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Regression models are widely used statistical procedures, and the validation of their assumptions plays a crucial role in the data analysis process. Unfortunately, validating assumptions usually depends on the availability of tests tailored to the specific model of interest. A novel Bayesian goodness-of-fit hypothesis testing approach is presented for a broad class of regression models the response variable of which is univariate and continuous. The proposed approach relies on a suitable transformation of the response variable and a Bayesian prior induced by a predictor-dependent mixture model. Hypothesis testing is performed via Bayes factor, the asymptotic properties of which are discussed. The method is implemented by means of a Markov chain Monte Carlo algorithm, and its performance is illustrated using simulated and real data sets.

This paper proposes a new method for choosing regression models which may produce multiple models with sufficient explanatory power and parsimony unlike the traditional model selection procedures that aim at obtaining a single best model. The method ensures interpretability of the resulting models even under strong multicollinearity. The algorithm proceeds in the forward stepwise manner with two requirements for the selected regression models to be fulfilled: goodness of fit and the magnitude of update in loss functions. For the latter criterion, the standardized update is newly introduced, which is closely related with the model selection criteria including the Mallows' C"p, Akaike information criterion and Bayesian information criterion. Simulation studies demonstrate that the proposed algorithm works well with and without strong multicollinearity and even with many explanatory variables. Application to real data is also provided.

Longitudinal studies involving categorical responses are extensively applied in many fields of research and are often fitted by the generalized estimating equations (GEE) approach and generalized linear mixed models (GLMMs). The assessment of model fit is an important issue for model inference. The purpose of this article is to extend Pan's (2002a) goodness-of-fit tests for GEE models with longitudinal binary data to the tests for logistic proportional odds models with longitudinal ordinal data. Two proposed methods based on Pearson chi-squared test and unweighted sum of residual squares are developed, and the approximate expectations and variances of the test statistics are easily computed. Four major variants of working correlation structures, independent, AR(1), exchangeable and unspecified, are considered to estimate the variances of the proposed test statistics. Simulation studies in terms of type I error rate and the power performance of the proposed tests are presented for various sample sizes. Furthermore, the approaches are demonstrated by two real data sets.

In areas such as biology, geology and meteorology data often occur as angles. This paper examines the use of Bayesian mixtures of triangular distributions for the semiparametric analysis of circular data. Applications to density estimation, goodness-of-fit testing and semiparametric regression are demonstrated.

Statistical models are often based on normal distributions and procedures for testing such distributional assumption are needed. Many goodness-of-fit tests are available. However, most of them are quite insensitive in detecting non-normality when the alternative distribution is symmetric. On the other hand all the procedures are quite powerful against skewed alternatives. A new test for normality based on a polynomial regression is presented. It is very effective in detecting non-normality when the alternative distribution is symmetric. A comparison between well known tests and this new procedure is performed by simulation study. Other properties are also investigated.

A number of methods have been proposed to estimate the period of a variable star; e.g., a recent approach uses smoothing spline regression to fit tentative periodic functions (light curves) and selects the period minimizing a robust goodness-of-fit criterion. These methods assume that measurement errors vary independently over time. Empirical evidence, however, indicates substantial temporal dependence, possibly related to changes in observing conditions. Dependence complicates the period analysis in several respects: selection of a ''best'' period among several local optima, estimation of the light curve, and evaluation of uncertainty about period and light curve estimates. This article presents methods designed to accommodate dependent errors. An analysis of several data sets shows that the proposed approach can produce substantially different and arguably better results compared with other methods.

Methods of regression diagnostics for functional regression models are developed which relate a functional response to predictor variables that can be multivariate vectors or random functions. For this purpose, a residual process is defined by subtracting the predicted from the observed response functions. This residual process is expanded into functional principal components (FPC), and the corresponding FPC scores are used as natural proxies for the residuals in functional regression models. For the case of a univariate covariate, a randomization test is proposed based on these scores to examine if the residual process depends on the covariate. If this is the case, it indicates lack of fit of the model. Graphical methods based on the FPC scores of observed and fitted functions can be used to complement more formal tests. The methods are illustrated with data from a recent study of Drosophila fruit flies regarding life-cycle gene expression trajectories as well as functional data from a dose-response experiment for Mediterranean fruit flies (Ceratitis capitata).

Logistic regression models are frequently used in epidemiological studies for estimating associations that demographic, behavioral, and risk factor variables have on a dichotomous outcome, such as disease being present versus absent. After the coefficients in a logistic regression model have been estimated, goodness-of-fit of the resulting model should be examined, particularly if the purpose of the model is to estimate probabilities of event occurrences. While various goodness-of-fit tests have been proposed, the properties of these tests have been studied under the assumption that observations selected were independent and identically distributed. Increasingly, epidemiologists are using large-scale sample survey data when fitting logistic regression models, such as the National Health Interview Survey or the National Health and Nutrition Examination Survey. Unfortunately, for such situations no goodness-of-fit testing procedures have been developed or implemented in available software. To address this problem, goodness-of-fit tests for logistic regression models when data are collected using complex sampling designs are proposed. Properties of the proposed tests were examined using extensive simulation studies and results were compared to traditional goodness-of-fit tests. A Stata ado function svylogitgof for estimating the F-adjusted mean residual test after svylogit fit is available at the author's website http://www.people.vcu.edu/~kjarcher/Research/Data.htm.

Binary outcomes are very common in medical studies. Logistic regression is typically used to analyze independent binary outcomes while generalized estimating equations regression methods (GEE) are often used to analyze correlated binary data. Several goodness-of-fit (GoF) statistics for the GEE methods have been developed recently. The objective of this study is to compare the power and Type I error rates of existing GEE GoF statistics using simulated data under different conditions. The number of clusters was varied in each condition. Different tested models included discrete, continuous, observation-specific and/or cluster-specific covariates. Two or three observations per cluster were generated with various correlations between observations. No single GEE GoF statistic performed best across all conditions. Generally, the larger the number of clusters, the more powerful the GEE GoF statistics. The GEE GoF statistics with correctly specified working correlation matrices tended to be robust in terms of Type I error rates and more powerful. For data with two observations per cluster, both Evans and Pan's statistics [1998. Goodness of fit in two models for clustered binary data. Ph.D. Dissertation, University of Massachusetts; 2002a. Goodness-of-fit tests for GEE with correlated binary data. Scand. J. Stat. 29(1), 101-110.] and Barnhart-Williamson's statistics [1998. Goodness-of-fit tests for GEE modeling with binary data. Biometrics 54, 720-729.] performed well for detecting the effect of the omitted interaction between two binary covariates. Barnhart-Williamson's statistics were generally the most powerful for detecting other types of interactions in models with at least one continuous covariate. For data with three observations per cluster, Evans and Pan's statistics performed best. fit in two models for clustered binary data. Ph.D. Dissertation, University of Massachusetts; 2002a. Goodness-of-fit tests for GEE with correlated binary data. Scand. J. Stat. 29(1), 101-110.] statistics performed best.

An intuitively appealing lack-of-fit test to assess the adequacy of a regression model is introduced together with a graphical diagnostic tool. The graphical method itself includes a formal testing procedure, and, it is particularly useful to detect the location of lack-of-fit. The procedure is based on regional residuals, using subsets of the space of the independent variables. A simulation study shows that, the proposed procedures in simple linear regression have similar power as those of some popular classical lack-of-fit tests. In case of local departures from the hypothesized regression model, the new tests are shown to be more powerful. Therefore, when it becomes difficult to discriminate between systematic deviations and noise, regional residual plots are very helpful in formally locating areas of lack-of-fit in the predictor space. Data examples illustrate the ability of the new methods to detect and to locate lack-of-fit.

This paper extends the Bayes marginal model plot (BMMP) model assessment technique from a traditional logistic regression setting to a multilevel application in the area of criminal justice. Convicted felons in the United States receive either a prison sentence or a less severe jail or non-custodial sentence. Researchers have identified many determinants of sentencing variation across the country, some individual such as type of crime and race, and some based on geographical units such as county crime rate. Multilevel rather than conventional regression should be used to quantify any interplay between such individual- and county-level effects since the covariates have a hierarchical structure. Questions arise, however, as to whether a multilevel model provides an adequate fit to the data, and whether the computational burden of a multilevel model over a conventional model is justified. Residual plots, traditionally used to assess regression models, are difficult to interpret with a binary response variable and multilevel covariates, as in this case. BMMPs, an alternative graphical technique, can be used to visualize goodness of fit in such settings. The plots clearly demonstrate the need to use multilevel modeling when analyzing data such as these.

In many Bradley-Terry models a more or less explicit assumption is that all decisions of the judges are independent. An assumption which might be questionable at least for the decisions of a given judge. In paired comparison studies, a judge chooses among objects several times, and in such cases, judgements made by the same judge are likely to be dependent. A log-linear representation for the Bradley-Terry model will be developed, which takes into account dependencies between judgements. The modelling of the dependencies is embedded in the analysis of multiple binomial responses, which has the advantage of interpretability in terms of conditional odds ratios. Furthermore, the modelling is done in the framework of generalized linear models, thus parameter estimation and the assessment of goodness of fit can be obtained in the standard way by using e.g. GLIM or another standard software.

This paper applies the Posterior Bayes Factor (Aitkin [1]) to two problems of model choice in logistic modelling of contingency tables. The problems are (i) choosing between different link functions for the same regression model, and (ii) assessing the goodness-of-fit of a logistic model with a continuous covariate with very small sample sizes at each covariate value. New and very simple results are obtained; numerical examples are given.

This paper presents a strategy for checking the goodness-of-fit of a regression model for survival data, based on analysis of residuals. In the event that a lack of fit is found, a method of stratification of the data is proposed that permits separate regression models to be fitted to each stratum. Recursive stratification yields a tree structure that provides information about the predictive power of the covariates as well as any interactions among them.