Covariate Selection for the Semiparametric Additive Risk Model

Abstract.  This paper considers covariate selection for the additive hazards model. This model is particularly simple to study theoretically and its practical implementation has several major advantages to the similar methodology for the proportional hazards model. One complication compared with the proportional model is, however, that there is no simple likelihood to work with. We here study a least squares criterion with desirable properties and show how this criterion can be interpreted as a prediction error. Given this criterion, we define ridge and Lasso estimators as well as an adaptive Lasso and study their large sample properties for the situation where the number of covariates p is smaller than the number of observations. We also show that the adaptive Lasso has the oracle property. In many practical situations, it is more relevant to tackle the situation with large p compared with the number of observations. We do this by studying the properties of the so-called Dantzig selector in the setting of the additive risk model. Specifically, we establish a bound on how close the solution is to a true sparse signal in the case where the number of covariates is large. In a simulation study, we also compare the Dantzig and adaptive Lasso for a moderate to small number of covariates. The methods are applied to a breast cancer data set with gene expression recordings and to the primary biliary cirrhosis clinical data.     

Efficient Estimation of Fixed and Time-varying Covariate Effects in Multiplicative Intensity Models

The proportional hazards assumption of the Cox model does sometimes not hold in practise. An example is a treatment effect that decreases with time. We study a general multiplicative intensity model allowing the influence of each covariate to vary non-parametrically with time. An efficient estimation procedure for the cumulative parameter functions is developed. Its properties are studied using the martingale structure of the problem. Furthermore, we introduce a partly parametric version of the general non-parametric model in which the influence of some of the covariates varies with time while the effects of the remaining covariates are constant. This semiparametric model has not been studied in detail before. An efficient procedure for estimating the parametric as well as the non-parametric components of this model is developed. Again the martingale structure of the model allows us to describe the asymptotic properties of the suggested estimators. The approach is applied to two different data sets, and a Monte Carlo simulation is presented.     

The Additive Risk Model for Estimation of Effect of Haplotype Match in BMT Studies

Abstract. In this article we consider a problem from bone marrow transplant (BMT) studies where there is interest on assessing the effect of haplotype match for donor and patient on the overall survival. The BMT study we consider is based on donors and patients that are genotype matched, and this therefore leads to a missing data problem. We show how Aalen's additive risk model can be applied in this setting with the benefit that the time‐varying haplomatch effect can be easily studied. This problem has not been considered before, and the standard approach where one would use the expected‐maximization (EM) algorithm cannot be applied for this model because the likelihood is hard to evaluate without additional assumptions. We suggest an approach based on multivariate estimating equations that are solved using a recursive structure. This approach leads to an estimator where the large sample properties can be developed using product‐integration theory. Small sample properties are investigated using simulations in a setting that mimics the motivating haplomatch problem.     

CUSTOMER RACIAL DISCRIMINATION IN MAJOR LEAGUE BASEBALL

Most studies of racial discrimination concentrate on identifying the existence rather than the source of discrimination. In contrast, this paper tests directly for racial discrimination by major league baseball customers by examining the market for baseball cards. In Tobit regressions, the market prices of cards consistently reflect player performance records and other demand variables. When interactive race-productivity variables are included in the models, a significant and complex pattern of fan discrimination against black pitchers and black hitters is found. Discrimination against latin players is not found.