Chapter 5 Simulation Studies
5.3 Simulation Results
The simulation results of coverage probability for the relative potency are given from Table 5.4 to Table 5.6. Table 5.7 presents summary results about the number of coverage probabilities that are within (0.8941, 0.9059). From Table 5.7, 94.9% (356/375) for censored rate=0.2 and 92.5% (347/375) of empirical coverage probabilities are within (0.8941, 0.9059). These results indicate that the confidence intervals by the proposed procedure can provide sufficient coverage probability.
The simulation results of empirical size and empirical power for the relative potency are given from Table 5.8 to Table 5.11. From Table 5.8 to Table 5.10, the empirical size at Δ＝-2 is approximately the same as atΔ=2. This may indicate that the power curve
may be symmetric about 0. Table 5.11 provides summary results about the numbers of empirical sizes are smaller than 0.0536. In general, the empirical size decreases as the number of the doses increases. 98% (147/150) for censored rate=0.2 and 94.6% (142/150)
53
for censored rate=0.3 of empirical sizes are smaller than 0.0536 as shown in Table 5.11.
Although the higher censored rate slightly reduces the size and power, the simulation results show that the proposed procedure can provide an excellent control of type I error rate at the nominal significance under the exponential distribution with a right and random censoring mechanism.
The results given from Table 5.8 to Table 5.10 show that the empirical power increases as the sample size increases on the true relative potency Δ approaches to 0.
In addition, the empirical power for censored rate of 0.2 is greater than that of 0.3. On the other hand, the empirical power increases as the number of doses increases. The magnitude of the slopes seems also has an impact on the empirical power. For example, the empirical power with slope between 0.3 and 0.7 are larger than those with slope -0.1 or -0.9. However, the impact of the magnitude of the slopes on power is less than that of the sample size. Furthermore, sufficient power can be provided even the sample size is small. For example when there are 3 doses, slope is -0.5, a sample size of 40 per group for a total sample size 120 can provided a power almost 90% when the true relative potency is 0.
The empirical power for the proposed method increases asΔincreases from -2 to 0.
It reaches the maximum atΔ= 0. AsΔincreases from 0 to 2, it is a decreasing function of Δ. Figure 5.3 presents a power curve when the slope = -0.3, number of doses is three,
54
and the sample per group is 40. It shows that the power curve is symmetric about 0.
In summary, our proposed procedure not only can provide sufficient coverage probability but also can adequately control the type I error rate at the nominal significance level. Moreover, the proposed method can provide sufficient power even when the sample size is relatively small for some combinations of the number of doses, magnitude of slopes, and the true relative potency.
55
Table 5.1: The results of empirical power for selection of doses based on relative potency for n=100, 3 doses
Slope -0.1 -0.3 -0.5 -0.7 -0.9
Hazard range
Censored rate Censored rate Censored rate Censored rate Censored rate 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.4~0.5 0.3824 0.2818 1.0000 0.9997 0.9998 0.9999 0.9726 0.9842 0.8333 0.8669 0.4~0.6 0.3868 0.2912 0.9999 0.9997 0.9995 0.9998 0.9584 0.9725 0.7752 0.8234
0.4~0.7 0.3895 0.2840 0.9999 0.9998 0.9996 0.9998 0.9549 0.9715 0.7645 0.8097
0.4~0.8 0.3813 0.2871 0.9999 0.9999 0.9999 0.9999 0.9560 0.9716 0.7772 0.8104
0.4~0.9 0.3791 0.2889 0.9999 1.0000 0.9997 0.9999 0.9637 0.9717 0.7929 0.8218 0.5~0.6 0.3783 0.2807 0.9997 0.9994 0.9924 0.9970 0.8508 0.9083 0.5512 0.6595
0.5~0.7 0.3659 0.2832 0.9992 0.9985 0.9890 0.9953 0.8063 0.8756 0.4801 0.6026
0.5~0.8 0.3609 0.2758 0.9994 0.9993 0.9839 0.9946 0.8007 0.8612 0.4736 0.5681
0.5~0.9 0.3677 0.2806 0.9996 0.999 0.9868 0.9936 0.8145 0.8640 0.4896 0.5686 0.6~0.7 0.3555 0.2627 0.9949 0.9977 0.9270 0.9744 0.5946 0.7443 0.2121 0.3758
0.6~0.8 0.3502 0.2678 0.9906 0.9972 0.8986 0.9580 0.5204 0.6837 0.1500 0.3116
0.6~0.9 0.3390 0.2700 0.9912 0.9956 0.8928 0.9493 0.5167 0.6589 0.1467 0.2872
0.7~0.8 0.321 0.2592 0.9340 0.9819 0.6944 0.8637 0.2252 0.4711 0.0000 0.0894
0.7~0.9 0.3193 0.2572 0.9074 0.9712 0.641 0.8244 0.1633 0.3825 0.0000 0.0298 0.8~0.9 0.2692 0.2494 0.6754 0.8902 0.2782 0.6067 0.0249 0.1433 0.0000 0.0000
56
Table 5.2: The results of empirical power for selection of doses based on relative potency for n=100 and 5 doses.
Slope -0.1 -0.3 -0.5 -0.7 -0.9
Hazard Censored rate Censored rate Censored rate Censored rate Censored rate range 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.4~0.5 0.7292 0.6511 1.0000 1.0000 0.9999 1.0000 0.9882 0.9925 0.8952 0.9238 0.4~0.6 0.7215 0.6495 1.0000 1.0000 0.9999 1.0000 0.9800 0.9896 0.8514 0.9021 0.4~0.7 0.7235 0.6524 1.0000 1.0000 0.9999 0.9999 0.9740 0.9846 0.8321 0.8760 0.4~0.8 0.7241 0.6489 1.0000 1.0000 0.9999 0.9999 0.9745 0.9835 0.8340 0.8666 0.4~0.9 0.7292 0.6487 1.0000 1.0000 0.9999 1.0000 0.9743 0.9858 0.8399 0.8641 0.5~0.6 0.7074 0.6392 1.0000 1.0000 0.9973 0.9995 0.9182 0.9570 0.6726 0.7714 0.5~0.7 0.7020 0.6321 1.0000 1.0000 0.9946 0.9989 0.8858 0.9274 0.6022 0.7088 0.5~0.8 0.6959 0.6388 0.9999 1.0000 0.9918 0.9977 0.8636 0.9136 0.5622 0.6605 0.5~0.9 0.6995 0.6333 1.0000 1.0000 0.9929 0.9972 0.8601 0.9072 0.5790 0.6601 0.6~0.7 0.6790 0.6140 0.9989 0.9999 0.9642 0.9900 0.7130 0.8339 0.3443 0.5248 0.6~0.8 0.6693 0.6151 0.9972 0.9997 0.9456 0.9772 0.6398 0.7728 0.2656 0.4369 0.6~0.9 0.6772 0.6106 0.9961 0.9991 0.9419 0.9739 0.6123 0.7434 0.2426 0.4031 0.7~0.8 0.6328 0.5892 0.9739 0.9956 0.8079 0.9310 0.3527 0.5927 0.0084 0.2192 0.7~0.9 0.6177 0.5844 0.9526 0.9916 0.7462 0.8997 0.2820 0.5143 0.0000 0.1375 0.8~0.9 0.5250 0.5401 0.7837 0.9494 0.4295 0.7423 0.0009 0.2669 0.0000 0.0000
57
Table 5.3: The results of empirical power for selection of doses based on relative potency for n=100 and 7 doses.
Slope -0.1 -0.3 -0.5 -0.7 -0.9
Hazard Censored rate Censored rate Censored rate Censored rate Censored rate range 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.4~0.5 0.8880 0.8381 1.0000 1.0000 1.0000 1.0000 0.9922 0.9959 0.9222 0.9456 0.4~0.6 0.8865 0.8384 1.0000 1.0000 0.9999 1.0000 0.9882 0.9932 0.8885 0.9228 0.4~0.7 0.8807 0.8296 1.0000 1.0000 1.0000 1.0000 0.9873 0.9904 0.8759 0.9111 0.4~0.8 0.8828 0.8275 1.0000 1.0000 1.0000 1.0000 0.9850 0.9896 0.8793 0.9030 0.4~0.9 0.8890 0.8307 1.0000 1.0000 1.0000 1.0000 0.9889 0.9911 0.8914 0.9108 0.5~0.6 0.8700 0.8299 1.0000 1.0000 0.9987 0.9999 0.9366 0.9673 0.7208 0.8051 0.5~0.7 0.8711 0.8240 1.0000 1.0000 0.9980 0.9993 0.9143 0.9496 0.6591 0.7598 0.5~0.8 0.8712 0.8252 1.0000 1.0000 0.9971 0.9989 0.9048 0.9389 0.6410 0.7293 0.5~0.9 0.8672 0.8247 1.0000 1.0000 0.9963 0.9985 0.9090 0.9427 0.6608 0.7328 0.6~0.7 0.8534 0.8061 0.9998 1.0000 0.9779 0.9925 0.7455 0.8577 0.4070 0.5638 0.6~0.8 0.8407 0.8043 0.9991 1.0000 0.9594 0.9888 0.6910 0.8168 0.3278 0.5013 0.6~0.9 0.8413 0.8001 0.9989 0.9998 0.9611 0.9868 0.6891 0.7966 0.3282 0.4709 0.7~0.8 0.8065 0.7808 0.9810 0.9969 0.8487 0.9466 0.4136 0.6382 0.0523 0.2694 0.7~0.9 0.7786 0.7716 0.9777 0.9933 0.8007 0.9252 0.3493 0.5790 0.0052 0.2053 0.8~0.9 0.6699 0.7259 0.8233 0.9620 0.4821 0.7812 0.0249 0.3278 0.0000 0.0019
58
Table 5.4: The coverage probability based on relative potency for 3 doses 3 Doses
censored rate=0.2 censored rate=0.3
△ △
Beta N -2 -1 0 1 2 -2 -1 0 1 2
-0.1
40 0.9004 0.8954 0.9020 0.8989 0.9020 0.8978 0.8960 0.8975 0.9004 0.8921 80 0.9015 0.9005 0.8960 0.8963 0.8977 0.9071 0.9009 0.8963 0.9009 0.8967 120 0.9037 0.9004 0.8985 0.8985 0.8979 0.8957 0.8987 0.8992 0.8997 0.9030 160 0.8976 0.8953 0.9003 0.8957 0.8983 0.9008 0.8975 0.9049 0.8978 0.9056 200 0.8999 0.8956 0.9026 0.9037 0.9054 0.8955 0.9047 0.9040 0.9002 0.9028
-0.3
40 0.8973 0.8933 0.8977 0.8988 0.8957 0.8939 0.9003 0.8987 0.8983 0.8960 80 0.8958 0.9016 0.8981 0.8963 0.9007 0.9001 0.9001 0.9018 0.9007 0.9052 120 0.8936 0.8998 0.8987 0.8967 0.8946 0.9036 0.9049 0.8973 0.8968 0.9010 160 0.9035 0.8974 0.9007 0.8975 0.8983 0.9010 0.9043 0.8953 0.8949 0.8942 200 0.9019 0.9005 0.9022 0.9009 0.8960 0.8978 0.8982 0.9002 0.9001 0.8983
-0.5
40 0.9052 0.9022 0.8920 0.8990 0.9033 0.8911 0.9023 0.8998 0.8963 0.9005 80 0.9011 0.9005 0.8982 0.8984 0.9004 0.9000 0.8959 0.9019 0.9038 0.8933 120 0.8956 0.9023 0.8932 0.9006 0.9037 0.8952 0.9000 0.8960 0.9041 0.8981 160 0.8982 0.9033 0.8973 0.8946 0.8989 0.8980 0.9035 0.9012 0.9037 0.9000 200 0.9020 0.8945 0.9042 0.8979 0.9047 0.9045 0.8982 0.8997 0.8962 0.8970
-0.7
40 0.8995 0.8941 0.9061 0.8959 0.9055 0.8994 0.9015 0.9007 0.8970 0.9087 80 0.9006 0.8981 0.8970 0.8965 0.8990 0.8984 0.8996 0.8930 0.8995 0.8994 120 0.8962 0.8978 0.8958 0.9006 0.9001 0.8983 0.8987 0.8989 0.8989 0.8978 160 0.9009 0.8996 0.8974 0.8974 0.8946 0.9011 0.9018 0.8978 0.8996 0.9052 200 0.9012 0.9008 0.8979 0.8995 0.8962 0.8981 0.9019 0.9028 0.8995 0.9050
-0.9
40 0.8944 0.9006 0.8827 0.9084 0.8931 0.9007 0.8888 0.9094 0.8965 0.8898 80 0.8973 0.8962 0.8977 0.8956 0.8996 0.9037 0.9032 0.8987 0.9044 0.8932 120 0.8969 0.9000 0.9014 0.8987 0.9027 0.8965 0.8969 0.8960 0.8992 0.8977 160 0.8978 0.8978 0.8979 0.9000 0.9006 0.8997 0.9002 0.8953 0.8946 0.9005 200 0.9017 0.9004 0.8988 0.8989 0.8991 0.9004 0.9045 0.8947 0.8981 0.8967
The shaded blocks indicate that the coverage probability is between 0.8941 and 0.9059.
59
Table 5.5: The coverage probability based on relative potency for 5 doses 5 Doses
censored rate=0.2 censored rate=0.3
△ △
Beta N -2 -1 0 1 2 -2 -1 0 1 2
-0.1
40 0.9045 0.8983 0.8934 0.9019 0.8958 0.8965 0.8964 0.8962 0.8941 0.8999 80 0.8979 0.8980 0.9019 0.8993 0.8998 0.9043 0.8992 0.8989 0.9034 0.8985 120 0.9015 0.9006 0.9025 0.8961 0.9031 0.9028 0.9008 0.8972 0.9007 0.8955 160 0.8972 0.8937 0.9017 0.9010 0.9012 0.8967 0.8992 0.8983 0.8964 0.8978 200 0.9005 0.8975 0.9023 0.9026 0.8983 0.9015 0.9028 0.9002 0.8987 0.9030
-0.3
40 0.8944 0.8991 0.8963 0.9005 0.9010 0.9002 0.8987 0.9014 0.9022 0.8967 80 0.8965 0.8993 0.8967 0.9017 0.8998 0.9037 0.8946 0.9006 0.9013 0.9013 120 0.9013 0.8949 0.8992 0.9007 0.8986 0.9017 0.8947 0.8946 0.8981 0.9005 160 0.8944 0.9028 0.8979 0.8960 0.8965 0.9017 0.9008 0.8936 0.9015 0.8989 200 0.9007 0.8983 0.9037 0.9055 0.9008 0.9027 0.8971 0.9048 0.8957 0.9000
-0.5
40 0.8966 0.9015 0.8974 0.9040 0.8962 0.8951 0.8996 0.8966 0.8970 0.8964 80 0.9004 0.8969 0.8929 0.8986 0.9012 0.8988 0.9012 0.8975 0.8949 0.8985 120 0.9005 0.8989 0.9035 0.8985 0.9022 0.8983 0.9007 0.8962 0.8982 0.8983 160 0.8989 0.9007 0.8957 0.8992 0.8965 0.9040 0.8998 0.8950 0.9004 0.9006 200 0.8981 0.8993 0.9006 0.8988 0.9050 0.9073 0.9014 0.8952 0.9014 0.8939
-0.7
40 0.8902 0.8981 0.8973 0.8938 0.9018 0.9017 0.9038 0.9001 0.9043 0.9086 80 0.8965 0.8984 0.8986 0.9056 0.8968 0.8997 0.9041 0.9044 0.8966 0.8935 120 0.9045 0.9041 0.9018 0.8959 0.8988 0.8985 0.8946 0.8981 0.8997 0.8939 160 0.9019 0.8984 0.9015 0.8961 0.9010 0.8981 0.8981 0.9005 0.8950 0.9021 200 0.9032 0.9014 0.8988 0.8979 0.9007 0.9005 0.9011 0.8961 0.9002 0.8986
-0.9
40 0.9120 0.8962 0.9003 0.8812 0.8953 0.9028 0.8948 0.9054 0.9072 0.8995 80 0.9039 0.9002 0.8947 0.8951 0.9025 0.9005 0.9002 0.8996 0.8979 0.9035 120 0.9012 0.8958 0.9017 0.9021 0.9045 0.9023 0.8970 0.8999 0.9026 0.8961 160 0.8978 0.8963 0.8948 0.8991 0.9007 0.8983 0.9018 0.8984 0.8985 0.8976 200 0.8994 0.9045 0.8989 0.8993 0.9032 0.9031 0.8961 0.8961 0.8959 0.9008
The shaded blocks indicate that the coverage probability is between 0.8941 and 0.9059.
60
Table 5.6: The coverage probability based on relative potency for 7 doses 7 Doses
censored rate=0.2 censored rate=0.3
△ △
Beta N -2 -1 0 1 2 -2 -1 0 1 2
-0.1
40 0.8985 0.8982 0.9008 0.9017 0.9047 0.8967 0.8967 0.9018 0.8971 0.8980 80 0.9017 0.8953 0.8979 0.8960 0.9038 0.8986 0.8983 0.8929 0.9025 0.8992 120 0.9005 0.9030 0.8992 0.9055 0.8987 0.8975 0.8997 0.8930 0.8950 0.9019 160 0.8992 0.9016 0.9004 0.8992 0.8988 0.9020 0.8982 0.9043 0.9025 0.9009 200 0.9033 0.9038 0.9006 0.9037 0.8963 0.9002 0.8955 0.8998 0.8954 0.9013
-0.3
40 0.8998 0.9006 0.8949 0.9002 0.9009 0.9024 0.9021 0.9031 0.9017 0.8970 80 0.8959 0.8968 0.8910 0.8974 0.9018 0.9014 0.8998 0.8973 0.8963 0.8957 120 0.9001 0.9010 0.9007 0.8970 0.9016 0.9035 0.9054 0.9072 0.9053 0.8982 160 0.8965 0.8972 0.9001 0.8991 0.8986 0.8929 0.8996 0.9048 0.9014 0.8998 200 0.9016 0.9000 0.9005 0.8982 0.8948 0.8987 0.9015 0.9002 0.9011 0.8998
-0.5
40 0.9032 0.9016 0.8977 0.8970 0.9027 0.8947 0.9022 0.8977 0.9021 0.8984 80 0.9016 0.9037 0.8976 0.8982 0.9004 0.8985 0.8996 0.9052 0.8968 0.8978 120 0.9020 0.8954 0.8997 0.8955 0.9060 0.8991 0.9020 0.8979 0.8958 0.8990 160 0.8980 0.9046 0.8970 0.9012 0.8980 0.8962 0.8979 0.9023 0.9018 0.8958 200 0.9002 0.9019 0.8982 0.9050 0.9003 0.9000 0.9001 0.8990 0.9036 0.8992
-0.7
40 0.9038 0.8996 0.8997 0.8986 0.9005 0.9001 0.9039 0.8990 0.8990 0.9062 80 0.9036 0.9033 0.8983 0.8969 0.8958 0.9000 0.9039 0.8920 0.8980 0.8951 120 0.8980 0.8933 0.8986 0.9010 0.9000 0.9060 0.9005 0.8994 0.8938 0.8995 160 0.9005 0.9003 0.9056 0.8958 0.9030 0.8983 0.8969 0.9044 0.8969 0.8981 200 0.8973 0.8965 0.8979 0.8986 0.8980 0.8940 0.9033 0.9023 0.8935 0.8980
-0.9
40 0.9057 0.8979 0.8960 0.8989 0.9040 0.8977 0.8984 0.9028 0.9047 0.9045 80 0.8983 0.8972 0.9044 0.9005 0.9005 0.8978 0.8980 0.8991 0.9019 0.8987 120 0.8973 0.8993 0.8990 0.8974 0.9009 0.8986 0.8966 0.8990 0.8990 0.9005 160 0.8980 0.8988 0.8950 0.8983 0.8971 0.8980 0.8975 0.8965 0.8951 0.8953 200 0.9010 0.8985 0.8984 0.8912 0.8997 0.8991 0.8982 0.9011 0.8986 0.8997
The shaded blocks indicate that the coverage probability is between 0.8941 and 0.9059.
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Table 5.7: A summary of coverage probabilities Censored rate=0.2
dose beta
total
-0.1 -0.3 -0.5 -0.7 -0.9
3 25 23 23 24 22 117
5 23 25 24 23 23 118
7 25 24 24 24 24 121
total 73 72 71 71 69 356
proportion 0.973 0.96 0.946 0.946 0.92 0.949 Censored rate=0.3
Dose beta
total
-0.1 -0.3 -0.5 -0.7 -0.9
3 23 24 23 23 21 114
5 25 24 23 22 24 118
7 23 23 25 19 25 115
Total 71 71 71 64 70 347
proportion 0.946 0.946 0.946 0.853 0.933 0.925
The entries are the numbers of coverage probability that are between 0.8941 and 0.9059.
62
Table 5.8: The empirical size and empirical power based on relative potency for 3 doses
3 Doses
censored rate=0.2 censored rate=0.3
△ △
Beta N -2 -1 0 1 2 -2 -1 0 1 2
-0.1
40 0.0026 0.0044 0.0067 0.0052 0.0023 0.0009 0.0006 0.0013 0.0012 0.0006 80 0.0358 0.1386 0.2164 0.1421 0.0375 0.0241 0.0923 0.1403 0.0955 0.0303 120 0.0489 0.2907 0.5265 0.2918 0.0487 0.0505 0.2426 0.4283 0.2451 0.0440 160 0.0532 0.3718 0.7196 0.3792 0.0504 0.0486 0.3355 0.6327 0.3390 0.0479 200 0.0503 0.4488 0.8376 0.4486 0.0489 0.0525 0.4097 0.7745 0.3999 0.0479
-0.3
40 0.0503 0.5368 0.9314 0.5439 0.0515 0.0529 0.5134 0.9101 0.5107 0.0476 80 0.0527 0.7965 0.9988 0.7974 0.0477 0.0491 0.7667 0.9976 0.7637 0.0479 120 0.0526 0.9181 1.0000 0.9199 0.0526 0.0454 0.8991 0.9999 0.8957 0.0479 160 0.0456 0.9683 1.0000 0.9688 0.0515 0.0500 0.9573 1.0000 0.9589 0.0530 200 0.0528 0.9909 1.0000 0.9897 0.0525 0.0526 0.9827 1.0000 0.9835 0.0486
-0.5
40 0.0397 0.6932 0.8971 0.6917 0.0412 0.0452 0.6937 0.9095 0.7012 0.0451 80 0.0452 0.9523 0.9980 0.9571 0.0478 0.0507 0.9519 0.9992 0.9500 0.0517 120 0.0521 0.9945 1.0000 0.9946 0.0476 0.0495 0.9923 1.0000 0.9926 0.0516 160 0.0526 0.9993 1.0000 0.9992 0.0490 0.0516 0.9987 1.0000 0.9991 0.0490 200 0.0496 1.0000 1.0000 0.9999 0.0455 0.0461 0.9999 1.0000 0.9998 0.0509
-0.7
40 0.0251 0.4454 0.5247 0.4424 0.0229 0.0264 0.4781 0.5663 0.4796 0.0277 80 0.0418 0.9007 0.9146 0.8933 0.0453 0.0459 0.9113 0.9269 0.9082 0.0449 120 0.0478 0.9849 0.9862 0.9850 0.0493 0.0501 0.9885 0.9887 0.9888 0.0481 160 0.0482 0.9977 0.9979 0.9978 0.0505 0.0499 0.9987 0.9989 0.9980 0.0486 200 0.0452 0.9995 1.0000 0.9998 0.0523 0.0513 0.9997 0.9999 0.9998 0.0451
-0.9
40 0.0067 0.1310 0.1507 0.1307 0.0057 0.0078 0.1681 0.1927 0.1762 0.0095 80 0.0327 0.6541 0.6654 0.6551 0.0329 0.0326 0.6876 0.6920 0.6858 0.0364 120 0.0464 0.8839 0.8758 0.8783 0.0458 0.0439 0.8945 0.8911 0.9004 0.0408 160 0.0442 0.9634 0.9575 0.9572 0.0457 0.0513 0.9700 0.9675 0.9648 0.0465 200 0.0481 0.9856 0.9843 0.9850 0.0497 0.0479 0.9909 0.9907 0.9893 0.0520
63
Table 5.9: The empirical size and empirical power based on relative potency for 5 doses
5 Doses
censored rate=0.2 censored rate=0.3
△ △
Beta N -2 -1 0 1 2 -2 -1 0 1 2
-0.1
40 0.0246 0.0839 0.1212 0.0814 0.0275 0.0172 0.0454 0.0674 0.0498 0.0205 80 0.0484 0.3185 0.5788 0.3177 0.0493 0.0452 0.2747 0.4929 0.2807 0.0490 120 0.0487 0.4434 0.8267 0.4471 0.0500 0.0495 0.4066 0.7630 0.4053 0.0503 160 0.0516 0.5398 0.9310 0.5326 0.0480 0.0510 0.4962 0.8921 0.5012 0.0538 200 0.0485 0.6131 0.9738 0.6144 0.0512 0.0490 0.5741 0.9540 0.5718 0.0472
-0.3
40 0.0507 0.6774 0.9860 0.6836 0.0481 0.0493 0.6460 0.9848 0.6602 0.0504 80 0.0495 0.9088 0.9999 0.9143 0.0489 0.0486 0.8972 1.0000 0.8969 0.0484 120 0.0459 0.9779 1.0000 0.9773 0.0498 0.0499 0.9734 1.0000 0.9708 0.0501 160 0.0544 0.9954 1.0000 0.9945 0.0508 0.0508 0.9937 1.0000 0.9937 0.0523 200 0.0485 0.9994 1.0000 0.9993 0.0514 0.0543 0.9982 1.0000 0.9990 0.0491
-0.5
40 0.0509 0.7985 0.9306 0.8046 0.0484 0.0507 0.8136 0.9425 0.8102 0.0458 80 0.0500 0.9848 0.9988 0.9853 0.0499 0.0494 0.9849 0.9991 0.9871 0.0531 120 0.0506 0.9993 1.0000 0.9988 0.0493 0.0538 0.9990 1.0000 0.9990 0.0532 160 0.0502 1.0000 1.0000 0.9999 0.0529 0.0476 1.0000 1.0000 1.0000 0.0516 200 0.0538 1.0000 1.0000 1.0000 0.0499 0.0499 1.0000 1.0000 1.0000 0.0523
-0.7
40 0.0294 0.5455 0.5916 0.5385 0.0298 0.0336 0.5877 0.6312 0.5858 0.0295 80 0.0464 0.9296 0.9377 0.9302 0.0506 0.0477 0.9445 0.9503 0.9469 0.0492 120 0.0457 0.9923 0.9916 0.9922 0.0515 0.0495 0.9936 0.9953 0.9945 0.0546 160 0.0485 0.9989 0.9988 0.9990 0.0490 0.0509 0.9996 0.9995 0.9990 0.0461 200 0.0491 0.9999 0.9997 1.0000 0.0492 0.0499 1.0000 1.0000 0.9997 0.0480
-0.9
40 0.0106 0.2091 0.2204 0.2112 0.0115 0.0134 0.2506 0.2659 0.2596 0.0152 80 0.0359 0.7146 0.7195 0.7163 0.0365 0.0390 0.7537 0.7546 0.7457 0.0384 120 0.0457 0.9139 0.9061 0.9183 0.0446 0.0451 0.9265 0.9324 0.9294 0.0477 160 0.0510 0.9746 0.9704 0.9708 0.0473 0.0501 0.9788 0.9807 0.9797 0.0488 200 0.0515 0.9916 0.9930 0.9923 0.0477 0.0480 0.9950 0.9955 0.9952 0.0499
64
Table 5.10: The empirical size and empirical power based on relative potency for 7 doses
7 Doses
censored rate=0.2 censored rate=0.3
△ △
Beta N -2 -1 0 1 2 -2 -1 0 1 2
-0.1
40 0.0435 0.1866 0.3122 0.1838 0.0402 0.0369 0.1500 0.2327 0.1463 0.0372 80 0.0488 0.4176 0.7863 0.4074 0.0484 0.0532 0.3814 0.7071 0.3704 0.0516 120 0.0517 0.5458 0.9362 0.5468 0.0510 0.0528 0.5086 0.8971 0.5111 0.0523 160 0.0485 0.6525 0.9839 0.6517 0.0503 0.0483 0.6055 0.9751 0.6099 0.0505 200 0.0491 0.7420 0.9959 0.7290 0.0512 0.0467 0.6872 0.9923 0.6865 0.0463
-0.3
40 0.0463 0.7728 0.9967 0.7853 0.0504 0.0499 0.7545 0.9969 0.7525 0.0515 80 0.0504 0.9631 1.0000 0.9626 0.0503 0.0482 0.9485 1.0000 0.9495 0.0585 120 0.0500 0.9944 1.0000 0.9949 0.0482 0.0489 0.9924 1.0000 0.9922 0.0533 160 0.0524 0.9992 1.0000 0.9989 0.0487 0.0537 0.9992 1.0000 0.9990 0.0510 200 0.0514 1.0000 1.0000 0.9999 0.0561 0.0523 0.9999 1.0000 1.0000 0.0497
-0.5
40 0.0475 0.8824 0.9614 0.8829 0.0469 0.0527 0.8847 0.9670 0.8831 0.0533 80 0.0522 0.9954 0.9998 0.9950 0.0483 0.0530 0.9944 0.9999 0.9956 0.0534 120 0.0489 0.9999 1.0000 0.9998 0.0467 0.0512 0.9997 1.0000 0.9998 0.0503 160 0.0513 1.0000 1.0000 1.0000 0.0535 0.0550 1.0000 1.0000 1.0000 0.0518 200 0.0485 1.0000 1.0000 1.0000 0.0499 0.0488 1.0000 1.0000 1.0000 0.0507
-0.7
40 0.0316 0.6488 0.6884 0.6531 0.0363 0.0372 0.6820 0.7106 0.6815 0.0360 80 0.0477 0.9647 0.9649 0.9646 0.0505 0.0502 0.9699 0.9749 0.9688 0.0520 120 0.0495 0.9969 0.9964 0.9969 0.0511 0.0465 0.9983 0.9977 0.9973 0.0518 160 0.0518 0.9995 1.0000 0.9997 0.0503 0.0525 0.9999 0.9996 1.0000 0.0505 200 0.0519 1.0000 1.0000 1.0000 0.0523 0.0546 1.0000 1.0000 1.0000 0.0522
-0.9
40 0.0154 0.3094 0.3182 0.3101 0.0163 0.0206 0.3510 0.3586 0.3410 0.0178 80 0.0393 0.7907 0.7988 0.7972 0.0375 0.0403 0.8165 0.8254 0.8180 0.0422 120 0.0448 0.9463 0.9455 0.9504 0.0435 0.0485 0.9553 0.9563 0.9555 0.0479 160 0.0524 0.9877 0.9844 0.9878 0.0519 0.0481 0.9906 0.9882 0.9893 0.0494 200 0.0503 0.9961 0.9974 0.9968 0.0510 0.0514 0.9981 0.9980 0.9987 0.0504
65
Table 5.11: A summary of empirical sizes.
Censored rate=0.2
dose beta
total
0.1 0.3 0.5 0.7 0.9
3 10 10 10 10 10 50
5 10 9 9 10 10 48
7 10 9 10 10 10 49
total 30 28 29 30 30 147
probability 1 0.93 0.967 1 1 0.98
Censored rate=0.3
dose beta
total
0.1 0.3 0.5 0.7 0.9
3 10 10 10 10 10 50
5 9 9 9 9 10 46
7 10 8 9 9 10 46
total 29 27 28 28 30 142
probability 0.967 0.9 0.93 0.93 1 0.946
The entries are number of sizes that are small than 0.0536
66
Figure 5.1: Flow chart of selection of dose levels
Repeated 10000 times
Set the doses by the following conditions
Dose=3 Dose=5 Dose=7 Ranges of hazard ratio are all combinations of 0.4
to 0.9 by 0.1 Censored rate=0.2 and 0.3 β=-0.1, -0.3, -0.5,-0.7, -0.9
N=40, 80, 120, 160, 200
Obtain the confidence interval of the methods based on relative potency
Empirical power
Calculate the probabilities by the log-hazard regression equations at each dose
f x ln x,
f x ln x,
R R R R
T T T T
λ α
λ α
67
Figure 5.2: Flow chart of simulation process
Repeated 10000 times
Empirical size Empirical power Coverage
probability
Set the doses by the following conditions
Dose=3 Dose=5 Dose=7 Ranges of hazard ratio are all combinations of 0.4
to 0.9 by 0.1 Censored rate=0.2 and 0.3 β=-0.1, -0.3, -0.5,-0.7, -0.9
N=40, 80, 120, 160, 200
Obtain the confidence interval of the methods based on relative potency Calculate the probabilities by the log-hazard regression equations at each dose
f x ln x,
f x ln x,
R R R R
T T T T
λ α
λ α
68
Figure 5.3: The empirical power curve of relative potency from -2.2 to 2.2 by 0.2 when the slope =-0.3, doses = 3 ,censored rate = 0.3 and n = 40.
69