Simulation Study II-3 Mild Censoring

Simulation II-3 was performed using survival data generated with θ = 2, λ = 1, and mild censoring (37.8 %). The results are presented in Table 5 and Figure 17 trough Figure 22. The value of the test statistic for proportional hazards assumption is 12.34 (p <.0001). This test statistic states that in true model, the effects of the covariate are varying over time.

The true regression coefficient, the estimated regression coefficient based on Gray’s time- varying coefficients model, and the estimated regression coefficient based on Cox PH model at each 9 time knot are presented in Table5. While the estimated regression coefficient based on Cox’s model are constant over time, .325 (p<.001), the estimated regression coefficient based on Gray’s model, (.063, .129, .230, .303, .358, .459, .538, .559, .538), and true regression coefficient, (.094, .154, .2.12, .292, .354, .410, .462, .522, .608), are changing over time. The estimated regression coefficients based on Gray’s model is close to true regression coefficient except for early time point. The results of Table 5were plotted in Figure 17and Figure 18. The estimates of survival functions were calculated and plotted based on Gray’s piecewise- constant time-varying model and Cox PH model at (Z = 2, 4, 6, 8) (Figure 19). Finally, pseudo residual plots and true-Gray residual plots along with smoothed average against the estimated survival rate based on fitted model at each 9 time point were conducted and presented in Figure 20, Figure 21, and Figure 22 to assess the goodness-of-fit for Gray’s time-varying coefficients model and Cox PH model.

Table 5: Simulation II-3 Compare true value of covariate effect β(t) to estimated covariate effect ˆβ(t) based on Gray’s time-varying coefficient model and Cox PH model; true value of

covariate effect β(t) = 2t, λ = 1, and 37.8 % censoring

Time Knots .047 .077 .106 .146 .177 .205 .231 .261 .304 True β(t): θ x Time (θ=2) .094 .154 .2.12 .292 .354 .410 .462 .522 .608 ˆ β(t) based on Gray’s model .063 .129 .230 .303 .358 .459 .538 .559 .538 ˆ β based on Cox model .353 .353 .353 .353 .353 .353 .353 .353 .353

Figure 17: Simulation II-3 Plot of log hazard ratio vs. time for the covariate Z based on Gray’s time-varying coefficients models and Cox PH model (θ=2, λ=1, and 34.6% censoring)

Figure 18: Simulation II-3 Plot of the true value of covariate effect β(t) , the estimated covariate effect ˆβ(t) based on Gray’s time-varying coefficient model and Cox PH model vs.

Figure 19: Simulation II-3 Plot of the estimated survival function based on Gray’s time-varying coefficients and Cox PH model,

Figure 20: Simulation II-3 Pseudo residual vs. the estimated survival function based on Gray’s time-varying coefficients model at each time knot: true value of covariate effect β(t) = 2t, λ = 1, and 34.6 % censoring

P seudoGray residual = ˆSi(t) − ˆSGray(t|Zi)

Figure 21: Simulation II-3 True-Gray residual vs. the estimated survival function based on Gray’s time-varying coefficients model at each time knot: true value of covariate effect β(t) = 2t, λ = 1, and 34.6 % censoring

Figure 22: Simulation II-3 Pseudo residual vs. the estimated survival function based on Cox PH model at each time knot: true value of covariate effect β(t) = 2t, λ = 1, and 34.6 % censoring

P seudoCox residual = ˆSi(t) − ˆSCox(t|Zi)

3.3.5 Simulation Study II-4 Heavy Censoring

Simulation II-4 was performed using survival data generated with θ = 2, λ = 1, and heavy censoring (71%). The results are presented in Table 6 and Figure 23trough Figure 28. The value of the test statistic for proportional hazards assumption is 5.427 (p = .023). This test statistic states that in true model, the effects of the covariate are varying over time.

The true regression coefficient, the estimated regression coefficient based on Gray’s time- varying coefficients model, and the estimated regression coefficient based on Cox PH model at each 9 time knot are presented in Table 6. While the estimated regression coefficient based on Cox’s model are constant over time as .310, the estimated regression coefficient based on Gray’s model, (.150, .192, .212, .236, .293, .410, .540, .583, .565), and true regression coefficient, (.054, .130, .172, .216, .282, .304, .356, .416, .516), are changing over time. The estimated regression coefficients based on Gray’s model is close to true regression coefficient except for early time points. The results of Table6were plotted in Figure23and Figure 24. The estimates of survival functions were calculated and plotted based on Gray’s piecewise- costant time-varying model and Cox PH model at (Z = 2, 4, 6, 8) (Figure 25). Finally, pseudo residual plots and true-Gray residual plots along with smoothed average against the estimated survival rate based on fitted model at each 9 time point are presented in Figure 26, Figure27, and Figure28to assess the goodness-of-fit for Gray’s time-varying coefficients model and Cox PH model.

Table 6: Simulation II-4 Compare true value of covariate effect β(t) to estimated covariate effect ˆβ(t) based on Gray’s time-varying coefficient model and Cox PH model; true value of

covariate effect β(t) = 2t, λ = 1, and 66 % censoring

Time Knots .027 .065 .086 .108 .141 .152 .178 .208 .258 True β(t): θ x Time (θ=2) .054 .130 .172 .216 .282 .304 .356 .416 .516 ˆ β(t) based on Gray’s model .150 .192 .212 .236 .293 .410 .540 .583 .565 ˆ β based on Cox model .326 .326 .326 .326 .326 .326 .326 .326 .326

Figure 23: Simulation II-4 Plot of log hazard ratio vs. time for the covariate Z based on Gray’s time-varying coefficients models and Cox PH model (θ=2, λ=1, and 66% censoring)

Figure 25: Simulation II-4 Plot of the estimated survival function based on Gray’s time-varying coefficients and Cox PH model, and true survival function at (z=2,4,6, and 8): ; true value of covariate effect β(t) = 2t, λ = 1, and 66 % censoring

Figure 26: Simulation II-4 Pseudo residual vs. the estimated survival function based on Gray’s time-varying coefficients model at each time knot: true value of covariate effect β(t) = 2t, λ = 1, and 66 % censoring

Figure 27: Simulation II-4 True-Gray residual vs. the estimated survival function based on Gray’s time-varying coefficients model at each time knot: true value of covariate effect β(t) = 2t, λ = 1, and 66 % censoring

T rueGray residual = ST rue(t|Zi) − ˆSGray(t|Zi)

Figure 28: Simulation II-4 Pseudo residual vs. the estimated survival function based on Cox PH model at each time knot: true value of covariate effect β(t) = 2t, λ = 1, and 66 % censoring

3.4 RESULTS FOR DIFFERENT THETA, LAMBDA, AND PERCENTAGE OF CENSORING

Table 7 and Table 8 present the results of true value of survival function and es- timated β based on Gray’s piecewise-constant time-varying coefficient model and Cox proportional hazards ratio model for different theta, lambda, and censoring percentage.