Nonparametric models and methods for designs with dependent censored data: part I.

We consider a nonparametric (NP) approach to the analysis of repeated measures designs with censored data. Using the NP model of Akritas and Arnold (1994, Journal of the American Statistical Association 89, 336-343) for marginal distributions, we present test procedures for the NP hypotheses of no main effects, no interaction, and no simple effects. This extends the existing NP methodology for such designs (Wei and Lachin, 1984, Journal of the American Statistical Association 79, 653-661). The procedures do not require any modeling assumptions and should be useful in cases where the assumptions of proportional hazards or location shift fail to be satisfied. The large-sample distribution of the test statistics is based on an i.i.d. representation for Kaplan-Meier integrals. The testing procedures apply also to ordinal data and to data with ties. Useful small-sample approximations are presented, and their performance is examined in a simulation study. Finally, the methodology is illustrated with two real life examples, one with censored and one with missing data. It is indicated that one of the data sets does not conform to any set of assumptions underlying the available methods and also that the present method provides a useful additional analysis even when data sets conform to modeling assumptions.

Analysis of covariance with incomplete data via semiparametric model transformations.

We propose a method for fitting semiparametric models such as the proportional hazards (PH), additive risks (AR), and proportional odds (PO) models. Each of these semiparametric models implies that some transformation of the conditional cumulative hazard function (at each t) depends linearly on the covariates. The proposed method is based on nonparametric estimation of the conditional cumulative hazard function, forming a weighted average over a range of t-values, and subsequent use of least squares to estimate the parameters suggested by each model. An approximation to the optimal weight function is given. This allows semiparametric models to be fitted even in incomplete data cases where the partial likelihood fails (e.g., left censoring, right truncation). However, the main advantage of this method rests in the fact that neither the interpretation of the parameters nor the validity of the analysis depend on the appropriateness of the PH or any of the other semiparametric models. In fact, we propose an integrated method for data analysis where the role of the various semiparametric models is to suggest the best fitting transformation. A single continuous covariate and several categorical covariates (factors) are allowed. Simulation studies indicate that the test statistics and confidence intervals have good small-sample performance. A real data set is analyzed.

Nonparametric inference in factorial designs with censored data.

A method is proposed for testing the hypotheses of no main effects and no interaction in factorial designs with several observations per cell. The method uses the fact that these hypotheses can be expressed in terms of a vector of contrasts. It is based on the observation that nonparametric estimation of these contrasts is no more difficult than estimation of the location difference in the two-sample problem. To implement the method with censored data, a new extension of the Hodges-Lehmann estimator is proposed. The estimator is simple to compute and its variance is easily evaluated. A simulation study examines the performance of the proposed estimation and testing method in the context of a two-by-two design, and a real data set from a three-way layout with heavy censoring is analyzed.