DIAGNOSTIC PLOTS TO REVEAL FUNCTIONAL FORM FOR COVARIATES IN MULTIPLICATIVE INTENSITY MODELS

We show how plots based on the residuals from a proportional hazards m odel may be used to reveal the correct functional form for covariates in the model. A smoothed plot of the martingale residuals was suggeste d for this purpose by Therneau, Grambsch, and Fleming (1990, Biometrik a 77, 147-160); however, its consistency required that the covariates be independent. They also noted that the plot could be biased for larg e covariate effects. We introduce two refinements which overcome these difficulties. The first is based on a ratio of scatter plot smooths, where the numerator is the smooth of the observed count plotted agains t the covariate, and the denominator is a smooth of the expected count . This is related to the Arjas goodness-of-fit plot (1988, Journal of the American Statistical Association 83, 204-212). The second techniqu e smooths the martingale residuals divided by the expected count, usin g expected count as a weight. This latter approach is related to a GLM partial residual plot, as well as to the iterative methods of Hastie and Tibshirani (1990, Biometrics 46, 1005-1016) and Gentleman and Crow ley (1991, Biometrics 47, 1283-1296). Applications to survival data se ts are given.