodology.     

Second-order least squares estimation of censored regression models

This paper proposes the second-order least squares estimation, which is an extension of the ordinary least squares method, for censored regression models where the error term has a general parametric distribution (not necessarily normal). The strong consistency and asymptotic normality of the estimator are derived under fairly general regularity conditions. We also propose a computationally simpler estimator which is consistent and asymptotically normal under the same regularity conditions. Finite sample behavior of the proposed estimators under both correctly and misspecified models are investigated through Monte Carlo simulations. The simulation results show that the proposed estimator using optimal weighting matrix performs very similar to the maximum likelihood estimator, and the estimator with the identity weight is more robust against the misspecification.     

Model selection for Lévy measures in diffusion processes with jumps from discrete observations

We deal with parametric inference and selection problems for jump components in discretely observed diffusion processes with jumps. We prepare several competing parametric models for the Lévy measure that might be misspecified, and select the best model from the aspect of information criteria. We construct quasi-information criteria (QIC), which are approximations of the information criteria based on continuous observations.     

Focused information criterion for locally misspecified vector autoregressive models

This paper investigates the focused information criterion and plug-in average for vector autoregressive models with local-to-zero misspecification. These methods have the advantage of focusing on a quantity of interest rather than aiming at overall model fit. Any (su?ciently regular) function of the parameters can be used as a quantity of interest. We determine the asymptotic properties and elaborate on the role of the locally misspecified parameters. In particular, we show that the inability to consistently estimate locally misspecified parameters translates into suboptimal selection and averaging. We apply this framework to impulse response analysis. A Monte Carlo simulation study supports our claims.     

Nonparametric estimation with recurrent competing risks data

Nonparametric estimators of component and system life distributions are developed and presented for situations where recurrent competing risks data from series systems are available. The use of recurrences of components’ failures leads to improved efficiencies in statistical inference, thereby leading to resource-efficient experimental or study designs or improved inferences about the distributions governing the event times. Finite and asymptotic properties of the estimators are obtained through simulation studies and analytically. The detrimental impact of parametric model misspecification is also vividly demonstrated, lending credence to the virtue of adopting nonparametric or semiparametric models, especially in biomedical settings. The estimators are illustrated by applying them to a data set pertaining to car repairs for vehicles that were under warranty.     

A unified approach to proving parametric bootstrap consistency for some goodness-of-fit tests

Because model misspecification can lead to inconsistent and inefficient estimators and invalid tests of hypotheses, testing for misspecification is critically important. We focus here on several general purpose goodness-of-fit tests which can be applied to assess the adequacy of a wide variety of parametric models without specifying an alternative model. Parametric bootstrap is the method of choice for computing the p-values of these tests however the proof of its consistency has never been rigourously shown in this setting. Using properties of locally asymptotically normal parametric models, we prove that under quite general conditions, the parametric bootstrap provides a consistent estimate of the null distribution of the statistics under investigation.     

MULTIPLE COMPARISONS OF LOG-LIKELIHOODS AND COMBINING NONNESTED MODELS WITH APPLICATIONS TO PHYLOGENETIC TREE SELECTION

We consider multiple comparisons of log-likelihood's to take account of the multiplicity of testings in selection of nonnested models. A resampling version of the Gupta procedure for the selection problem is used to obtain a set of good models, which are not significantly worse than the maximum likelihood model; i.e., a confidence set of models. Our method is to test which model is better than the other, while the object of the classical testing methods is to find the correct model. Thus the null hypotheses behind these two approaches are very different. Our method and the other commonly used approaches, such as the approximate Bayesian posterior, the bootstrap selection probability, and the LR test against the full model, are applied to the selection of molecular phylogenetic tree of mammal species. Tree selection is a version of the model-based clustering, which is an example of nonnested model selection. It is shown that the structure of the tree selection problem is equivalent to that of the variable selection problem of the multiple regression with some constraints on the combinations of the variables. It turns out that the LR test rejects all the possible trees because of the misspecification of the models, whereas our method gives a reasonable confidence set. For a better understanding of the uncertainty in the selection, we combine the maximum likelihood estimates (MLE's) of the trees to obtain the full model that includes the trees as the submodels by using a linear approximation of the parametric models. The MLE of the phylogeny is then represented as a network of species rather than a tree. A geometrical interpretation of the problem is also discussed.     

Choosing the link function and accounting for link uncertainty in generalized linear models using Bayes factors

One important component of model selection using generalized linear models (GLM) is the choice of a link function. We propose using approximate Bayes factors to assess the improvement in fit over a GLM with canonical link when a parametric link family is used. The approximate Bayes factors are calculated using the Laplace approximations given in [32], together with a reference set of prior distributions. This methodology can be used to differentiate between different parametric link families, as well as allowing one to jointly select the link family and the independent variables. This involves comparing nonnested models and so standard significance tests cannot be used. The approach also accounts explicitly for uncertainty about the link function. The methods are illustrated using parametric link families studied in [12] for two data sets involving binomial responses. The first author was supported by Sonderforschungsbereich 386 Statistische Analyse Diskreter Strukturen, and the second author by NIH Grant 1R01CA094212-01 and ONR Grant N00014-01-10745.