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In many epidemiology studies, family data with survival endpoints are collected to investigate the association between risk factors and disease incidence. Sometimes the risk of the disease may change when a certain risk factor exceeds a certain threshold. Finding this threshold value could be important for disease risk prediction and diseases prevention. In this work, we propose a change-point proportional hazards model for clustered event data. The model incorporates the unknown threshold of a continuous variable as a change point in the regression. The marginal pseudo-partial likelihood functions are maximized for estimating the regression coefficients and the unknown change point. We develop a supremum test based on robust score statistics to test the existence of the change point. The inference for the change point is based on the m out of n bootstrap. We establish the consistency and asymptotic distributions of the proposed estimators. The finite-sample performance of the proposed method is demonstrated via extensive simulation studies. Finally, the Strong Heart Family Study dataset is analyzed to illustrate the methods.

For patients on dialysis, hospitalizations remain a major risk factor for mortality and morbidity. We use data from a large national database, United States Renal Data System (USRDS), to model time-varying effects of hospitalization risk factors as functions of time since initiation of dialysis. To account for the three-level hierarchical structure in the data where hospitalizations are nested in patients and patients are nested in dialysis facilities, we propose a multilevel mixed effects varying coefficient model (MME-VCM) where multilevel (patient- and facility-level) random effects are used to model the dependence structure of the data. The proposed MME-VCM also includes multilevel covariates, where baseline demographics and comorbidities are among the patient-level factors, and staffing composition and facility size are among the facility-level risk factors. To address the challenge of high-dimensional integrals due to the hierarchical structure of the random effects, we propose a novel two-step approximate EM algorithm based on the fully exponential Laplace approximation. Inference for the varying coefficient functions and variance components is achieved via derivation of the standard errors using score contributions. The finite sample performance of the proposed estimation procedure is studied through simulations.

Mendelian randomization (MR) is a type of instrumental variable (IV) analysis that uses genetic variants as IVs for a risk factor to study its causal effect on an outcome. Extensive investigations on the performance of IV analysis procedures, such as the one based on the two-stage least squares (2SLS) procedure, have been conducted under the one-sample scenario, where measures on IVs, the risk factor, and the outcome are assumed to be available for each study participant. Recent MR analysis usually is performed with data from two independent or partially overlapping genetic association studies (two-sample setting), with one providing information on the association between the IVs and the outcome, and the other on the association between the IVs and the risk factor. We investigate the performance of 2SLS in the two-sample based MR when the IVs are weakly associated with the risk factor. We derive closed form formulas for the bias and mean squared error (MSE) of the 2SLS estimate, and verify them with numeric simulations under realistic circumstances. Using these analytic formulas we can study the pros and cons of conducting MR analysis under one-sample and two-sample settings, and assess the impact of having overlapping samples. We also propose and validate a bias-corrected estimator for the causal effect.

The generalized impact fraction (also called the generalized attributable fraction) was introduced by Walter (1980, American Journal of Epidemiology 112, 409-416) and Morgenstern and Bursic (1982, Journal of Community Health 7, 292-309) as a measure that generalizes the population attributable fraction (attributable risk). It is defined as the fractional reduction of a disease resulting from changing the current distribution of a risk factor to some modified distribution. We show that the point and variance estimator derived by Greenland and Drescher (1993, Biometrics 49, 865-872) for fixed shift functions can be extended to situations where the shift is a probabilistic function of the actual exposure value. The formulas are applicable for case-control designs where the cases are simply randomly selected and the controls are chosen in one of three ways: simple random sampling, stratified random sampling, and frequency matching.

Bruzzi et al. (1985, American Journal of Epidemiology 122, 904-914) provided a general logistic-model-based estimator of the attributable fraction for case-control data, and Benichou and Gail (1990, Biometrics 46, 991-1003) gave an implicit-delta-method variance formula for this estimator. The Bruzzi et al. estimator is not, however, the maximum likelihood estimator (MLE) based on the model, as it uses the model only to construct the relative risk estimates, and not the covariate-distribution estimate. We here provide maximum likelihood estimators for the attributable fraction in cohort and case-control studies, and their asymptotic variances. The case-control estimator generalizes the estimator of Drescher and Schill (1991, Biometrics 47, 1247-1256). We also present a limited simulation study which confirms earlier work that better small-sample performance is obtained when the confidence interval is centered on the log-transformed point estimator rather than the original point estimator.

In this paper we present a proportional hazards model to determine the effect of predictors on repeated measures data with ordinal response that is monotonic over time. The model can handle both categorical and continuous predictors. It naturally models the time spent in the response states. The assumption of proportional hazards can be partially assessed if replicates (individuals with identical covariate vectors) are available. Censoring can be incorporated into the model, although it does require additional modelling if the censoring is informative.

For desired power of detecting the association between a disease and several potential risk factors in case-control studies, Yang 1978 and Halperin 1980 proposed a probit risk model and derived approximate sample size formulas, respectively. The lack of direct interpretation of the parameters in the probit risk model and the complexity of these approximate sample size formulas, however, limit the application of Yang's and Halperin's procedures in practice. In this paper, we assume an exponential risk model for the incidence rate. Under our model assumptions, we derive an exact sample size calculation procedure that is simpler and easier to use than Yang's and Halperin's.

The proportion of tumor-bearing animals in a chronic bioassay is often used as a basis for assessing risk at human exposure levels that are below the experimental range. However, such a crude experimental tumor rate can be biased by dose-related differences in the nontumor mortality. To adjust tumor rates for competing mortality, Kodell, Gaylor, and Chen (1986, Biometrics 42, 867-873) propose a standardized tumor rate, calculated separately in each dose group from dose-specific estimates of the tumor prevalence and mortality functions. This paper extends the approach of Kodell et al, by developing a method of modeling the standardized rate as a function of dose. An advantage of this approach is that it leads to estimates that are monotone in dose. In addition, by modeling the lifetime risk as a function of dose directly, it is possible to obtain estimates for the risk at dose levels outside the experimental range, such as for low-dose extrapolation, and estimation of the "tumorigenic dose 50" (TD50). A semiparametric model is developed, as well as the nonparametric alternative of isotonic regression. Simulations are provided to show the relative bias and precision of the two approaches to that of Kodell et al.

We introduce a test for the equality of two survival distributions against the specific alternative of crossing hazards. Although this kind of alternative is somewhat rare, designing a test specifically aimed at detecting such departures from the null hypothesis in this direction leads to powerful procedures, upon which we can call in those few cases where such departures are suspected. Furthermore, the proposed test and an approximate version of the test are seen to suffer only moderate losses in power, when compared with their optimal counterparts, should the alternative be one of proportional hazards. Our interest in the problem is motivated by clinical studies on the role of acute graft versus host disease as a risk factor in leukemic children and we discuss the analysis of this study in detail. The model we use in this work is a special case of the one introduced by Anderson and Senthilselvan (1982. Applied Statistics 31, 44-51). We propose overcoming an inferential problem stemming from their model by using the methods of Davies (1977, Biometrika 64, 247-254; 1987, Biometrika 74, 33-43) backed up by resampling techniques. We also look at an approach relying directly on resampling techniques. The distributional aspects of this approach under the null hypothesis are interesting but, practically, its behaviour is such that its use cannot be generally recommended. Outlines of the necessary asymptotic theory are presented and for this we use the tools of martingale theory.

A method is presented for approximating the influence of individual cases upon regression coefficient estimates obtained from the Cox proportional hazards model. Observations can thus be identified which may greatly influence statistical inferences regarding the effects of prognostic factors upon survival time. An example from a cancer clinical trial is given.

A general system of log-linear modeling is proposed for analysis of competing-risks data with discrete covariates. The instantaneous cause-specific failure rates, approximated by step-functions, are analyzed by techniques for multidimensional contingency tables. Censored observations are accommodated. Counts of failures of each type, and the amount of follow-up, are summarized in two arrays in which each cell denotes a distinct combination of failure type, time interval and covariate value. Maximum likelihood estimators for the parameters of the model are derived by iterative proportional fitting; the resulting estimates of the number of failures in each cell are used for goodness-of-fit tests. The principal advantages of this approach are its simple display of data, its computational ease for the fitting and comparison of models and its provision of explicit goodness-of-fit tests. Interpretation of the models is facilitated by reference to several alternative models for survivorship and competing risks. The basic model is extended to incorporate stochastic covariates whose values change during follow-up, and to accommodate quantitative covariates.

Breslow (1981, Biometrika 68, 73-84) has shown that the Mantel-Haenszel odds ratio is a consistent estimator of a common odds ratio in sparse stratifications. For cohort studies, however, estimation of a common risk ratio or risk difference can be of greater interest. Under a binomial sparse-data model, the Mantel-Haenszel risk ratio and risk difference estimators are consistent in sparse stratifications, while the maximum likelihood and weighted least squares estimators are biased. Under Poisson sparse-data models, the Mantel-Haenszel and maximum likelihood rate ratio estimators have equal asymptotic variances under the null hypothesis and are consistent, while the weighted least squares estimators are again biased; similarly, of the common rate difference estimators the weighted least squares estimators are biased, while the estimator employing "Mantel-Haenszel" weights is consistent in sparse data. Variance estimators that are consistent in both sparse data and large strata can be derived for all the Mantel-Haenszel estimators.

Many asymptotic formulas for estimating the variance of and confidence intervals for odds ratios in case-control studies are available, but most have limitations. The method suggested by Cornfield (1956, Proceedings of the Third Berkeley Symposium 4, 135-148), for example, is iterative and requires calculation of exact variances if the number of subjects within each stratum is small. This paper presents a new asymptotic, noniterative variance estimate for the Mantel-Haenszel odds ratio which, based on a series of Monte Carlo experiments, compares favorably with other noniterative estimates when used for confidence interval estimation.

Longitudinal studies provide the opportunity to relate chronic disease risk factors in children to the same risk factors in adults. The epidemiologic concept of tracking allows the quantification of that relationship. Tracking of a risk factor is defined as the maintenance of relative rank over a given time span. A U statistic is defined, which is based on a set of individual growth curves and which estimates the degree of tracking observed. The properties of this estimator are discussed, and applications presented.