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.     

Testing goodness of fit of parametric models for censored data

We propose and study a goodness‐of‐fit test for left‐censored, right‐censored, and interval‐censored data assuming random censorship. Main motivation comes from dietary exposure assessment in chemical risk assessment, where the determination of an appropriate distribution for concentration data is of major importance. We base the new goodness‐of‐fit test procedure proposed in this paper on the order selection test. As part of the testing procedure, we extend the null model to a series of nested alternative models for censored data. Then, we use a modified AIC model selection to select the best model to describe the data. If a model with one or more extra parameters is selected, then we reject the null hypothesis. As an alternative to the use of the asymptotic null distribution of the test statistic, we define a bootstrap‐based procedure. We illustrate the applicability of the test procedure on data of cadmium concentrations and on data from the Signal Tandmobiel study and demonstrate its performance characteristics through simulation studies. Copyright © 2012 John Wiley & Sons, Ltd.     

A new test and graphical tool to assess the goodness of fit of logistic regression models

A prognostic model is well calibrated when it accurately predicts event rates. This is first determined by testing for goodness of fit with the development dataset. All existing tests and graphic tools designed for the purpose suffer several drawbacks, related mainly to the subgrouping of observations or to heavy dependence on arbitrary parameters. We propose a statistical test and a graphical method to assess the goodness of fit of logistic regression models, obtained through an extension of similar techniques developed for external validation. We analytically computed and numerically verified the distribution of the underlying statistic. Simulations on a set of realistic scenarios show that this test and the well‐known Hosmer–Lemeshow approach have similar type I error rates. The main advantage of this new approach is that the relationship between model predictions and outcome rates across the range of probabilities can be represented in the calibration belt plot, together with its statistical confidence. By readily spotting any deviations from the perfect fit, this new graphical tool is designed to identify, during the process of model development, poorly modeled variables that call for further investigation. This is illustrated through an example based on real data. Copyright © 2015 John Wiley & Sons, Ltd.     

A comparison of goodness of fit tests for age-related reference ranges

Age-related reference ranges based on cubic spline or kernel-based smoothing methods lack a parametric framework with which to assess goodness of fit, as the distribution of the log-likelihood ratio comparing models with different degrees of smoothing is not known in general. Several methods have been proposed to assess goodness of fit in this case: Healy's grid test, Royston's Q tests and van Buuren's worm plot. Here we compare the performance of the various methods, including an extension of the Q tests, and show how they can be used, with other information including the appearance of the fitted curves, to inform the choice of model for age-related reference ranges of height and weight in a large Dutch data set.     

Adapting the logical basis of tests for Hardy‐Weinberg Equilibrium to the real needs of association studies in human and medical genetics

The standard procedure to assess genetic equilibrium is a χ² test of goodness‐of‐fit. As is the case with any statistical procedure of that type, the null hypothesis is that the distribution underlying the data is in agreement with the model. Thus, a significant result indicates incompatibility of the observed data with the model, which is clearly at variance with the aim in the majority of applications: to exclude the existence of gross violations of the equilibrium condition. In current practice, we try to avoid this basic logical difficulty by increasing the significance bound to the P‐value (e.g. from 5 to 10%) and inferring compatibility of the data with Hardy Weinberg Equilibrium (HWE) from an insignificant result. Unfortunately, such direct inversion of a statistical testing procedure fails to produce a valid test of the hypothesis of interest, namely, that the data are in sufficiently good agreement with the model under which the P‐value is calculated. We present a logically unflawed solution to the problem of establishing (approximate) compatibility of an observed genotype distribution with HWE. The test is available in one‐ and two‐sided versions. For both versions, we provide tools for exact power calculation. We demonstrate the merits of the new approach through comparison with the traditional χ² goodness‐of‐fit test in 2×60 genotype distributions from 43 published genetic studies of complex diseases where departure from HWE was noted in either the case or control sample. In addition, we show that the new test is useful for the analysis of genome‐wide association studies. Genet. Epidemiol. 33:569–580, 2009. © 2009 Wiley‐Liss, Inc.     

Politics and health outcomes: A path analytic approach

Social and health policies and political participation are associated with each political tradition related to public health outcomes. However, there is a lack of evidence for the relationship between policy and outcomes. This study seeks to determine the relationship between politics, labour and welfare state indicators, economic inequality, and health outcome indicators. Data to test the model was obtained from the Turkish Statistical Institute (TurkStat) that belongs to the 81 provinces of Turkey. Path analysis was used to model the associations between policy, labour and welfare states, economic inequality, and health outcomes. To test the goodness of fit of the model, multiple criteria of model fit indices were utilised. The fit of the respecified path analytic model data is good (normed fit index [NFI] is 0.91, comparative fit index [CFI] is 0.92, goodness of fit index [GFI] is 0.91, and adjusted goodness of fit index [AGFI] is 0.93). Study results illustrate a strong relationship between voter partisanship, employment rate, satisfaction from both social security and health services, and life expectancy at birth and mortality. These results represent an important step towards understanding the elusive relationship between policy and health outcomes. Designing socially inclusive policies, considering labour market opportunities, and enhancing the population's well‐being are advisable strategies for policymakers who wish to optimise public health outcomes.     

Standardizing the power of the Hosmer–Lemeshow goodness of fit test in large data sets

The Hosmer–Lemeshow test is a commonly used procedure for assessing goodness of fit in logistic regression. It has, for example, been widely used for evaluation of risk‐scoring models. As with any statistical test, the power increases with sample size; this can be undesirable for goodness of fit tests because in very large data sets, small departures from the proposed model will be considered significant. By considering the dependence of power on the number of groups used in the Hosmer–Lemeshow test, we show how the power may be standardized across different sample sizes in a wide range of models. We provide and confirm mathematical derivations through simulation and analysis of data on 31,713 children from the Collaborative Perinatal Project. We make recommendations on how to choose the number of groups in the Hosmer–Lemeshow test based on sample size and provide example applications of the recommendations. Copyright © 2012 John Wiley & Sons, Ltd.     

Quadratic inference functions in marginal models for longitudinal data

The quadratic inference function (QIF) is a new statistical methodology developed for the estimation and inference in longitudinal data analysis using marginal models. This method is an alternative to the popular generalized estimating equations approach, and it has several useful properties such as robustness, a goodness‐of‐fit test and model selection. This paper presents an introductory review of the QIF, with a strong emphasis on its applications. In particular, a recently developed SAS MACRO QIF is illustrated in this paper to obtain numerical results. Copyright © 2009 John Wiley & Sons, Ltd.     

Understanding MCP‐MOD dose finding as a method based on linear regression

MCP‐MOD is a testing and model selection approach for clinical dose finding studies. During testing, contrasts of dose group means are derived from candidate dose response models. A multiple‐comparison procedure is applied that controls the alpha level for the family of null hypotheses associated with the contrasts. Provided at least one contrast is significant, a corresponding set of “good” candidate models is identified. The model generating the most significant contrast is typically selected. There have been numerous publications on the method. It was endorsed by the European Medicines Agency. The MCP‐MOD procedure can be alternatively represented as a method based on simple linear regression, where “simple” refers to the inclusion of an intercept and a single predictor variable, which is a transformation of dose. It is shown that the contrasts are equal to least squares linear regression slope estimates after a rescaling of the predictor variables. The test for each contrast is the usual t statistic for a null slope parameter, except that a variance estimate with fewer degrees of freedom is used in the standard error. Selecting the model corresponding to the most significant contrast P value is equivalent to selecting the predictor variable yielding the smallest residual sum of squares. This criteria orders the models like a common goodness‐of‐fit test, but it does not assure a good fit. Common inferential methods applied to the selected model are subject to distortions that are often present following data‐based model selection.     

Penalized estimation for proportional hazards models with current status data

We provide a simple and practical, yet flexible, penalized estimation method for a Cox proportional hazards model with current status data. We approximate the baseline cumulative hazard function by monotone B‐splines and use a hybrid approach based on the Fisher‐scoring algorithm and the isotonic regression to compute the penalized estimates. We show that the penalized estimator of the nonparametric component achieves the optimal rate of convergence under some smooth conditions and that the estimators of the regression parameters are asymptotically normal and efficient. Moreover, a simple variance estimation method is considered for inference on the regression parameters. We perform 2 extensive Monte Carlo studies to evaluate the finite‐sample performance of the penalized approach and compare it with the 3 competing R packages: C1.coxph, intcox, and ICsurv. A goodness‐of‐fit test and model diagnostics are also discussed. The methodology is illustrated with 2 real applications.     

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.     

零频数过多模型在亚健康状态研究中的应用

探讨零频数过多(ZI)模型在亚健康症状数研究中的应用.应用Stata 11.0软件拟合ZI模型分析亚健康症状数的危险因素,并用d系数、Vuong检验、O检验、似然比拟合优度检验比较ZI模型与传统负二项回归模型、Poisson回归模型的拟合效果.α=0.939,Vuong检验Z=32.08,P＜0.0001,表明此数据的零频数过多.亚健康症状数的(-x)=2.90,s=3.85,过度离散统计量0=308.011,P＜0.001,s2＞(-x),表明存在过度离散.从4个模型中的拟合优度看,零频数过多的负二项回归(ZINB)模型log likelihood最大,AIC最小,说明ZINB模型的拟合效果最佳.当计数资料中出现过多的零频数时(如亚健康症状数资料),应用ZINB模型能够获得最佳的拟合效果.

Abstract：

To explore the goodness of fit about the zero-inflated (ZI) models in analyzing data related to sub-health symptoms in which the counts are non-negative integers. ZI models are conducted with Stata 11.0. The coefficient of a, Vuong test, O test and likelihood test are used to compare the goodness of fit for ZI models with the common used models such as passion model,negative binomial model. When a is 0.939, and the Z statistic of Vuong test is 32.08, P＜0.0001,which shows that there are too many zeros. The mean number of sub-health symptoms is 2.90, s=3.85, 0=308.011, P＜0.001, s2＞(-x), indicating that the data are over-dispersed. In addition, the optimum goodness of fit is found in zero-inflated negative binomial (ZINB) model with the largest log likelihood and the smallest AIC. ZINB seems the optimal model to study those over-dispersed count data with too many zeros.     

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.     

Heteroscedastic transformation cure regression models

Cure models have been applied to analyze clinical trials with cures and age‐at‐onset studies with nonsusceptibility. Lu and Ying (On semiparametric transformation cure model. Biometrika 2004; 91:331?‐343. DOI: 10.1093/biomet/91.2.331) developed a general class of semiparametric transformation cure models, which assumes that the failure times of uncured subjects, after an unknown monotone transformation, follow a regression model with homoscedastic residuals. However, it cannot deal with frequently encountered heteroscedasticity, which may result from dispersed ranges of failure time span among uncured subjects' strata. To tackle the phenomenon, this article presents semiparametric heteroscedastic transformation cure models. The cure status and the failure time of an uncured subject are fitted by a logistic regression model and a heteroscedastic transformation model, respectively. Unlike the approach of Lu and Ying, we derive score equations from the full likelihood for estimating the regression parameters in the proposed model. The similar martingale difference function to their proposal is used to estimate the infinite‐dimensional transformation function. Our proposed estimating approach is intuitively applicable and can be conveniently extended to other complicated models when the maximization of the likelihood may be too tedious to be implemented. We conduct simulation studies to validate large‐sample properties of the proposed estimators and to compare with the approach of Lu and Ying via the relative efficiency. The estimating method and the two relevant goodness‐of‐fit graphical procedures are illustrated by using breast cancer data and melanoma data. Copyright © 2016 John Wiley & Sons, Ltd.     

Tests of calibration and goodness‐of‐fit in the survival setting

To access the calibration of a predictive model in a survival analysis setting, several authors have extended the Hosmer–Lemeshow goodness‐of‐fit test to survival data. Grønnesby and Borgan developed a test under the proportional hazards assumption, and Nam and D'Agostino developed a nonparametric test that is applicable in a more general survival setting for data with limited censoring. We analyze the performance of the two tests and show that the Grønnesby–Borgan test attains appropriate size in a variety of settings, whereas the Nam‐D'Agostino method has a higher than nominal Type 1 error when there is more than trivial censoring. Both tests are sensitive to small cell sizes. We develop a modification of the Nam‐D'Agostino test to allow for higher censoring rates. We show that this modified Nam‐D'Agostino test has appropriate control of Type 1 error and comparable power to the Grønnesby–Borgan test and is applicable to settings other than proportional hazards. We also discuss the application to small cell sizes. Copyright © 2015 John Wiley & Sons, Ltd.     

A joint frailty model to estimate the recurrence process and the disease‐specific mortality process without needing the cause of death

In chronic diseases, such as cancer, recurrent events (such as relapses) are commonly observed; these could be interrupted by death. With such data, a joint analysis of recurrence and mortality processes is usually conducted with a frailty parameter shared by both processes. We examined a joint modeling of these processes considering death under two aspects: ‘death due to the disease under study' and ‘death due to other causes', which enables estimating the disease‐specific mortality hazard. The excess hazard model was used to overcome the difficulties in determining the causes of deaths (unavailability or unreliability); this model allows estimating the disease‐specific mortality hazard without needing the cause of death but using the mortality hazards observed in the general population. We propose an approach to model jointly recurrence and disease‐specific mortality processes within a parametric framework. A correlation between the two processes is taken into account through a shared frailty parameter. This approach allows estimating unbiased covariate effects on the hazards of recurrence and disease‐specific mortality. The performance of the approach was evaluated by simulations with different scenarios. The method is illustrated by an analysis of a population‐based dataset on colon cancer with observations of colon cancer recurrences and deaths. The benefits of the new approach are highlighted by comparison with the ‘classical' joint model of recurrence and overall mortality. Moreover, we assessed the goodness of fit of the proposed model. Comparisons between the conditional hazard and the marginal hazard of the disease‐specific mortality are shown, and differences in interpretation are discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

An application of a Hill‐based response surface model for a drug combination experiment on lung cancer

Combination chemotherapy with multiple drugs has been widely applied to cancer treatment owing to enhanced efficacy and reduced drug resistance. For drug combination experiment analysis, response surface modeling has been commonly adopted. In this paper, we introduce a Hill‐based global response surface model and provide an application of the model to a 512‐run drug combination experiment with three chemicals, namely AG490, U0126, and indirubin‐3 ′ ‐monoxime (I‐3‐M), on lung cancer cells. The results demonstrate generally improved goodness of fit of our model from the traditional polynomial model, as well as the original Hill model on the basis of fixed‐ratio drug combinations. We identify different dose–effect patterns between normal and cancer cells on the basis of our model, which indicates the potential effectiveness of the drug combination in cancer treatment. Meanwhile, drug interactions are analyzed both qualitatively and quantitatively. The distinct interaction patterns between U0126 and I‐3‐M on two types of cells uncovered by the model could be a further indicator of the efficacy of the drug combination. Copyright © 2014 John Wiley & Sons, Ltd.     

Assessing the goodness of fit of personal risk models

We describe a flexible family of tests for evaluating the goodness of fit (calibration) of a pre‐specified personal risk model to the outcomes observed in a longitudinal cohort. Such evaluation involves using the risk model to assign each subject an absolute risk of developing the outcome within a given time from cohort entry and comparing subjects’ assigned risks with their observed outcomes. This comparison involves several issues. For example, subjects followed only for part of the risk period have unknown outcomes. Moreover, existing tests do not reveal the reasons for poor model fit when it occurs, which can reflect misspecification of the model's hazards for the competing risks of outcome development and death. To address these issues, we extend the model‐specified hazards for outcome and death, and use score statistics to test the null hypothesis that the extensions are unnecessary. Simulated cohort data applied to risk models whose outcome and mortality hazards agreed and disagreed with those generating the data show that the tests are sensitive to poor model fit, provide insight into the reasons for poor fit, and accommodate a wide range of model misspecification. We illustrate the methods by examining the calibration of two breast cancer risk models as applied to a cohort of participants in the Breast Cancer Family Registry. The methods can be implemented using the Risk Model Assessment Program, an R package freely available at http://stanford.edu/~ggong/rmap/ . Copyright © 2014 John Wiley & Sons, Ltd.     

The Vicarious Trauma Scale: Confirmatory factor analysis and psychometric properties with a sample of victim advocates

Vicarious trauma is referred to as the detrimental change in the manner that professionals understand and interpret material, as a result of exposure to second‐hand traumatic material (McCann & Pearlman [1990] Journal of Traumatic Stress, 3:131). According to Aparicio et al. (Health & Social Work, 2013, 38:199), vicarious trauma comprises both affective and cognitive components and, while it is distinct from posttraumatic stress disorder (PTSD), it is associated with similar symptoms, including re‐experiencing and avoiding traumatic material and experiencing depressed mood. The purpose of this study was to analyse the psychometric properties of the Victim Trauma Scale (VTS) and provide additional support, supplementing the findings of Aparicio et al. (2013), but instead using victim advocates as participants (n = 142). The survey was open between February 2016 and February 2017. More than 96% of participants were in paid employment positions, as more than 80% reporting working 40 or more hours a week. Aparicio et al. (2013) found that the VTS was two‐dimensional (affective and cognitive); however, after examining the goodness of fit of the two‐factor model using a confirmatory factor analysis (CFA) approach, this study concluded that the two‐dimensional model was not a good fit. Due to the poor goodness of fit of the two‐factor model and the post hoc EFA resulting in a one‐factor model, our data do not support the findings of Aparicio et al. (2013). Further, the findings suggest the VTS is an acceptable measure of vicarious trauma, as demonstrated by the high internal consistency and the single‐factor loading.