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Statistical methods for analyzing recurrent events have attracted significant attention. The majority of existing works consider situations in which subjects are observed over time periods and events of interest that occurred during the course of follow-up are recorded. In some applications, a subject may leave the study for a period of time and then resume due to various reasons. During the absence, which is referred to as an intermittent gap in this study, it may be impossible to observe a recording of the event. A naive analysis disregards gaps and considers events to be a typical recurrent event dataset. However, this may result in biased estimations and misleading results. In this study, we build an additive rates model for recurrent event data considering intermittent gaps. We provide the asymptotic theories behind the proposed model, as well as the goodness of fit between observed and modeled values. Simulation studies reveal that the estimations perform well if intermittent gaps are taken into account. In addition, we utilized the longitudinal cohort of elderly patients who have type 2 diabetes and at least one record of a severe recurrent complication, hypoglycemia, from the National Health Insurance Research Database in Taiwan to demonstrate the proposed method.

The integrated discrimination improvement (IDI) is commonly used to compare two risk prediction models; it summarizes the extent a new model increases risk in events and decreases risk in non-events. The IDI averages risks across events and non-events and is therefore susceptible to Simpson's paradox. In some settings, adding a predictive covariate to a well calibrated model results in an overall negative (positive) IDI. However, if stratified by that same covariate, the strata-specific IDIs are positive (negative). Meanwhile, the calibration (observed to expected ratio and Hosmer-Lemeshow Goodness of Fit Test), area under the receiver operating characteristic curve, and Brier score improve overall and by stratum. We ran extensive simulations to investigate the impact of an imbalanced covariate upon metrics (IDI, area under the receiver operating characteristic curve, Brier score, and R2), provide an analytic explanation for the paradox in the IDI, and use an investigative metric, a Weighted IDI, to better understand the paradox. In simulations, all instances of the paradox occurred under stratum-specific mis-calibration, yet there were mis-calibrated settings in which the paradox did not occur. The paradox is illustrated on Cancer Genomics Network data by calculating predictions based on two versions of BRCAPRO, a Mendelian risk prediction model for breast and ovarian cancer. In both simulations and the Cancer Genomics Network data, overall model calibration did not guarantee stratum-level calibration. We conclude that the IDI should only assess model performance among a clinically relevant subset when stratum-level calibration is strictly met and recommend calculating additional metrics to confirm the direction and conclusions of the IDI. Copyright © 2016 John Wiley & Sons, Ltd.

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. Gronnesby 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 Gronnesby–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 Gronnesby–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.

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.

Meta-analysis for adverse events resulting from medical interventions has many challenges, in part due to small numbers of such events within primary studies. Furthermore, variability in drug dose, potential differences between drugs within the same pharmaceutical class and multiple indications for a specific treatment can all add to the complexity of the evidence base. This paper explores the use of synthesis methods, incorporating mixed treatment comparisons, to estimate the risk of adverse events for a medical intervention, while acknowledging and modelling the complexity of the structure of the evidence base. The motivating example was the effect on malignancy of three anti-tumour necrosis factor (anti-TNF) drugs (etanercept, adalimumab and infliximab) indicated to treat rheumatoid arthritis. Using data derived from 13 primary studies, a series of meta-analysis models of increasing complexity were applied. Models ranged from a straightforward comparison of anti-TNF against non-anti-TNF controls, to more complex models in which a treatment was defined by individual drug and its dose. Hierarchical models to allow ‘borrowing strength’ across treatment classes and dose levels, and models involving constraints on the impact of dose level, are described. These models provide a flexible approach to estimating sparse, often adverse, outcomes associated with interventions. Each model makes its own set of assumptions, and approaches to assessing goodness of fit of the various models will usually be extremely limited in their effectiveness, due to the sparse nature of the data. Both methodological and clinical considerations are required to fit realistically complex models in this area and to evaluate their appropriateness. Copyright © 2014 John Wiley & Sons, Ltd.

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 healthcarerelated 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 GEEbased 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.

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.

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.

A natural way of modelling relative survival through regression analysis is to assume an additive form between the expected population hazard and the excess hazard due to the presence of an additional cause of mortality. Within this context, the existing approaches in the parametric, semiparametric and non-parametric setting are compared and discussed. We study the additive excess hazards models, where the excess hazard is on additive form. This makes it possible to assess the importance of time-varying effects for regression models in the relative survival framework. We show how recent developments can be used to make inferential statements about the non-parametric version of the model. This makes it possible to test the key hypothesis that an excess risk effect is time varying in contrast to being constant over time. In case some covariate effects are constant, we show how the semiparametric additive risk model can be considered in the excess risk setting, providing a better and more useful summary of the data. Estimators have explicit form and inference based on a resampling scheme is presented for both the non-parametric and semiparametric models. We also describe a new suggestion for goodness of fit of relative survival models, which consists on statistical and graphical tests based on cumulative martingale residuals. This is illustrated on the semiparametric model with proportional excess hazards. We analyze data from the TRACE study using different approaches and show the need for more flexible models in relative survival.

In many epidemiological studies it is common to resort to regression models relating incidence of a disease and its risk factors. The main goal of this paper is to consider inference on such models with error-prone observations and variances of the measurement errors changing across observations. We suppose that the observations follow a bivariate normal distribution and the measurement errors are normally distributed. Aggregate data allow the estimation of the error variances. Maximum likelihood estimates are computed numerically via the EM algorithm. Consistent estimation of the asymptotic variance of the maximum likelihood estimators is also discussed. Test statistics are proposed for testing hypotheses of interest. Further, we implement a simple graphical device that enables an assessment of the model's goodness of fit. Results of simulations concerning the properties of the test statistics are reported. The approach is illustrated with data from the WHO MONICA Project on cardiovascular disease.

The attributable fraction (AF) is often used to explore the policy implications of an association between a disease and an exposure. To date, there have been no proposed estimators of AF in the context of partial questionnaire designs (PQD). The PQD, first proposed in a public health context by Wacholder is often used to enhance response rates in questionnaires. It involves eliciting responses from each subject on preassigned subsets of questions, thereby reducing the burden of response. We propose a computationally efficient method of estimating logistic (or more generally, binary) regression parameters from a PQD model where there is non-response to the questionnaire and the rates of non-response differ between sub-populations. Assuming a log-linear model for the distribution of missing covariates, we employ the methods of Wacholder to motivate consistent estimating equations, and weight each subject's contribution to the estimating function by the inverse probability of responding to the questionnaire. We also propose techniques for goodness-of-fit to assist in model selection. We then use the PQD regression estimates to derive an estimate of AF similar to that proposed by Bruzzi. Finally, we demonstrate our methods using data obtained from a study on adult occupational asthma, conducted within a Massachusetts HMO. Although we concentrate on a particular type of missing data mechanism, other missing data techniques can be incorporated into AF estimation in a similar manner.

The World Health Organization (WHO), in collaboration with a number of research institutions worldwide, is developing new child growth standards. As part of a broad consultative process for selecting the best statistical methods, WHO convened a group of statisticians and child growth experts to review available methods, develop a strategy for assessing their strengths and weaknesses, and discuss methodological issues likely to be faced in the process of constructing the new growth curves. To select the method(s) to be used, the group proposed a two-stage decision-making process. First, to select a few relevant methods based on a list of set criteria and, second, to compare the methods using available tests or other established procedures. The group reviewed 30 methods for attained growth curves. Using the pre-defined criteria, a few were selected combining five distributions and two smoothing techniques. Because the number of selected methods was considered too large to be fully tested, a preliminary study was recommended to evaluate goodness of fit of the five distributions. Methods based on distributions with poor performance will be eliminated and the remaining methods fully tested and compared. Copyright © 2005 John Wiley & Sons, Ltd.

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.

A new goodness-of-fit test for the logistic regression model is proposed. It exploits the property of this model that when it is correct, i.e. not misspecified, the parameter estimates are (asymptotically) invariant under reweighting the observations by weights wi that are a function of the binary (0/1) outcomes yi. Misspecification of the model can thus be concluded when parameter estimates change under reweighting. A local test, considering weights of the form wi=(1 + eyi) is explored. The test is especially suitable for case–control studies but may be used in other contexts as well. Copyright © 2005 John Wiley & Sons, Ltd.

Markov regression models describe the way in which a categorical response variable changes over time for subjects with different explanatory variables. Frequently it is difficult to measure the response variable on equally spaced discrete time intervals. Here we propose a Pearson-type goodness-of-fit test for stationary Markov regression models fitted to panel data. A parametric bootstrap algorithm is used to study the distribution of the test statistic. The proposed technique is applied to examine the fit of a Markov regression model used to identify markers for disease progression in psoriatic arthritis.

In this paper we use simulations to compare the performance of new goodness-of-fit tests based on weighted statistical processes to three currently available tests: the Hosmer-Lemeshow decile-of-risk test; the Pearson chi-square, and the unweighted sum-of-squares tests. The simulations demonstrate that all tests have the correct size. The power for all tests to detect lack-of-fit due to an omitted quadratic term with a sample of size 100 is close to or exceeds 50 per cent to detect moderate departures from linearity and is over 90 per cent for these same alternatives for sample size 500. All tests have low power with sample size 100 to detect lack-of-fit due to an omitted interaction between a dichotomous and continuous covariate, while the power exceeds 80 per cent to detect extreme interaction with a sample size of 500. The power is low to detect any alternative link function with sample size 100 and for most alternative links for sample size 500. Only in the case of sample size 500 and an extremely asymmetric link function is the power over 80 per cent. The results from these simulations show that no single test, new or current, performs best in detecting lack-of-fit due to an omitted covariate or incorrect link function. However, one of the new weighted tests has power comparable to other tests in all settings simulated and had the highest power in the difficult case of an omitted interaction term. We illustrate the tests within the context of a model for factors associated with abstinence from drug use in a randomized trial of residential treatment programmes. We conclude the paper with a summary and specific recommendations for practice.

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 χ2-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. Copyright © 2002 John Wiley & Sons, Ltd.

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 age-specific reference interval is a commonly used screening tool in medicine. It involves estimation of extreme quantile curves (such as the 5th and 95th centiles) of a reference distribution of clinically normal individuals. It is crucial that models used to estimate such intervals fit the data extremely well. However, few procedures to assess goodness-of-fit have been proposed in the literature, and even fewer have been evaluated systematically. Here we consider procedures based on the distribution of the Z-scores (standardized residuals) from a model and on Pearson chi(2) statistics for observed and expected counts in groups defined by age and the estimated reference centile curves. Two of the procedures (Q and grid tests) are mainly inferential, whereas the third (permutation bands and B-tests) is essentially graphical. We obtain approximations to the null distributions of several relevant test statistics and examine their size and power for a range of models based on real data sets. We recommend Q-tests in all situations where Z-scores are available since they are general, simple to calculate and usually have the highest power among the three classes of test considered. For the cases considered the grid tests are always inferior to the Q- and B- tests.

Suppose we use generalized estimating equations to estimate a marginal regression model for repeated binary observations. There are no established summary statistics available for assessing the adequacy of the fitted model. In this paper we propose a goodness-of-fit test statistic which has an approximate chi-squared distribution when we have specified the model correctly. The proposed statistic can be viewed as an extension of the Hosmer and Lemeshow goodness-of-fit statistic for ordinary logistic regression to marginal regression models for repeated binary responses. We illustrate the methods using data from a study of mental health service utilization by children. The repeated responses are a set of binary measures of service use. We fit a marginal logistic regression model to the data using generalized estimating equations, and we apply the proposed goodness-of-fit statistic to assess the adequacy of the fitted model.

Recent work has shown that there may be disadvantages in the use of the chi-square-like goodness-of-fit tests for the logistic regression model proposed by Hosmer and Lemeshow that use fixed groups of the estimated probabilities. A particular concern with these grouping strategies based on estimated probabilities, fitted values, is that groups may contain subjects with widely different values of the covariates. It is possible to demonstrate situations where one set of fixed groups shows the model fits while the test rejects fit using a different set of fixed groups. We compare the performance by simulation of these tests to tests based on smoothed residuals proposed by le Cessie and Van Houwelingen and Royston, a score test for an extended logistic regression model proposed by Stukel, the Pearson chi-square and the unweighted residual sum-of-squares. These simulations demonstrate that all but one of Royston's tests have the correct size. An examination of the performance of the tests when the correct model has a quadratic term but a model containing only the linear term has been fit shows that the Pearson chi-square, the unweighted sum-of-squares, the Hosmer-Lemeshow decile of risk, the smoothed residual sum-of-squares and Stukel's score test, have power exceeding 50 per cent to detect moderate departures from linearity when the sample size is 100 and have power over 90 per cent for these same alternatives for samples of size 500. All tests had no power when the correct model had an interaction between a dichotomous and continuous covariate but only the continuous covariate model was fit. Power to detect an incorrectly specified link was poor for samples of size 100. For samples of size 500 Stukel's score test had the best power but it only exceeded 50 per cent to detect an asymmetric link function. The power of the unweighted sum-of-squares test to detect an incorrectly specified link function was slightly less than Stukel's score test. We illustrate the tests within the context of a model for factors associated with low birth weight.

We present a general approach based on chi-square goodness-of-fit tests for performing tests of significance about the kappa statistic. We make recommendations for the particular goodness-of-fit test to use under different situations. We also compare the test statistics in terms of the generated confidence intervals about the kappa statistic.

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.

Backcalculation has been used to estimate the rate of past HIV infection and to predict future AIDS incidence. In this study we examine another use: estimating the probability of progression from HIV infection to AIDS as a function of time from infection. Given observed AIDS incidence data, the technique of backcalculation estimates the most likely number of persons infected with HIV in the past. Assumptions about probability of progression from HIV infection to AIDS are necessary. By varying these assumptions and examining the resulting goodness of fit to the AIDS incidence data, we can theoretically estimate parameters of progression. We report on implementation of this method and examine its practical utility in deciding among four competing progression models specified on a priori grounds. The four specific models comprise three Weibull distributions with medians of 8, 10, and 12 years, respectively, and one model that begins as a Weibull with 8 year median but where the hazard is level after 3.5 years. To employ asymptotic maximum likelihood methods, we define a two parameter family of progression models that includes all four a priori models. One parameter sets the scale for an initial Weibull progression (the shape parameter being fixed for all models), and the other specifies a levelling point after which the hazard remains constant. AIDS incidence data from Canada's national surveillance system provided the empiric data for this evaluation. First we corrected these data for reporting delay by Poisson modelling of the delay distribution. We used three parametric families of infection curves: step-function, log-logistic, and logistic. The results support the hypothesis of an early levelling of the hazard function. When we fixed the scale parameter to that of the Weibull curve with 8 year median, the maximum likelihood estimate of the levelling point was 2.7 years, and a clearly superior fit was produced compared to a pure Weibull progression with the same scale parameter (likelihood ratio chi-square of 10.6 on 1 degree of freedom, p = 0.001). The maximum was indistinguishable in fit from the levelling point of 3.5 years hypothesized in advance (chi-square = 0.30, d.f. = 1, p = 0.58). Backcalculation, however, could not determine the Weibull scale parameter itself because the likelihood was quite flat as a function of this parameter. We conclude that one must determine the parameters governing the initial shape of the hazard function from other kinds of data.(ABSTRACT TRUNCATED AT 400 WORDS)

Screening for a disorder may be carried out by assessing the risk that an individual is affected given the values of variables whose distributions alter when the disorder is present. An optimal screening policy is obtained by identifying those individuals whose risk is greater than some cut-off value. This paper summarizes the way in which risk is derived from the likelihood ratio of being affected by the disorder, and compares three different methods of estimating the likelihood ratio, namely direct estimation, logistic regression and distribution modelling. For continuous variables that have a multivariate normal distribution, screening by risk is equivalent to the use of quadratic discrimination. The paper shows how estimates of the risk and associated detection and false positive rates can be derived for a screening policy which uses specified risk cut-offs. Screening by risk has the counter-intuitive property that as the separation in the distribution of screening variables between affected and unaffected individuals increases, the detection and false positive rates may both increase. The approach is explored using data on antenatal screening for Down's syndrome. The method of choice is model-based; the model is described and tested for goodness of fit. Complications arising from outliers and non-normality must be overcome before an appropriate assessment of risk can be made. The concept of shrinkage is used to estimate the detection of false positive rates that may be expected in a new data set.

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.

To evaluate the calibration of a disease risk prediction tool, the quantity E/O, i.e. the ratio of the expected to the observed number of events, is usually computed. However, because of censoring, or more precisely because of individuals who drop out before the termination of the study, this quantity is generally unavailable for the complete population study and an alternative estimate has to be computed. In this paper, we present and compare four methods to do this. We show that two of the most commonly used methods generally lead to biased estimates. Our arguments are first based on some theoretic considerations. Then, we perform a simulation study to highlight the magnitude of biases. As a concluding example, we evaluate the calibration of an existing predictive model for breast cancer on the E3N-EPIC cohort.

The cumulative logit or the proportional odds regression model is commonly used to study covariate effects on ordinal responses. This paper provides some graphical and numerical methods for checking the adequacy of the proportional odds regression model. The methods focus on evaluating functional misspecification for specific covariate effects, but misspecification of the link function can also be dealt with under the same framework. For the logistic regression model with binary responses, Arbogast and Lin (Statist. Med. 2005; 24:229-247) developed similar graphical and numerical methods for assessing the adequacy of the model using the cumulative sums of residuals. The paper generalizes their methods to ordinal responses and illustrates them using an example from the VA Normative Aging Study. Simulation studies comparing the performance of the different diagnostic methods indicate that some of the graphical methods are more powerful in detecting model misspecification than the Hosmer-Lemeshow-type goodness-of-fit statistics for the class of models studied.

In studies of the natural history of HIV-1 infection, the time scale of primary interest is the time since infection. Unfortunately, this time is very often unknown for HIV infection and using the follow-up time instead of the time since infection is likely to provide biased results because of onset confounding. Laboratory markers such as the CD4 T-cell count carry important information concerning disease progression and can be used to predict the unknown date of infection. Previous work on this topic has made use of only one CD4 measurement or based the imputation on incident patients only. However, because of considerable intrinsic variability in CD4 levels and because incident cases are different from prevalent cases, back calculation based on only one CD4 determination per person or on characteristics of the incident sub-cohort may provide unreliable results. Therefore, we propose a methodology based on the repeated individual CD4 T-cells marker measurements that use both incident and prevalent cases to impute the unknown date of infection. Our approach uses joint modelling of the time since infection, the CD4 time path and the drop-out process. This methodology has been applied to estimate the CD4 slope and impute the unknown date of infection in HIV patients from the Swiss HIV Cohort Study. A procedure based on the comparison of different slope estimates is proposed to assess the goodness of fit of the imputation. Results of simulation studies indicated that the imputation procedure worked well, despite the intrinsic high volatility of the CD4 marker.

In preventing diseases such as cancers and osteoporosis, statistical models are often used to identify subjects with high risks. The ratio of the expected (or predicted) number of cases in the target population and the observed numbers of cases (the E/O ratio) is a useful quantity to evaluate the goodness of fit of the model. The model is usually evaluated on a sample taken from the target population and, in the literature, statistical inferences on the E/O ratio often assume that the expected number is a constant and the observed number follows a Poisson distribution. In this paper, we introduce a nonparametric method that takes into account the variability of the predicted number due to sampling and its correlation with the observed number. By estimating the variance of the estimated E/O ratio more accurately, this nonparametric approach offers better inferences. In addition, we propose to use an F-statistic to test the goodness of a model across subgroups defined by certain risk factors.

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.

Applications frequently involve logistic regression analysis with clustered data where there are few positive outcomes in some of the independent variable categories. For example, an application is given here that analyzes the association of asthma with various demographic variables and risk factors using data from the third National Health and Nutrition Examination Survey, a weighted multi stage cluster sample. Although there are 742 asthma cases in all (out of 18 395 individuals), for one of the categories of one of the independent variables there are only 25 asthma cases (out of 695 individuals). Generalized Wald and score hypothesis tests, which use appropriate cluster-level variance estimators, and a bootstrap hypothesis test have been proposed for testing logistic regression coefficients with cluster samples. When there are few positive outcomes, simulations presented in this paper show that these tests can sometimes have either inflated or very conservative levels. A simulation-based method is proposed for testing logistic regression coefficients with cluster samples when there are few positive outcomes. This testing methodology is shown to compare favorably with the generalized Wald and score tests and the bootstrap hypothesis test in terms of maintaining nominal levels. The proposed method is also useful when testing goodness-of-fit of logistic regression models using deciles-of-risk tables. Published in 2007 by John Wiley & Sons, Ltd.

Additive regression models are preferred over multiplicative models in the analysis of relative survival data. Such preferences are mainly grounded in practical experience with mostly cancer registries data, where the basic assumption of the additivity of hazards is more likely to be met. Also, the interpretation of coefficients is more meaningful in additive than in multiplicative models. Nonetheless, the question of goodness of fit of the assumed model must still be addressed, and while there is an abundance of methods to check the goodness of fit of multiplicative models, the respective arsenal for additive models is almost empty. We propose here a variety of procedures for testing the null hypothesis of a good fit. These are based on partial residuals defined similarly to Schoenfeld residuals familiar for Cox model diagnostics. The tests have appropriate sizes under the null hypothesis, and good power under different alternatives. We investigate their performance through simulations and apply the methods to data from a study into survival of colon cancer patients.

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.

Current methods for statistical analysis of twin studies focus on continuous and dichotomous data, while only limited methodology exists for analysing multinomial data. As a consequence, investigators are often tempted to collapse multinomial data into two categories simply to facilitate the analysis. We address this problem by developing and evaluating two approaches to the assessment of twin correlation for an outcome variable having more than two nominal categories. One method developed is an extension of the goodness-of-fit approach, while the other method is based on large sample normal theory. Procedures for confidence interval construction are developed and compared using Monte Carlo simulation. The results show that either method may be safely used for confidence interval construction provided the number of twin pairs is large (> or =100) but that in smaller sample sizes the goodness-of-fit procedure is to be preferred on the grounds of validity. Other inference problems are also discussed, including point estimation, hypothesis testing and sample size estimation. An example is included.

We propose a goodness-of-fit test statistic for linear regression with heterogeneous variance, which is asymptotically chi-square if the given model is correct. The test statistic is computed as a quadratic form of observed minus predicted responses. We apply the method to a linear regression for an ordinal categorical response, the wheezing status of a child (no wheeze, wheeze with cold, wheeze apart from cold) as a function of maternal smoking and city of residence.

Many mathematical representations are possible both for the frequency distribution of survival time and for the effect on that distribution of explanatory variables. A short review is given of the main types of model the techniques available for model choice the consequences of assuming a particular form and methods for assessing goodness of fit. Effects of model misspecification are discussed briefly. (EXCERPT)

We develop the application of age, period and cohort models to the representation of tables of age- and period-specific rates. A derivation is given by way of a familiar graphical technique. The identifiability problem is discussed, identification techniques are reviewed and a new approach is recommended that is based upon the success of the three two-variable submodels. Other constraints are introduced that enhance interpretation. Examples are given for two sites of cancer. This approach is contrasted with other methods designed to demonstrate trends. Finally, standard errors of the parameters and tests of goodness of fit are discussed.