PHSELECT Procedure

Let be a vector of covariates. The linear predictor and its standard error estimate are, respectively,

The predicted cumulative hazard at a time point t is

and the predicted survival probability is

The martingale residual for the ith observation is

The deviance residual is a transform of the martingale residual,

The square root shrinks large negative martingale residuals, and the logarithmic transformation expands martingale residuals that are close to unity.

The Schoenfeld (1982) residual vector is calculated per event time. If the ith observation is uncensored, the Schoenfeld residual vector is

Otherwise, the elements of the Schoenfeld residual vector are set to missing values. Under the proportional hazards assumption, the Schoenfeld residuals have the sample path of a random walk; therefore, they are useful in assessing time trend or lack of proportionality. Harrell (1986) proposed a z-transform of the Pearson correlation between these residuals and the rank order of the failure time as a test statistic for nonproportional hazards.

The score residual vector for the ith observation is

The score residuals are a decomposition of the first partial derivative of the log likelihood. They are useful in assessing the influence of each observation on individual parameter estimates. Therneau, Grambsch, and Fleming (1990) have considered a Kolmogorov-type test based on the cumulative sum of the residuals for detecting nonproportional hazards.

The vector of weighted Schoenfeld residuals, , is computed as

where is the total number of events and is the vector of Schoenfeld residuals for the i observation. The weighted Schoenfeld residuals are useful in assessing the proportional hazards assumption. The idea is that most of the common alternatives to the proportional hazards can be cast in terms of a time-varying coefficient model,

where and are hazard rates. Let and be the jth component of and , respectively. Grambsch and Therneau (1994) suggest using a smoothed plot of () versus to discover the functional form of the time-varying coefficient . A zero slope indicates that the coefficient does not vary with time.

The DFBETA diagnostics approximate the changes in the parameter estimates, , when the ith observation is left out. The jth component of can be used to assess any unusual effect of the ith observation on . The exact computation of involves refitting the model each time a subject is omitted. Cain and Lange (1984) derived the following approximation of as weighted score residuals,

where is the score residual vector of the ith observation.

The LD statistic approximates the likelihood displacement, which is the amount by which minus twice the log likelihood (), under a fitted model, changes when each observation in turn is left out. When the ith subject is omitted, the likelihood displacement is

where is the vector of parameter estimates that you obtain by fitting the model without the ith observation. Instead of refitting the model without the ith observation, Pettitt and Bin Daud (1989) propose that the likelihood displacement for the ith observation be approximated by

where is the score residual vector of the ith observation.