Joint inference for nonlinear mixed-effects models and time to event at the presence of missing data

In many longitudinal studies, the individual characteristics associated with the repeated measures may be possible covariates of the time to an event of interest, and thus, it is desirable to model the time-to-event process and the longitudinal process jointly. Statistical analyses may be further complicated in such studies with missing data such as informative dropouts. This article considers a nonlinear mixed-effects model for the longitudinal process and the Cox proportional hazards model for the time-to-event process. We provide a method for simultaneous likelihood inference on the 2 models and allow for nonignorable data missing. The approach is illustrated with a recent AIDS study by jointly modeling HIV viral dynamics and time to viral rebound.     

Noniterative adjustment to regression estimators with population-based auxiliary information for semiparametric models

Disease registries, surveillance data, and other datasets with extremely large sample sizes become increasingly available in providing population-based information on disease incidence, survival probability, or other important public health characteristics. Such information can be leveraged in studies that collect detailed measurements but with smaller sample sizes. In contrast to recent proposals that formulate additional information as constraints in optimization problems, we develop a general framework to construct simple estimators that update the usual regression estimators with some functionals of data that incorporate the additional information. We consider general settings that incorporate nuisance parameters in the auxiliary information, non-i.i.d. data such as those from case-control studies, and semiparametric models with infinite-dimensional parameters common in survival analysis. Details of several important data and sampling settings are provided with numerical examples.     

An Empirical Comparison of Parametric and Semiparametric Cure Models

Parametric and semiparametric cure models have been proposed for cure proportion estimation in cancer clinical research. In this paper, several parametric and semiparametric models are compared, and their estimation methods are discussed within the framework of the EM algorithm. We show that the semiparametric PH cure model can achieve efficiency levels similar to those of parametric cure models, provided that the failure time distribution is well specified and uncured patients have an increasing hazard rate. Therefore the semiparametric model is a viable alternative to parametric cure models. When the hazard rate of uncured patients is rapidly decreasing, the estimates from the semiparametric cure model tend to have large variations and biases. However, all other models also tend to have large variations and biases in this case.     

High‐Dimensional Cox Models: The Choice of Penalty as Part of the Model Building Process

The Cox proportional hazards regression model is the most popular approach to model covariate information for survival times. In this context, the development of high‐dimensional models where the number of covariates is much larger than the number of observations ( $p \,{\gg }\, n$ ) is an ongoing challenge. A practicable approach is to use ridge penalized Cox regression in such situations. Beside focussing on finding the best prediction rule, one is often interested in determining a subset of covariates that are the most important ones for prognosis. This could be a gene set in the biostatistical analysis of microarray data. Covariate selection can then, for example, be done by L₁‐penalized Cox regression using the lasso (Tibshirani ( 1997 ). Statistics in Medicine 16 , 385–395). Several approaches beyond the lasso, that incorporate covariate selection, have been developed in recent years. This includes modifications of the lasso as well as nonconvex variants such as smoothly clipped absolute deviation (SCAD) (Fan and Li ( 2001 ). Journal of the American Statistical Association 96 , 1348–1360; Fan and Li ( 2002 ). The Annals of Statistics 30 , 74–99). The purpose of this article is to implement them practically into the model building process when analyzing high‐dimensional data with the Cox proportional hazards model. To evaluate penalized regression models beyond the lasso, we included SCAD variants and the adaptive lasso (Zou ( 2006 ). Journal of the American Statistical Association 101 , 1418–1429). We compare them with “standard” applications such as ridge regression, the lasso, and the elastic net. Predictive accuracy, features of variable selection, and estimation bias will be studied to assess the practical use of these methods. We observed that the performance of SCAD and adaptive lasso is highly dependent on nontrivial preselection procedures. A practical solution to this problem does not yet exist. Since there is high risk of missing relevant covariates when using SCAD or adaptive lasso applied after an inappropriate initial selection step, we recommend to stay with lasso or the elastic net in actual data applications. But with respect to the promising results for truly sparse models, we see some advantage of SCAD and adaptive lasso, if better preselection procedures would be available. This requires further methodological research.     

Time allocation of the parasitoidAphidius colemani (Hymenoptera: Aphidiidae) foraging forAphis gossypii (Homoptera: Aphidae) on cucumber leaves

The time allocation of individualAphidius colemani female parasitoids foraging forAphis gossypii nymphs on cucumber leaves has been investigated. Apart from experiences on the current leaf (such as density of hosts on the current leaf, density of hosts on a neighboring leaf and encounters with hosts on the current leaf), the effect of previous leaf visits on the time allocation was studied. Behavioral records were analyzed by means of the proportional hazards model, to determine the tendency of leaving the current leaf. The leaving tendency decreased only on leaves with a high host density (100 aphids), thus increasing the giving up time since the latest encounter. Rejection of aphids had no influence on the leaving tendency. To assess the effect of the number of hosts encountered on the leaving tendency, we considered three classes: 0–30 encounters, 31–100 encounters, and 100 or more encounters with hosts. The effect of the number of hosts encountered differed at different aphid densities. When fewer than 10 aphids were present the leaving tendency was much greater after 30 encounters than beforehand. At a density of 100 aphids the leaving tendency was lower than at the other aphid densities and increased only after 100 encounters. The density of hosts on a neighboring leaf, ranging from 0 to 100 hosts, had a negligible effect on the leaving tendency. Repeated visits to leaves with 10 unparasitized aphids resulted in an increase in the leaving tendency after 10 visits. It is argued that the parasitoids have some innate expectancy of host availability and that they concentrate on high-density patches.     

Fit and Frailties in Proportional Hazards Regression

We exploit a conjectured equivalence between proportional hazards models with frailties and a particular subclass of non proportional hazards models, specifically those with declining effects, to address the question of fit. A goodness of fit test of the proportional hazards assumption against an alternative of declining regression effect is equivalent to a test for the presence of frailties. Such tests are now widely available in standard software. Although a number of tests of the proportional hazards assumption have been developed there is no test that directly formulates the alternative in terms of a non‐specified monotonic decline in regression effect and that enables a quantification of this in terms of a simple index. The index we obtain lies between zero and one such that, for any given set of covariates, values of the index close to one indicate that the fit cannot essentially be improved by allowing the possibility of regression effects to decline. Values closer to zero and away from one indicate that the fit can be improved by relaxing the proportional hazards constraint in this particular direction. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Models of Independent Action for Combination Therapy

Two methods of applying Finney's concept of simple independent action are suggested for examining possible synergism in combination therapy. In one model the treatments act independently at one time to cure the disease. In the other model the treatments act independently continuously to prevent the disease from progressing to a terminal outcome. The implications of these models for the interaction terms of a proportional hazards model, a log-linear model on death rates, and a logistic model are discussed.     

ENDORSE: a prognostic model for endocrine therapy in estrogen‐receptor‐positive breast cancers

Advanced and metastatic estrogen receptor‐positive (ER⁺) breast cancers are often endocrine resistant. However, endocrine therapy remains the primary treatment for all advanced ER⁺ breast cancers. Treatment options that may benefit resistant cancers, such as add‐on drugs that target resistance pathways or switching to chemotherapy, are only available after progression on endocrine therapy. Here we developed an endocrine therapy prognostic model for early and advanced ER⁺ breast cancers. The endocrine resistance (ENDORSE) model is composed of two components, each based on the empirical cumulative distribution function of ranked expression of gene signatures. These signatures include a feature set associated with long‐term survival outcomes on endocrine therapy selected using lasso‐regularized Cox regression and a pathway‐based curated set of genes expressed in response to estrogen. We extensively validated ENDORSE in multiple ER⁺ clinical trial datasets and demonstrated superior and consistent performance of the model over clinical covariates, proliferation markers, and multiple published signatures. Finally, genomic and pathway analyses in patient data revealed possible mechanisms that may help develop rational stratification strategies for endocrine‐resistant ER⁺ breast cancer patients.     

Goodness of Fit Tests for Proportional Hazards Regression Models

We present a test of goodness of fit for the proportional hazard regression model. The test is based on a score statistic for testing against local mixture alternatives. Contrary to the findings of several other authors, we detect a significant lack of fit in Freireich's leukemia data.     

Estimating predictors for long- or short-term survivors

Suppose that the response variable in a well-executed clinical or observational study to evaluate a treatment is the time to a certain event, and a set of baseline covariates or predictors was collected for each study patient. Furthermore, suppose that a significant number of study patients had nontrivial, long-term adverse effects from the treatment. A commonly posed question is how to use these covariates from the study to identify future patients who would (or would not) benefit from the treatment. In this article, we present "point" and "interval" estimates for the set of covariate or predictor vectors associated with a specific patient survival status, e.g., long- (or short-) term survival, in the presence of censoring. These estimates can be easily displayed on a two-dimensional plane, even for the case with high-dimensional covariate vectors. These simple numerical and graphical procedures provide useful information for patient management and/or the design of future studies, which are key issues in pharmacogenomics with genetic markers. The new proposal is illustrated with a data set from a cancer study for treating multiple myeloma.     

Heterogeneous treatment effects in stratified clinical trials with time‐to‐event endpoints

When analyzing clinical trials with a stratified population, homogeneity of treatment effects is a common assumption in survival analysis. However, in the context of recent developments in clinical trial design, which aim to test multiple targeted therapies in corresponding subpopulations simultaneously, the assumption that there is no treatment‐by‐stratum interaction seems inappropriate. It becomes an issue if the expected sample size of the strata makes it unfeasible to analyze the trial arms individually. Alternatively, one might choose as primary aim to prove efficacy of the overall (targeted) treatment strategy. When testing for the overall treatment effect, a violation of the no‐interaction assumption renders it necessary to deviate from standard methods that rely on this assumption. We investigate the performance of different methods for sample size calculation and data analysis under heterogeneous treatment effects. The commonly used sample size formula by Schoenfeld is compared to another formula by Lachin and Foulkes, and to an extension of Schoenfeld's formula allowing for stratification. Beyond the widely used (stratified) Cox model, we explore the lognormal shared frailty model, and a two‐step analysis approach as potential alternatives that attempt to adjust for interstrata heterogeneity. We carry out a simulation study for a trial with three strata and violations of the no‐interaction assumption. The extension of Schoenfeld's formula to heterogeneous strata effects provides the most reliable sample size with respect to desired versus actual power. The two‐step analysis and frailty model prove to be more robust against loss of power caused by heterogeneous treatment effects than the stratified Cox model and should be preferred in such situations.     

Modeling clustered long‐term survivors using marginal mixture cure model

There is a great deal of recent interests in modeling right‐censored clustered survival time data with a possible fraction of cured subjects who are nonsusceptible to the event of interest using marginal mixture cure models. In this paper, we consider a semiparametric marginal mixture cure model for such data and propose to extend an existing generalized estimating equation approach by a new unbiased estimating equation for the regression parameters in the latency part of the model. The large sample properties of the regression effect estimators in both incidence and the latency parts are established. The finite sample properties of the estimators are studied in simulation studies. The proposed method is illustrated with a bone marrow transplantation data and a tonsil cancer data.