A goodness‐of‐fit test for the proportional odds regression model

We examine goodness‐of‐fit tests for the proportional odds logistic regression model—the most commonly used regression model for an ordinal response variable. We derive a test statistic based on the Hosmer–Lemeshow test for binary logistic regression. Using a simulation study, we investigate the distribution and power properties of this test and compare these with those of three other goodness‐of‐fit tests. The new test has lower power than the existing tests; however, it was able to detect a greater number of the different types of lack of fit considered in this study. Moreover, the test allows for the results to be summarized in a contingency table of observed and estimated frequencies, which is a useful supplementary tool to assess model fit. We illustrate the ability of the tests to detect lack of fit using a study of aftercare decisions for psychiatrically hospitalized adolescents. The test proposed in this paper is similar to a recently developed goodness‐of‐fit test for multinomial logistic regression. A unified approach for testing goodness of fit is now available for binary, multinomial, and ordinal logistic regression models. Copyright © 2012 John Wiley & Sons, Ltd.     

Group testing regression models with dilution submodels

Group testing, where specimens are tested initially in pools, is widely used to screen individuals for sexually transmitted diseases. However, a common problem encountered in practice is that group testing can increase the number of false negative test results. This occurs primarily when positive individual specimens within a pool are diluted by negative ones, resulting in positive pools testing negatively. If the goal is to estimate a population‐level regression model relating individual disease status to observed covariates, severe bias can result if an adjustment for dilution is not made. Recognizing this as a critical issue, recent binary regression approaches in group testing have utilized continuous biomarker information to acknowledge the effect of dilution. In this paper, we have the same overall goal but take a different approach. We augment existing group testing regression models (that assume no dilution) with a parametric dilution submodel for pool‐level sensitivity and estimate all parameters using maximum likelihood. An advantage of our approach is that it does not rely on external biomarker test data, which may not be available in surveillance studies. Furthermore, unlike previous approaches, our framework allows one to formally test whether dilution is present based on the observed group testing data. We use simulation to illustrate the performance of our estimation and inference methods, and we apply these methods to 2 infectious disease data sets.     

The sum of standardized residuals: Goodness‐of‐fit test for binary response models

We propose a new goodness‐of‐fit statistic for evaluating generalized linear models with binary responses on the basis of the sum of standardized residuals. We derive the asymptotic distribution of the sum of standardized residuals statistic and argue that, despite its relative simplicity, it typically outperforms many of the more sophisticated currently used goodness‐of‐fit statistics.     

A goodness‐of‐fit test for the ordered stereotype model

This paper presents a new goodness‐of‐fit test for an ordered stereotype model used for an ordinal response variable. The proposed test is based on the well‐known Hosmer–Lemeshow test and its version for the proportional odds regression model. The latter test statistic is calculated from a grouping scheme assuming that the levels of the ordinal response are equally spaced which might be not true. One of the main advantages of the ordered stereotype model is that it allows us to determine a new uneven spacing of the ordinal response categories, dictated by the data. The proposed test takes the use of this new adjusted spacing to partition data. A simulation study shows good performance of the proposed test under a variety of scenarios. Finally, the results of the application in two examples are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Propensity score‐based diagnostics for categorical response regression models

For binary or categorical response models, most goodness‐of‐fit statistics are based on the notion of partitioning the subjects into groups or regions and comparing the observed and predicted responses in these regions by a suitable chi‐squared distribution. Existing strategies create this partition based on the predicted response probabilities, or propensity scores, from the fitted model. In this paper, we follow a retrospective approach, borrowing the notion of balancing scores used in causal inference to inspect the conditional distribution of the predictors, given the propensity scores, in each category of the response to assess model adequacy. We can use this diagnostic under both prospective and retrospective sampling designs, and it may ascertain general forms of misspecification. We first present simple graphical and numerical summaries that can be used in a binary logistic model. We then generalize the tools to propose model diagnostics for the proportional odds model. We illustrate the methods with simulation studies and two data examples: (i) a case‐control study of the association between cumulative lead exposure and Parkinson's disease in the Boston, Massachusetts, area and (ii) and a cohort study of biomarkers possibly associated with diabetes, from the VA Normative Aging Study. Copyright © 2013 John Wiley & Sons, Ltd.     

Multilocus association testing with penalized regression

In multilocus association analysis, since some markers may not be associated with a trait, it seems attractive to use penalized regression with the capability of automatic variable selection. On the other hand, in spite of a rapidly growing body of literature on penalized regression, most focus on variable selection and outcome prediction, for which penalized methods are generally more effective than their nonpenalized counterparts. However, for statistical inference, i.e. hypothesis testing and interval estimation, it is less clear how penalized methods would perform, or even how to best apply them, largely due to lack of studies on this topic. In our motivating data for a cohort of kidney transplant recipients, it is of primary interest to assess whether a group of genetic variants are associated with a binary clinical outcome, acute rejection at 6 months. In this article, we study some technical issues and alternative implementations of hypothesis testing in Lasso penalized logistic regression, and compare their performance with each other and with several existing global tests, some of which are specifically designed as variance component tests for high-dimensional data. The most interesting, and perhaps surprising, conclusion of this study is that, for low to moderately high-dimensional data, statistical tests based on Lasso penalized regression are not necessarily more powerful than some existing global tests. In addition, in penalized regression, rather than building a test based on a single selected "best" model, combining multiple tests, each of which is built on a candidate model, might be more promising.     

Goodness-of-fit tests for ordinal response regression models

It is well documented that the commonly used Pearson chi-square and deviance statistics are not adequate for assessing goodness-of-fit in logistic regression models when continuous covariates are modelled. In recent years, several methods have been proposed which address this shortcoming in the binary logistic regression setting or assess model fit differently. However, these techniques have typically not been extended to the ordinal response setting and few techniques exist to assess model fit in that case. We present the modified Pearson chi-square and deviance tests that are appropriate for assessing goodness-of-fit in ordinal response models when both categorical and continuous covariates are present. The methods have good power to detect omitted interaction terms and reasonable power to detect failure of the proportional odds assumption or modelling the wrong functional form of a continuous covariate. These tests also provide immediate information as to where a model may not fit well. In addition, the methods are simple to understand and implement, and are non-specific. That is, they do not require prespecification of a type of lack-of-fit to detect.     

Regression analysis for multiple‐disease group testing data

Group testing, where individual specimens are composited into groups to test for the presence of a disease (or other binary characteristic), is a procedure commonly used to reduce the costs of screening a large number of individuals. Group testing data are unique in that only group responses may be available, but inferences are needed at the individual level. A further methodological challenge arises when individuals are tested in groups for multiple diseases simultaneously, because unobserved individual disease statuses are likely correlated. In this paper, we propose new regression techniques for multiple‐disease group testing data. We develop an expectation–solution based algorithm that provides consistent parameter estimates and natural large‐sample inference procedures. We apply our proposed methodology to chlamydia and gonorrhea screening data collected in Nebraska as part of the Infertility Prevention Project and to prenatal infectious disease screening data from Kenya. Copyright © 2013 John Wiley & Sons, Ltd.     

Misspecification tests for binomial and beta-binomial models

The IOS test of Presnell and Boss (J. Am. Stat. Assoc. 2004; 99(465):216-227) is a general-purpose goodness-of-fit test based on the ratio of in-sample and out-of-sample likelihoods. For large samples, the IOS statistic can be approximated by a multiplicative contrast between two estimates of the information matrix, and in this way the IOS test is connected to White's (Econometrica 1982; 50:1-26) information matrix test, or IM test, which is based directly on the difference of two estimates of the information matrix. In this paper, we compare the performance of IOS to that of the IM test and of other goodness-of-fit tests for binomial and beta-binomial models, in both examples and simulations. Our findings suggest that IOS is strongly competitive, not only against the IM test but also against tests designed for specific binomial and beta-binomial models.     

Estimating the prevalence of multiple diseases from two‐stage hierarchical pooling

Testing protocols in large‐scale sexually transmitted disease screening applications often involve pooling biospecimens (e.g., blood, urine, and swabs) to lower costs and to increase the number of individuals who can be tested. With the recent development of assays that detect multiple diseases, it is now common to test biospecimen pools for multiple infections simultaneously. Recent work has developed an expectation–maximization algorithm to estimate the prevalence of two infections using a two‐stage, Dorfman‐type testing algorithm motivated by current screening practices for chlamydia and gonorrhea in the USA. In this article, we have the same goal but instead take a more flexible Bayesian approach. Doing so allows us to incorporate information about assay uncertainty during the testing process, which involves testing both pools and individuals, and also to update information as individuals are tested. Overall, our approach provides reliable inference for disease probabilities and accurately estimates assay sensitivity and specificity even when little or no information is provided in the prior distributions. We illustrate the performance of our estimation methods using simulation and by applying them to chlamydia and gonorrhea data collected in Nebraska. Copyright © 2016 John Wiley & Sons, Ltd.     

Multinomial goodness-of-fit tests for logistic regression models.

We examine the properties of several tests for goodness-of-fit for multinomial logistic regression. One test is based on a strategy of sorting the observations according to the complement of the estimated probability for the reference outcome category and then grouping the subjects into g equal-sized groups. A g x c contingency table, where c is the number of values of the outcome variable, is constructed. The test statistic, denoted as Cg, is obtained by calculating the Pearson chi2 statistic where the estimated expected frequencies are the sum of the model-based estimated logistic probabilities. Simulations compare the properties of Cg with those of the ungrouped Pearson chi2 test (X2) and its normalized test (z). The null distribution of Cg is well approximated by the chi2 distribution with (g-2) x (c-1) degrees of freedom. The sampling distribution of X2 is compared with a chi2 distribution with n x (c-1) degrees of freedom but shows erratic behavior. With a few exceptions, the sampling distribution of z adheres reasonably well to the standard normal distribution. Power simulations show that Cg has low power for a sample of 100 observations, but satisfactory power for a sample of 400. The tests are illustrated using data from a study of cytological criteria for the diagnosis of breast tumors.     

Global goodness-of-fit tests in logistic regression with sparse data

The logistic regression model has become the standard analysing tool for binary responses in medical statistics. Methods for assessing goodness-of-fit, however, are less developed where this problem is especially pronounced in performing global goodness-of-fit tests with sparse data, that is, if the data contain only a small numbers of observations for each pattern of covariate values. In this situation it has been known for a long time that the standard goodness-of-fit tests (residual deviance and Pearson chi-square) behave unsatisfactorily if p-values are calculated from the chi(2)-distribution. As a remedy in this situation the Hosmer-Lemeshow test is frequently recommended; it relies on a new grouping of the observations to avoid sparseness, where this grouping depends on the estimated probabilities from the model. It has been shown, however, that the Hosmer-Lemeshow test also has some deficiencies, for example, it depends heavily on the calculating algorithm and thus different implementations might lead to different conclusions regarding the fit of the model. We present some alternative tests from the statistical literature which should also perform well with sparse data. Results from a simulation study are given which show that there exist some goodness-of-fit tests (for example, the Farrington test) that have good properties regarding size and power and that even outperform the Hosmer-Lemeshow test. We illustrate the various tests with an example from dermatology on occupational hand eczema in hairdressers.     

Goodness‐of‐fit test for proportional subdistribution hazards model

This paper concerns using modified weighted Schoenfeld residuals to test the proportionality of subdistribution hazards for the Fine–Gray model, similar to the tests proposed by Grambsch and Therneau for independently censored data. We develop a score test for the time‐varying coefficients based on the modified Schoenfeld residuals derived assuming a certain form of non‐proportionality. The methods perform well in simulations and a real data analysis of breast cancer data, where the treatment effect exhibits non‐proportional hazards. Copyright © 2013 John Wiley & Sons, Ltd.     

Two goodness-of-fit tests for logistic regression models with continuous covariates.

Assessing goodness-of-fit in logistic regression models can be problematic, in that commonly used deviance or Pearson chi-square statistics do not have approximate chi-square distributions, under the null hypothesis of no lack of fit, when continuous covariates are modelled. We present two easy to implement test statistics similar to the deviance and Pearson chi-square tests that are appropriate when continuous covariates are present. The methodology uses an approach similar to that incorporated by the Hosmer and Lemeshow goodness-of-fit test in that observations are classified into distinct groups according to fitted probabilities, allowing sufficient cell sizes for chi-square testing. The major difference is that the proposed tests perform this grouping within the cross-classification of all categorical covariates in the model and, in some situations, allow for a more powerful assessment of where model predicted and observed counts may differ. A variety of simulations are performed comparing the proposed tests to the Hosmer-Lemeshow test.     

Comparison of hypertabastic survival model with other unimodal hazard rate functions using a goodness‐of‐fit test

We studied the problem of testing a hypothesized distribution in survival regression models when the data is right censored and survival times are influenced by covariates. A modified chi‐squared type test, known as Nikulin‐Rao‐Robson statistic, is applied for the comparison of accelerated failure time models. This statistic is used to test the goodness‐of‐fit for hypertabastic survival model and four other unimodal hazard rate functions. The results of simulation study showed that the hypertabastic distribution can be used as an alternative to log‐logistic and log‐normal distribution. In statistical modeling, because of its flexible shape of hazard functions, this distribution can also be used as a competitor of Birnbaum‐Saunders and inverse Gaussian distributions. The results for the real data application are shown. Copyright © 2017 John Wiley & Sons, Ltd.     

An improved test of latent‐variable model misspecification in structural measurement error models for group testing data

We consider structural measurement error models for group testing data. Likelihood inference based on structural measurement error models requires one to specify a model for the latent true predictors. Inappropriate specification of this model can lead to erroneous inference. We propose a new method tailored to detect latent‐variable model misspecification in structural measurement error models for group testing data. Compared with the existing diagnostic methods developed for the same purpose, our method shows vast improvement in the power to detect latent‐variable model misspecification in group testing design. We illustrate the implementation and performance of the proposed method via simulation and application to a real data example. Copyright © 2009 John Wiley & Sons, Ltd.     

On assessing model fit for distribution‐free longitudinal models under missing data

The generalized estimating equation (GEE), a distribution‐free, or semi‐parametric, approach for modeling longitudinal data, is used in a wide range of behavioral, psychotherapy, pharmaceutical drug safety, and healthcare‐related research studies. Most popular methods for assessing model fit are based on the likelihood function for parametric models, rendering them inappropriate for distribution‐free GEE. One rare exception is a score statistic initially proposed by Tsiatis for logistic regression (1980) and later extended by Barnhart and Willamson to GEE (1998). Because GEE only provides valid inference under the missing completely at random assumption and missing values arising in most longitudinal studies do not follow such a restricted mechanism, this GEE‐based score test has very limited applications in practice. We propose extensions of this goodness‐of‐fit test to address missing data under the missing at random assumption, a more realistic model that applies to most studies in practice. We examine the performance of the proposed tests using simulated data and demonstrate the utilities of such tests with data from a real study on geriatric depression and associated medical comorbidities. Copyright © 2013 John Wiley & Sons, Ltd.     

A comparison of goodness of fit tests for the logistic GEE model

Generalized estimating equations have become a popular regression method for analysing clustered binary data. Methods to assess the goodness of fit of the fitted models have recently been developed. However, evaluations and comparisons of these methods are limited. We discuss these methods and develop two additional statistics to evaluate goodness of fit. We evaluate the performance of each of the statistics with respect to type I error rates and power in a simulation study. Guidance is provided regarding appropriate use of the statistics under various scenarios.