*************************** * Biostatistics 513 * Exercise Set 8, 2002 *************************** 1(a) We can use the tests of the PH assumption that are based on the Schoenfeld residuals. We find: Test of proportional hazards assumption Time: Time ---------------------------------------------------------------- | rho chi2 df Prob>chi2 ------------+--------------------------------------------------- logbil | 0.14146 2.33 1 0.1271 logalb | -0.01932 0.05 1 0.8254 age | -0.03642 0.15 1 0.7032 logpro | -0.17921 3.25 1 0.0714 edema | -0.14801 2.65 1 0.1033 ------------+--------------------------------------------------- Therefore, we find no significant departure from the PH assumption (using the nominal 5% significance level). We do find that log(prothrombin time), edema, and log(bilirubin) yield trends that approach statistical significance. 1(b) I decided to use the values 2.30 and 2.40 for log(prothrombin) to create 3 groups since these were simple values of the predictor and lead to a reasonable proportion in the lowest and highest groups (about 25% for each, with 50% in the second category). SEE the web page for code to generate the log-minus-log plot. This plot does not appear to be parallel for groups 2 and 3. The separation seems to get smaller with increased time. The curve for the first group (lowest level) appears approximately parallel to the second group. As groups 2 and 3 seem to be non-parallel, this plot suggests that the PH assumption is violated for this predictor. 1(c) SEE the web page for code to generate the 5 residual plots. These residual plots are "scaled Schoenfeld" residual plots produced by STATA. The scaled residuals actually display an estimate of beta(t) by taking a smooth curve through the residuals. Thus, these residual plots will not be centered at zero, but rather at approximately beta -- the value assuming PH. The key patterns we would look for are either increasing or decreasing trends in these plots, indicating either increasing or decreasing hazard ratios over time. Summary of the plots: log(bili): A slight increase over time. However, the magnitude of the increase is relatvely small: from about 0.8 to 1.0. PH looks acceptable here. log(alb): Trend looks flat -- PH assumption is fine. age: A slight decrease over time, but reasonably flat -- PH assumption is fine. log(pro): A decrease over time. Notice the scale -- the early level of the trend is approximately 5 and decreasing to less than one. This is a large change in the comparison over time. PH assumption looks violated, with a large relative hazard early in time, but the relative hazard decreases at later times. edema: Some early decrease. Note also that the change in the level is from approximately 1.0 to a value smaller than 1.0. Suggestive of a violation of the PH assumption but not entirely clear. (optional) The wrong value is a prothrombin time of 17.1 -- all of the records were checked and it was found that this should have been 10.1. Notice that this value has a big impact on the coefficient of log(pro) -- try to drop this case and refit the model. 2(a) The following is a table of the hazard ratios estimated via Cox regression using clinic as a stratifying variable. These hazard ratios are for a model with only the covariate listed (ie. a single predictor): Crude hazard ratios: ------------------------------------------------------------------------------ | | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- female | .8854169 .1731552 -0.622 0.534 .6035091 1.299008 ------------------------------------------------------------------------------ age | .9813881 .0197619 -0.933 0.351 .9434098 1.020895 ------------------------------------------------------------------------------ employ | .646265 .1237861 -2.279 0.023 .4439876 .9406983 ------------------------------------------------------------------------------ benzo | .7600031 .132281 -1.577 0.115 .5403324 1.06898 ------------------------------------------------------------------------------ amph | 1.226355 .2790442 0.897 0.370 .7851167 1.915571 ------------------------------------------------------------------------------ alc | 1.234586 .2080928 1.250 0.211 .8872562 1.717883 ------------------------------------------------------------------------------ dose | .9659974 .0062278 -5.366 0.000 .953868 .9782811 ------------------------------------------------------------------------------ * each hazard ratio is obtained stratifying on clinic. This table suggests that predictors of retention loss (failure for this analysis) are: younger age (not significant); unemployment (p=0.023); no history of benzodiazepine/barbiturate abuse (p=0.115); a history of amphetamine/cocaine abuse (not signififcant); a history of alcohol abuse (p=0.211); and a lower dose of methadone (p<0.001). 2(b) A model using all variables (without interactions) yields: ------------------------------------------------------------------------------ | | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- female | .9975521 .2097731 -0.012 0.991 .6605994 1.506375 age | .987356 .0205294 -0.612 0.541 .9479279 1.028424 employ | .6968842 .1464536 -1.718 0.086 .4616113 1.05207 benzo | .717695 .1309526 -1.818 0.069 .5019117 1.026248 amph | 1.13407 .2803717 0.509 0.611 .6985547 1.841107 alc | 1.223356 .2147707 1.148 0.251 .8671959 1.725793 dose | .9675486 .0063383 -5.036 0.000 .9552051 .9800515 ------------------------------------------------------------------------------ Stratified by clinic 2(c) The stratified Cox regression assumes that each of the variables listed in the table above has a hazard ratio (controlling for the other variables in the model) that is constant over time. For example, the hazard comparing benzo=1 to benzo=0 is estimated as 0.718 for any time (controlling for clinic and the other predictor variables). In the stratified model we assume PH for each of the covariates, but do not assume common baseline survivals for the stratifying variable (clinic). Finally, all Cox regression models assume that the censoring is independent of the failure time -- this is the same assumption that is required for the Kaplan-Meier estimator. 2(d) Dose is the only variable that remains significant at the alpha=0.05 level. Other variables are suggestive (employ, benzo) but not nominally significant. 2(e) We can use the PH test to assess the assumption: Test of proportional hazards assumption Time: Time ---------------------------------------------------------------- | rho chi2 df Prob>chi2 ------------+--------------------------------------------------- female | 0.08399 0.91 1 0.3398 age | 0.05999 0.59 1 0.4437 employ | -0.14640 3.11 1 0.0777 benzo | -0.02553 0.10 1 0.7562 amph | 0.01353 0.03 1 0.8618 alc | -0.00765 0.01 1 0.9256 dose | 0.13167 2.35 1 0.1251 ------------+--------------------------------------------------- global test | 5.64 7 0.5820 ---------------------------------------------------------------- The global test suggest that the PH model is satisfied for all of the variables.