Volume 2, Issue 1

A natural choice of time scale for analyzing recurrent event data is the ``gap" (or soujourn) time between successive events. In many situations it is reasonable to assume correlation exists between the successive events experienced by a given subject. This paper looks at the problem of extending the accelerated failure time (AFT) model to the case of dependent recurrent event data via intensity modeling. Specifically, the accelerated gap times model of Strawderman (2005), a semiparametric intensity model for independent gap time data, is extended to the case of multiplicative gamma frailty. As argued in Aalen & Husebye (1991), incorporating frailty captures the heterogeneity between subjects and the ``hazard" portion of the intensity model captures gap time variation within a subject. Estimators are motivated using semiparametric efficiency theory and lead to useful generalizations of the rank statistics considered in Strawderman (2005). Several interesting distinctions arise in comparison to the Cox-Andersen-Gill frailty model (e.g., Nielsen et al, 1992; Klein, 1992). The proposed methodology is illustrated by simulation and data analysis.

Many statistical problems involve the learning of an importance/effect of a variable for predicting an outcome of interest based on observing a sample of $n$ independent and identically distributed observations on a list of input variables and an outcome. For example, though prediction/machine learning is, in principle, concerned with learning the optimal unknown mapping from input variables to an outcome from the data, the typical reported output is a list of importance measures for each input variable. The approach in prediction has been to learn the unknown optimal predictor from the data and derive, for each of the input variables, the variable importance from the obtained fit. In this article we propose a new approach which involves for each variable separately 1) defining variable importance as a real valued parameter, 2) deriving the efficient influence curve and thereby optimal estimating function for this parameter in the assumed (possibly nonparametric) model, and 3) develop a corresponding double robust locally efficient estimator of this variable importance, obtained by substituting for the nuisance parameters in the optimal estimating function data adaptive estimators. We illustrate this methodology in the context of prediction, and obtain in this manner double robust locally optimal estimators of marginal variable importance, accompanied with p-values and confidence intervals. In addition, we present a model based and machine learning approach to estimate covariate-adjusted variable importance. Finally, we generalize this methodology to variable importance parameters for time-dependent variables.

Blood doping has been challenging the scientific community since the early 1970's, where it was demonstrated that blood transfusion significantly improves physical performance. Here, we present through 3 applications how statistical classification techniques can assist the implementation of effective tests to deter blood doping in elite sports. In particular, we developed a new indirect and universal test of blood doping, called Abnormal Blood Profile Score (ABPS), based on the statistical classification of indirect biomarkers of altered erythropoiesis. Up to 601 hematological profiles have been compiled in a reference database. Twenty-one of them were obtained from blood samples withdrawn from professional athletes convicted of blood doping by other direct tests. Discriminative training algorithms were used jointly with cross-validation techniques to map these labeled reference profiles to target outputs. The strict cross-validation procedure facilitates the adherence to medico-legal standards mandated by the World Anti Doping Agency (WADA). The test has a sensitivity to recombinant erythropoietin (rhEPO) abuse up to 3 times better than current generative models, independently whether the athlete is currently taking rhEPO or has stopped the treatment. The test is also sensitive to any form of blood transfusion, autologous transfusion included. We finally conclude why a probabilistic approach should be encouraged for the evaluation of evidence in anti-doping area of investigation.

In the case of incomplete data we give general relationships between the first and second derivatives of the loglikelihood relative to the full and the incomplete observation set-ups. In the case where these quantities are easy to compute for the full observation set-up we propose to compute their analogue for the incomplete observation set-up using the above mentioned relationships: this involves numerical integrations. Once we are able to compute these quantities, Newton-Raphson type algorithms can be applied to find the maximum likelihood estimators, together with estimates of their variances. We detail the application of this approach to parametric multiplicative frailty models and we show that the method works well in practice using both a real data and a simulated example. The proposed algorithm outperforms a Newton-Raphson type algorithm using numerical derivatives.

Comparing two large multivariate distributions is potentially complicated at least for the following reasons. First, some variable/level combinations may have a redundant difference in prevalence between groups in the sense that the difference can be completely explained in terms of lower-order combinations. Second, the total number of variable/level combinations to compare between groups is very large, and likely computationally prohibitive. In this paper, for both the paired and independent sample case, an approximate comparison method is proposed, along with a computationally efficient algorithm, that estimates the set of variable/level combinations that have a non-redundant different prevalence between two populations. The probability that the estimate contains one or more false or redundant differences is asymptotically bounded above by any pre-specified level for arbitrary data-generating distributions. The method is shown to perform well for finite samples in a simulation study, and is used to investigate HIV-1 genotype evolution in a recent AIDS clinical trial.

Van der Laan (2005) proposed a targeted method used to construct variable importance measures coupled with respective statistical inference. This technique involves determining the importance of a variable in predicting an outcome. This method can be applied as inverse probability of treatment weighted (IPTW) or double robust inverse probability of treatment weighted (DR-IPTW) estimators. The variance and respective p-value of the estimate are calculated by estimating the influence curve. This article applies the Van der Laan (2005) variable importance measures and corresponding inference to HIV-1 sequence data. In this application, the method is targeted at every codon position. In this data application, protease and reverse transcriptase codon positions on the HIV-1 strand are assessed to determine their respective variable importance, with respect to an outcome of viral replication capacity. We estimate the DR-IPTW W-adjusted variable importance measure for a specified set of potential effect modifiers W. In addition, simulations were performed on two separate datasets to examine the DR-IPTW estimator.

Optimal designs of dose levels in order to estimate parameters from a model for binary response data have a long and rich history. These designs are based on parametric models. Here we consider fully nonparametric models with interest focused on estimation of smooth functionals using plug-in estimators based on the nonparametric maximum likelihood estimator. An important application of the results is the derivation of the optimal choice of the monitoring time distribution function for current status observation of a survival distribution. The optimal choice depends in a simple way on the dose-response function and the form of the functional. The results can be extended to allow dependence of the monitoring mechanism on covariates.

We provide a method for calculating the sample size required to attain a given average power (the ratio of rejected hypotheses to the number of false hypotheses) and a given false discovery rate (the number of incorrect rejections divided by the number of rejections) in adaptive versions of the Benjamini-Hochberg method of multiple testing. The method works in an asymptotic sense as the number of hypotheses grows to infinity and under quite general conditions, and it requires data from a pilot study. The consistency of the method follows from several results in classical areas of nonparametric statistics developed in a new context of "weak" dependence.

We consider the inverse problem of estimating a survival distribution when the survival times are only observed to be in one of the intervals of a random bisection of the time axis. We are particularly interested in the case that high-dimensional and/or time-dependent covariates are available, and/or the survival events and censoring times are only conditionally independent given the covariate process. The method of estimation consists of regularizing the survival distribution by taking the primitive function or smoothing, estimating the regularized parameter by using estimating equations, and finally recovering an estimator for the parameter of interest.

We propose an improved Akaike information criterion (AICc) for generalized log-gamma regression models, which include the extreme-value and normal regression models as special cases. Moreover, we extend our proposed criterion to situations when the data contain censored observations. Monte Carlo results show that AICc outperforms the classical Akaike information criterion (AIC), and an empirical example is presented to illustrate its usefulness.

Suppose one observes a sample of independent and identically distributed observations from a particular data generating distribution. Suppose that one is concerned with estimation of a particular pathwise differentiable Euclidean parameter. A substitution estimator evaluating the parameter of a given likelihood based density estimator is typically too biased and might not even converge at the parametric rate: that is, the density estimator was targeted to be a good estimator of the density and might therefore result in a poor estimator of a particular smooth functional of the density. In this article we propose a one step (and, by iteration, k-th step) targeted maximum likelihood density estimator which involves 1) creating a hardest parametric submodel with parameter epsilon through the given density estimator with score equal to the efficient influence curve of the pathwise differentiable parameter at the density estimator, 2) estimating epsilon with the maximum likelihood estimator, and 3) defining a new density estimator as the corresponding update of the original density estimator. We show that iteration of this algorithm results in a targeted maximum likelihood density estimator which solves the efficient influence curve estimating equation and thereby yields a locally efficient estimator of the parameter of interest, under regularity conditions. In particular, we show that, if the parameter is linear and the model is convex, then the targeted maximum likelihood estimator is often achieved in the first step, and it results in a locally efficient estimator at an arbitrary (e.g., heavily misspecified) starting density.We also show that the targeted maximum likelihood estimators are now in full agreement with the locally efficient estimating function methodology as presented in Robins and Rotnitzky (1992) and van der Laan and Robins (2003), creating, in particular, algebraic equivalence between the double robust locally efficient estimators using the targeted maximum likelihood estimators as an estimate of its nuisance parameters, and targeted maximum likelihood estimators. In addition, it is argued that the targeted MLE has various advantages relative to the current estimating function based approach. We proceed by providing data driven methodologies to select the initial density estimator for the targeted MLE, thereby providing data adaptive targeted maximum likelihood estimation methodology. We illustrate the method with various worked out examples.

The Prostate Cancer Prevention Trial (PCPT) recently demonstrated a significant reduction in prostate cancer incidence of about 25% among men taking finasteride compared to men taking placebo. However, the effect of finasteride on the natural history of prostate cancer is not well understood. We adapted a convolution model developed by Pinsky (2001) to characterize the natural history of prostate cancer in the presence and absence of finasteride. The model was applied to data from 10,995 men in PCPT who had disease status determined by interim diagnosis of prostate cancer or end-of-study biopsy. Prostate cancer cases were either screen-detected by Prostate-Specific Antigen (PSA), biopsy-detected at the end of the study, or clinically detected, that is, detected by methods other than PSA screening. The hazard ratio (HR) for the incidence of preclinical disease on finasteride versus placebo was 0.42 (95% CI: 0.20-0.58). The progression from preclinical to clinical disease was relatively unaffected by finasteride, with mean sojourn time being 16 years for placebo cases and 18.5 years for finasteride cases (p-value for difference = 0.2). We conclude that finasteride appears to affect prostate cancer primarily by preventing the emergence of new, preclinical tumors with little impact on established, latent disease.