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Extensions of the Kaplan-Meier estimator have been developed to illustrate the relationship between a time-varying covariate of interest and survival. In particular, Snapinn et al and Xu et al developed estimators to display survival for patients who always have a certain value of a time-varying covariate. These estimators properly handle time-varying covariates, but their clinical interpretation is limited. It is of greater clinical interest to display survival for patients whose covariates lie along certain defined paths. In this article, we propose extensions of Snapinn et al and Xu et al's estimators, providing crude and covariate-adjusted estimates of the survival function for patients defined by covariate paths. We also derive analytical variance estimators. We demonstrate the utility of these estimators with medical examples and a simulation study., (© 2021 John Wiley & Sons, Ltd.)

Relating time-varying biomarkers of Alzheimer's disease to time-to-event using a Cox model is complicated by the fact that Alzheimer's disease biomarkers are sparsely collected, typically only at study entry; this is problematic since Cox regression with time-varying covariates requires observation of the covariate process at all failure times. The analysis might be simplified by using study entry as the time origin and treating the time-varying covariate measured at study entry as a fixed baseline covariate. In this paper, we first derive conditions under which using an incorrect time origin of study entry results in consistent estimation of regression parameters when the time-varying covariate is continuous and fully observed. We then derive conditions under which treating the time-varying covariate as fixed at study entry results in consistent estimation. We provide methods for estimating the regression parameter when a functional form can be assumed for the time-varying biomarker, which is measured only at study entry. We demonstrate our analytical results in a simulation study and apply our methods to data from the Rush Religious Orders Study and Memory and Aging Project and data from the Alzheimer's Disease Neuroimaging Initiative., (Copyright © 2017 John Wiley & Sons, Ltd.)

Outcome-based sampling is an efficient study design for rare conditions, such as glioblastoma. It is often used in conjunction with matching, for increased efficiency and to potentially avoid bias due to confounding. A study was conducted at the Massachusetts General Hospital that involved retrospective sampling of glioblastoma patients with respect to multiple-ordered disease states, as defined by three categories of overall survival time. To analyze such studies, we posit an adjacent categories logit model and exploit its allowance for prospective analysis of a retrospectively sampled study and its advantageous removal of set and level specific nuisance parameters through conditioning on sufficient statistics. This framework allows for any sampling design and is not limited to one level of disease within each set, such as in previous publications. We describe how this ordinal conditional model can be fit using standard conditional logistic regression procedures. We consider an alternative pseudo-likelihood approach that potentially offers robustness under partial model misspecification at the expense of slight loss of efficiency under correct model specification for small sample sizes. We apply our methods to the Massachusetts General Hospital glioblastoma study., (Copyright © 2015 John Wiley & Sons, Ltd.)

We consider a clinical trial of a potentially lethal disease in which patients are randomly assigned to two treatment groups and are followed for a fixed period of time; a continuous endpoint is measured at the end of follow-up. For some patients; however, death (or severe disease progression) may preclude measurement of the endpoint. A statistical analysis that includes only patients with endpoint measurements may be biased. An alternative analysis includes all randomized patients, with rank scores assigned to the patients who are available for the endpoint measurement on the basis of the magnitude of their responses and with 'worst-rank' scores assigned to those patients whose death precluded the measurement of the continuous endpoint. The worst-rank scores are worse than all observed rank scores. The treatment effect is then evaluated using the Wilcoxon-Mann-Whitney test. In this paper, we derive closed-form formulae for the power and sample size of the Wilcoxon-Mann-Whitney test when missing measurements of the continuous endpoints because of death are replaced by worst-rank scores. We distinguish two approaches for assigning the worst-rank scores. In the tied worst-rank approach, all deaths are weighted equally, and the worst-rank scores are set to a single value that is worse than all measured responses. In the untied worst-rank approach, the worst-rank scores further rank patients according to their time of death, so that an earlier death is considered worse than a later death, which in turn is worse than all measured responses. In addition, we propose four methods for the implementation of the sample size formulae for a trial with expected early death. We conduct Monte Carlo simulation studies to evaluate the accuracy of our power and sample size formulae and to compare the four sample size estimation methods., (Copyright © 2014 John Wiley & Sons, Ltd.)

High-grade gliomas are the most common primary brain tumors in adults and are typically diagnosed using histopathology. However, these diagnostic categories are highly heterogeneous and do not always correlate well with survival. In an attempt to refine these diagnoses, we make several immunohistochemical measurements of YKL-40, a gene previously shown to be differentially expressed between diagnostic groups. We propose two latent class models for classification and variable selection in the presence of high-dimensional binary data, fit by using Bayesian Markov chain Monte Carlo techniques. Penalization and model selection are incorporated in this setting via prior distributions on the unknown parameters. The methods provide valid parameter estimates under conditions in which standard supervised latent class models do not, and outperform two-stage approaches to variable selection and parameter estimation in a variety of settings. We study the properties of these methods in simulations, and apply these methodologies to the glioma study for which identifiable three-class parameter estimates cannot be obtained without penalization. With penalization, the resulting latent classes correlate well with clinical tumor grade and offer additional information on survival prognosis that is not captured by clinical diagnosis alone. The inclusion of YKL-40 features also increases the precision of survival estimates. Fitting models with and without YKL-40 highlights a subgroup of patients who have glioblastoma (GBM) diagnosis but appear to have better prognosis than the typical GBM patient., (Copyright © 2012 John Wiley & Sons, Ltd.)

Familial aggregation studies are a common first step in the identification of genetic determinants of disease. If aggregation is found, more refined genetic studies may be undertaken. Complex ascertainment schemes are frequently employed to ensure that the sample contains a sufficient number of families with multiple affected members, as required to detect aggregation. For example, an eligibility criterion for a family might be that both the mother and daughter have disease. Adjustments must be made for ascertainment to avoid bias. We propose adjusting for complex ascertainment schemes through a joint model for the outcomes of disease and ascertainment. This approach improves upon previous simplifying assumptions regarding the ascertainment process., (Copyright 2008 John Wiley & Sons, Ltd.)

Subjects in tumour studies are often misclassified with respect to histologic features that are not routinely recorded in diagnostic reports and that display heterogeneity within tumours. Pathologic analysis of the tumours may miss the feature of interest if the pathologist was not alerted to detail the microscopic feature of interest or if it is not present in the selected specimens. In this setting, only the subjects for whom the outcome is not found are potentially misclassified. Analyses of associations between the observed, potentially misclassified, outcome and a second outcome are invalid if the probability of misclassification depends on the second outcome. Three natural tests of association based on the observed data depend on different numbers of nuisance parameters. Most promising is a test based on the ratio of proportions of the observed feature. We illustrate this test using a study of the association of imaging parameters with genetic features in subjects with oligodendroglioma, a common brain tumour. In this study, calcification, a feature related to the imaging parameters, was potentially misclassified as not present.

In many settings, one would expect that the hazard for a terminal event would change with the occurrence of an intermediate event. For example, in an AIDS clinical trial, it is of interest to assess whether there is a difference between treatments in the hazards for death prior to drop in Karnofsky performance score and in the hazards subsequent to the drop in Karnofsky score. Tests for the effect of treatment on these hazard functions, separately or jointly, are useful in conjunction with tests of overall survival. We consider four Cox regression models for the hazard function, constructed by allowing for various combinations of time-dependent stratification and time-dependent covariates, both of which are based on the occurrence of the intermediate event. Assuming a Markov transition model from the intermediate to the terminal event, partial likelihoods can be used for inference, enabling the use of standard statistical software for computation. We develop analytic approximations for the power of the derived score tests for treatment differences in the hazard functions and evaluate them through simulations. We apply our results to AIDS Clinical Trials Group (ACTG) protocol 021., (Copyright 2004 John Wiley & Sons, Ltd.)

The distribution of time between an intermediate event and a terminal event is frequently of interest both in studying the behaviour of populations and in predicting outcomes for individuals. Current methods for estimating this latency distribution have either imposed assumptions on the hazard for the terminal event following the occurrence of the intermediate event or on the parametric form of the hazards. Here, local likelihood estimation is applied to the underlying hazard functions of a three-state process in which the time of the intermediate event may be interval censored and the time of the terminal event is either observed or right censored. Smooth non-parametric estimates of the latency distribution, along with bootstrap confidence intervals, are calculated, and tests for the comparison of two latency distributions are presented. The method is applied to two studies: a cohort of haemophiliacs who were infected with HIV by contaminated blood factor and followed for AIDS onset, and an AIDS clinical trial in which the intermediate event is the time to 50 per cent drop in CD4 count and the terminal event is AIDS or death. Simulations are presented to assess the performance of the estimation procedure., (Copyright 2002 John Wiley & Sons, Ltd.)

We discuss the use of local likelihood methods to fit proportional hazards regression models to right and interval censored data. The assumed model allows for an arbitrary, smoothed baseline hazard on which a vector of covariates operates in a proportional manner, and thus produces an interpretable baseline hazard function along with estimates of global covariate effects. For estimation, we extend the modified EM algorithm suggested by Betensky, Lindsey, Ryan and Wand. We illustrate the method with data on times to deterioration of breast cosmeses and HIV-1 infection rates among haemophiliacs., (Copyright 2002 John Wiley & Sons, Ltd.)

A data augmentation algorithm is presented for estimating the hazard function and pointwise variability intervals based on interval censored data. The algorithm extends that proposed by Tanner and Wong for grouped right censored data to interval censored data. It applies multiple imputation and local likelihood methods to obtain smooth non-parametric estimates for the hazard function. This approach considerably simplifies the problem of estimation for interval censored data as it transforms it into the more tractable problem of estimation for right censored data. The method is illustrated for two real data sets: times to breast cosmesis deterioration and times to HIV-1 infection for individuals with haemophilia. Simulations are presented to assess the effects of various parameters on the estimates and their variances., (Copyright 2000 John Wiley & Sons, Ltd.)

We derive a non-parametric maximum likelihood estimator for bivariate interval censored data using standard techniques for constrained convex optimization. Our approach extends those taken for univariate interval censored data. We illustrate the estimator with bivariate data from an AIDS study., (Copyright 1999 John Wiley & Sons, Ltd.)

Non-parametric tests of independence, as well as accompanying measures of association, are essential tools for the analysis of bivariate data. Such tests and measures have been developed for uncensored and right censored failure time data, but have not been developed for interval censored failure time data. Bivariate interval censored data arise in AIDS studies in which screening tests for early signs of viral and bacterial infection are done at clinic visits. Because of missed clinic visits, the actual times of first positive screening tests are interval censored. To handle such data, we propose an extension of Kendall's coefficient of concordance. We apply it to data from an AIDS study that recorded times of shedding of cytomegalovirus (CMV) and times of colonization of mycobacterium avium complex (MAC). We examine the performance of our proposed measure through a simulation study., (Copyright 1999 John Wiley & Sons, Ltd.)

In designing experiments, investigators frequently can specify an important effect that they wish to detect with high power, without the ability to provide an equally certain assessment of the variance of the response. If the experiment is designed based on a guess of the variance, an under-powered study may result. To remedy this problem, there have been several procedures proposed that obtain estimates of the variance from the data as they accrue and then recalculate the sample size accordingly. One class of procedures is fully sequential in that it assesses after each response whether the current sample size yields the desired power based on the current estimate of the variance. This approach is efficient, but it is not practical or advisable in many situations. Another class of procedures involves only two or three stages of sampling and recalculates the sample size based on the observed variance at designated times, perhaps coinciding with interim efficacy analyses. The two-stage approach can result in substantial oversampling, but it is feasible in many situations, whereas the three-stage approach corrects the problem of oversampling, but is less feasible. We propose a procedure that aims to combine the advantages of both the fully sequential and the two-stage approaches. This quasi-sequential procedure involves only two stages of sampling and it applies to the stopping rule from the fully sequential procedure to data beyond the initial sample which we obtain via multiple imputation. We show through simulations that when the initial sample size is substantially less than the correct sample size, the mean squared error of the final sample size calculated from the quasi-sequential procedure can be considerably less than that from the two-stage procedure. We compare the distributions of these recalculated sample sizes and discuss our findings for alternative procedures, as well.

Longitudinal data are commonly analysed using mixed-effects models in which the population growth curve and individual subjects' growth curves are assumed to be known functions of time. Frequently, polynomial functions are assumed. In practice, however, polynomials may not fit the data and a mechanistic model that could suggest a non-linear function might not be known. Recent, more flexible approaches to these data approximate the underlying population mean curve or the individual subjects' curves using smoothing splines or kernel-based functions. I apply the local likelihood estimation method of Tibshirani and Hastie and estimate smooth population and individual growth curves by assuming that they are approximately linear or quadratic functions of time within overlapping neighbourhoods. This method requires neither complete data, nor that measurements are made at the same time points for each individual. For descriptive purposes, this approach is easy to implement with standard software. Inference for the resulting curve is facilitated by the theory of estimating equations. I illustrate the methods with data sets containing longitudinal measurements of serum neopterin in an AIDS clinical trial, measurements of ultrafiltration rates of high flux membrane dialysers for haemodialysis, and measurements of the volume of air expelled by individuals.

For ethical and efficiency concerns one often wishes to design a clinical trial to stop early if there is a strong treatment effect or if there is strong evidence of no treatment effect. There is a large literature to address the design of sequential trials for detecting treatment differences. There has been less attention paid to the design of trials for detecting lack of a treatment difference and most of the designs proposed have been ad hoc modifications of the traditional designs. In the context of fixed sample tests, various authors have proposed basing the decision to stop in favour of the null hypothesis, H0, on conditional power calculations for the end of the trial given the current data. Here I extend this procedure to the popular sequential designs: the O'Brien-Fleming test and the repeated significance test. I derive explicit boundaries for monitoring the test statistic useful for visualizing the impact of the parameters on the operating characteristics of the tests and thus for the design of the tests. Also, they facilitate the use of boundary crossing methods for approximations of power. I derive appropriate boundaries retrospectively for two clinical trials: one that concluded with no treatment difference (AIDS Clinical Trials Group protocol 118) and one that stopped early for positive effect (Beta-Blocker Heart Attack Trial). Finally, I compare the procedures based on the different upper boundaries and assess the impact of allowing for early stopping in favour of H0 in numerical examples.