************************** * Biostatistics 513 * Exercise Set 7, 2002 *************************** 1(a) I used STATA for this since it makes the 95% CI's for us automatically: Here are the values around 3 years, and 6 years: Beg. Net Survivor Std. Time Total Fail Lost Function Error [95% Conf. Int.] ------------------------------------------------------------------------------- stage34=0 2.5 43 0 1 0.8795 0.0461 0.7513 0.9440 2.6 42 0 1 0.8795 0.0461 0.7513 0.9440 3.2 41 1 1 0.8581 0.0497 0.7250 0.9297 5.5 22 0 1 0.6415 0.0731 0.4798 0.7646 5.9 21 0 2 0.6415 0.0731 0.4798 0.7646 6 19 1 0 0.6077 0.0767 0.4411 0.7385 6.1 18 0 1 0.6077 0.0767 0.4411 0.7385 stage34=1 2.3 21 1 0 0.5000 0.0791 0.3383 0.6419 2.9 20 0 1 0.5000 0.0791 0.3383 0.6419 3.2 19 1 0 0.4737 0.0792 0.3140 0.6175 5 10 1 1 0.3537 0.0797 0.2040 0.5068 5.1 8 0 1 0.3537 0.0797 0.2040 0.5068 6.3 7 1 0 0.3032 0.0828 0.1543 0.4666 From this we can obtain the following: stage34 = 0 stage34 = 1 S(3yr) 0.8795 (0.7513, 0.9440) 0.5000 (0.3383, 0.6419) S(6yr) 0.6077 (0.4411, 0.7385) 0.3537 (0.2040, 0.5068) This implies that there is a statistically significant difference in the 3 year survival for stage34=1 compared to stage34=0. However, the confidence intervals for 6 year survival for these groups overlap indicating that although the estimated 6 year survival is lower for stage34=1, only 35% survival beyond 6 years, this is not significantly different from the 6 year survival for stage34=0, 61%. 1(b) Here are the log-rank and Wilcoxon (Breslow) tests: Log-rank test for equality of survivor functions ------------------------------------------------ | Events stage34 | observed expected --------+------------------------- 0 | 22 32.58 1 | 28 17.42 --------+------------------------- Total | 50 50.00 chi2(1) = 10.13 Pr>chi2 = 0.0015 Wilcoxon (Breslow) test for equality of survivor functions ---------------------------------------------------------- | Events Sum of stage34 | observed expected ranks --------+-------------------------------------- 0 | 22 32.58 -802 1 | 28 17.42 802 --------+-------------------------------------- Total | 50 50.00 0 chi2(1) = 14.06 Pr>chi2 = 0.0002 Each of these tests rejects the null hypothesis that the two groups have equal survival curves. We find that the Wilcoxon test statistic is larger, which is expected since we saw a large difference in the survival curves for early times, and this statistic places more weight on the comparison for early times. 1(c) Here are the log-rank and the Wilcoxon (Breslow) tests for stage: Log-rank test for equality of survivor functions ------------------------------------------------ | Events stage | observed expected ------+------------------------- 1 | 15 22.57 2 | 7 10.01 3 | 17 14.08 4 | 11 3.34 ------+------------------------- Total | 50 50.00 chi2(3) = 22.76 Pr>chi2 = 0.0000 Wilcoxon (Breslow) test for equality of survivor functions ---------------------------------------------------------- | Events Sum of stage | observed expected ranks ------+-------------------------------------- 1 | 15 22.57 -579 2 | 7 10.01 -223 3 | 17 14.08 244 4 | 11 3.34 558 ------+-------------------------------------- Total | 50 50.00 0 chi2(3) = 23.18 Pr>chi2 = 0.0000 Again we reject the null hypothesis that the groups have the same survival functions. Here the test statistics are quite similar, which is consistent with the KM curves that show clear differences between these groups both ealy in time and at later times. 1(d) Cox regression with dummy variables for stage: LR chi2(3) = 16.26 Log likelihood = -189.08124 Prob > chi2 = 0.0010 ------------------------------------------------------------------------------ _t | _d | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- Istage_2 | 1.067972 .489604 0.143 0.886 .4348436 2.622932 Istage_3 | 1.844227 .655076 1.723 0.085 .9193153 3.69968 Istage_4 | 5.600403 2.350266 4.105 0.000 2.46039 12.74778 ------------------------------------------------------------------------------ This regression estimates the hazard for stage=2 to be 1.068 times the hazard for stage=1 patients. This small increase is not significant as the confidence interval for this comparison contains 1.0 (95% CI 0.435, 2.623). The estimated hazard ratio comparing stage=3 to stage=1 is 1.844 with a 95% confidence interval 0.919, 3.700. Thus although there is an increased risk of death for stage=3 patients relative to stage=1 patients, this increase is not significant at the nominal 5% level. Finally, the hazard ratio comparing stage=4 to stage=1 patients is 5.600 which is statistically significant with a 95% CI (2.460, 12.748). These summaries are in agreement with what the KM plots show: stage 1 and stage 2 are similar; and stage 3 has poorer survival rates while stage 4 patients fare worst of all. 1(e) Here is a Cox regression that uses stage=1,2,3,4 as the predictor variable in a linear log hazard model: LR chi2(1) = 13.10 Log likelihood = -190.66133 Prob > chi2 = 0.0003 ------------------------------------------------------------------------------ _t | _d | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- stage | 1.657713 .2338867 3.582 0.000 1.257226 2.185774 ------------------------------------------------------------------------------ In this model we estimate the following comparisons: stage 2 versus stage 1: hazard ratio of 1.658 stage 3 versus stage 1: hazard ratio of 1.658*1.658 = 2.749 stage 4 versus stage 1: hazard ratio of 1.658*1.658*1.658 = 4.558 This model estimates a higher relative risk for stage 2 and stage 3 patients compared to the dummy variable (unrestricted) model. Also, stage 4 is estimated to have somewhat lower relative risk. This model assumes that a 1 unit difference in stage results in an exp(b)=1.658 increase in the hazard of death. This assumes that 2 versus 1, 3 versus 2, and 4 versus 3, all have the same hazard ratio. 1(f) A likelihood ratio test computes: LR test = 2*( -189.08 - -190.66 ) = 3.16 This test has 2 degrees of freedom ( 3 parameters for the dummy variable model versus 1 parameter for the linear model). The 5% critical value for a chi-square(2) is 5.99. Therefore, since the test statistic is not larger than this critical value we would not reject the null hypothesis and conclude that there is not enough evidence to reject the linear model. Note: I might have reservations using this linear model since the fitted hazards differ systematically from the dummy variable model. However, we do not have enough evidence to reject the linear model in favor of the dummy variable model. This says that the variation we see (away from the linear model) is possibly due to chance. 1(g) SEE the web page for the plots. 2(a) None of the p(PH) values indicate a violation of the PH assumption. 2(b) Use: P = platelet A = age S = sex Based on model 1 we have: h( t, X ) = h0(t) exp( 0.470*P +0.000*A + 0.183*S -0.008*P*A -0.503*P*S ) The hazard ratios are obtained from the Cox regression part (the hazard divided by the baseline hazard, or just the "exp" part above). 2(c) Compare (P=1, A=40, S=0) to (P=0, A=40, S=0): exp( 0.470*(1) + 0.000*(40) +0.183*(0) -0.008*(1)*(40) -0.503*(1)*(0) ) ----------------------------------------------------------------------- exp( 0.470*(0) + 0.000*(40) +0.183*(0) -0.008*(0)*(40) -0.503*(0)*(0) ) = exp( 0.15 ) ----------- exp( 0.00 ) = 1.162 Compare (P=1, A=50, S=1) to (P=0, A=50, S=1): exp( 0.470*(1) + 0.000*(50) +0.183*(1) -0.008*(1)*(50) -0.503*(1)*(1) ) ----------------------------------------------------------------------- exp( 0.470*(0) + 0.000*(50) +0.183*(1) -0.008*(0)*(50) -0.503*(0)*(1) ) = exp(-0.250 ) ----------- exp( 0.183 ) = 0.649 2(d) We can execute a likelihood ratio test comparing model 2 to model 1: terms # parameters -2*log L --------------------------------------------------------------------------- model 1: P + A + S + P*A + P*S 5 306.080 model 2: P + A + S 2 306.505 LR test = 306.505 - 306.080 = 0.425 Comparing this test statistic to a chi-square(df=2) yields a non-significant p-value (clearly since the critical value for chi-square(2) is 5.99, or if you make the p-value calculation you obtain p=0.809). Therefore, we conclude that we fail to reject the null hypothesis that the coefficients of P*A and P*S are zero. 2(e) We can summarize these models as follows: terms platelet=1 vs platelet=0 HR --------------------------------------------------------------------------- model 5: P exp(-0.694) = 0.500 (1/0.500 = 2.000) model 4: P + S exp(-0.705) = 0.494 (1/0.494 = 2.024) model 3: P + A exp(-0.706) = 0.494 (1/0.494 = 2.024) model 2: P + A + S exp(-0.725) = 0.484 (1/0.484 = 2.066) Since the model that adjusts for A and S yields an estimated hazard ratio (HR) of 0.484, and this is not meaningfully different from the unadjusted estimate (HR=0.500), we conclude that it is not necessary to adjust for A and S confounding. However, in presenting these results we would likely choose the model that does the adjustment so that readers can see that we have accounted for these (potentially important) variables.