On the Use of Indicator Variables for Studying the Time-Dependence of Parameters in a Response-Time Model

The study of the dependence of response-time data on a multivariate regressor variable in the presence of arbitrary censoring has been approached in a number of ways. The exponential regression model proposed byr Feigl and Zelen 1965 and extended by Zippin and Armitage 1966 and by Mantel and Myers 1971 to the case of arbitrarily right censored data relates the reciprocal of the exponential parameter, i.e. the expected survival time, to a linear function of the regressor variables. Later, Glasser 1967 proposed an exponential model in which the logarithm of the exponential parameter was assumed to be a linear function of the regressor variables. In both formulations the rather stringent assumption of a constant hazard may be dropped by the assumption of a more general response-time distribution such as the Weibull, gamma or Gompertz, each of which contains the exponential as a special case. The nonparametric model proposed by Cox 1972 admits an arbitrary response-time distribution and, for discrete data, becomes a logistic regression model. An alternative version of Cox's discrete model has beenl proposed by Kalbfleisch and Prentice 1973. These approaches have the advantage of not specifying the hazard function in advance and, as such, are more robust than the above parametric methods. Their major drawback, however, is the computational difficulties in the presence of tied response times. In many practical situations the data are recorded in such a way as to make this a very real problem and serious enough to implv that an alternative procedure may be desirable. This logistic regression model was also used by Myers et al. 1973 in conjunction with the assumption of a constant hazard. The model they considered incorporated concomitant information by assuming that the probability of responding within a unit time period followed a logistic regression function, while the actual time to response followed a particular distributional form. They chose a form which assumed a time-independent risk of responding-the exponential for a continuous time process or geometric for discrete time. This approach was extended by Hankey and Mantel 1974 by the addition of a time function to the logistic regression function. This tinme function was approximated by a low order polynomial. Inherent in these exponential and logistic regression models is the assumption that the effects of the covariates are independent of time. The exponential model of Feigl and Zelen relates the expected survival time to the concomitant information and, since the exponential distribution is without memory, the expected remaining survival timne given survival up to some time point T has the same relationship to the concomitant information no matter what the value of T. The logistic regression methods that have been proposed allow the underlying hazard to be a function of time but the relative effects of the covariates