Simulation Studies
4.1 Data generation
In this section, we conduct simulation studies to assess the finite sample prop-erties of our proposed estimators. We consider the following AFT model,
log ˜T = β0+ β1X + β2Z + ,
where the covariate X was generated from a standard normal distribution, Z was from a Bernoulli distribution with P (Z = 1) = 0.5 and followed a standard nor-mal distribution, above leading to ˜T following to that a log-normal distribution.
We set β0 = 2, β1 = 1 and β2 = 1 for data generation. The censoring time C was generated from a uniform distribution [0, c] with c chosen to depend on the desired percentage of censoring. We considerd censoring rates of approximately 60% and 80% with the corresponding c values, 19.55 and 7.18, respectively. For our PDS design, we partitioned all the cases into three strata by quantiles q1 and q3 of X. We then randomly select an SRS sample of size n0 from N = 2000. We considered two pairs of the cutpoints, (0.3, 0.7) and (0.1, 0.9) quantiles,
respec-tively, to investigate the impact of different cutpoints for our PDS design. In the second phase, we set c1 = c3 = 80% or 90% and we denote ˆβP DS1 for 80% and βˆP DS2 for 90%. We compare ˆβP DS1 and ˆβP DS2 with two competing estimators:
(i) ˆβSRS, from a simple random sample with the same sample size as the PDS design, and
(ii) ˆβODS, the estimator from the ODS design (Zhou et al., 2002). To obtained the ODS sample, we partitioned all the cases into three strata by the quan-tiles of their failure time (B1S B2S B3). The cutpoint is either (0.3, 0.7) or (0.1, 0.9) quantiles, similar to the PDS design. The supplemental samples of the ODS design are drawn from ˜T ∈ Bk, k = 1, 3, and δ = 1. We estimate the estimator with R package, aftgee
Results are based on 2000 independent simulation runs.
4.2 Results
In Tables 4.1 and 4.2, we fixed the size of the simple random sample, whereas we considered balanced sizes for the two tails in Table 4.1 and unbalanced size in Table 4.2. Below is our summary.
(i) ˆβP DS1, ˆβP DS2, ˆβODS and ˆβSRS are all consistent.
(ii) The sample standard deviation is close to the estimated of standard errors based on the 2000 simulations, which is also as expected. Then we compared the proposed estimator with other estimator by the estimated of standard errors.
(iii) ˆβP DS1, ˆβP DS2 and ˆβODS, are more efficient than ˆβSRS since ˆβSRS obtained the largest standard error.
(iv) When the overall sample sizes increase, ˆβP DS1 and ˆβP DS2 are more efficient than ˆβODS because standard error of ˆβODS is larger than ˆβP DS1 and ˆβP DS2. (v) As the sample size increases from 100 to 300, all the standard errors decrease,
as expected.
(vi) For ˆβP DS2 and ˆβP DS1 in all the designs, we found that the standard error of ˆβP DS2 is smaller than that of ˆβP DS1 so that ˆβP DS2 is more efficient than βˆP DS1.
Due to the low number of failures, we considered the supplemental sample drawn by the proportions. In Tables 4.3, 4.4 and 4.5, we presented results for different sampling proportions balanced and unbalanced of the supplemental samples out of the PDS sample. Note that the total sample size is not fixed. Then, we compared proposed estimator with the simple random sample estimator. We summarize our findings below.
(i) The standard errors of ˆβP DS1 and ˆβP DS2 are smaller than ˆβSRS, except in some cases with censoring rate of 60%. Mostly ˆβP DS1 and ˆβP DS2 are more efficient than ˆβSRS.
(ii) The bias for the proportion (0.5, 0.8) is smaller than other proportions.
In Table 4.6, we consider the situation where the supplemental samples are drawn from those failed earlier or much later, i.e., the cutpoint (0.1, 0.9). We note that
(i) in the case with cutpoint (0.1,0.9) and censoring rate 80%, the bias of ˆβP DS1 and ˆβP DS2 are smaller than those with censoring rate 60%;
(ii) the efficiency gains are higher when the cutpoint is further out i.e., (0.1, 0.9);
(iii) as the simple random sample size increases, the number of failures is much smaller. Thus we can rarely draw the cases with cutpoint (0.1, 0.9) quan-tiles. We instead consider the cutpoint quantiles of (0.2, 0.8) when the random sample size is greater than 200. The results are similar to the aforementioned.
(iv) Under some settings,such as the sample size of the SRS being 100 and the censoring rate of 60%, ˆβP DS obtained slightly the larger standard errors than ˆβSRS. We will have further discussion later.
In Table 4.7, we consider the design with asymmetric cutpoint quantiles,(0.1, 0.7). We compared the results with Table 4.1 and summarize our findings below.
(i) ˆβP DS1 and ˆβP DS2 are still more efficient than ˆβSRS since ˆβSRS obtained the largest standard error under both symmetric and asymmetric cutpoints (ii) The bias of ˆβP DS with (0.1, 0.7) quantiles is smaller than ˆβP DS with (0.3,
0.7) quantiles.
(iii) The bias of ˆβODS with (0.1, 0.7) quantiles is larger than ˆβODS with (0.3, 0.7) quantiles.
(iv) When the simple sample size is 300, if is hard to draw the supplemental sam-ple for fail and the probability is 90%. Then, we instead consider cutpoint quantiles to (0.2, 0.7). Most results are similar to the cutpoint quantiles (0.1, 0.7).
We also consider that followed a Gumble distribution in Table 4.8. We com-pared the results of Gumble distribution with the results of Normal distribution.
The results are very similar to those in Table 4.1. It concludes that our proposed estimators are robust under different error distributions.
Table 4.1: Results are based on the model log T = β
1X + β
2Z, where X ∼ N (0, 1) and Z ∼ B(0.5), and the cutpoints for the design were 0.3 and 0.7 sample quantiles.
β1 = 1 β2 = 1
(n0, n1, n3) Censoring Mean ESE SSD Mean ESE SSD (100,10,10) 0.6 βˆSRS 1.017 0.137 0.141 1.017 0.249 0.248
βˆODS 0.989 0.119 0.119 1.011 0.239 0.240 βˆP DS1 1.113 0.122 0.124 1.015 0.250 0.223 βˆP DS2 1.104 0.119 0.123 1.076 0.243 0.235 0.8 βˆSRS 1.026 0.189 0.210 1.021 0.337 0.376 βˆODS 0.981 0.133 0.130 1.027 0.259 0.269 βˆP DS1 1.097 0.153 0.157 1.037 0.306 0.283 βˆP DS2 1.096 0.150 0.162 1.051 0.304 0.301 (200,10,10) 0.6 βˆSRS 1.006 0.099 0.100 1.003 0.180 0.183 βˆODS 0.983 0.106 0.105 1.011 0.211 0.209 βˆP DS1 1.099 0.093 0.091 1.027 0.182 0.169 βˆP DS2 1.085 0.087 0.089 1.077 0.173 0.171 0.8 βˆSRS 1.010 0.140 0.147 1.020 0.249 0.259 βˆODS 0.978 0.114 0.113 1.017 0.219 0.231 βˆP DS1 1.086 0.118 0.119 1.045 0.231 0.218 βˆP DS2 1.079 0.115 0.117 1.074 0.222 0.215 (300,10,10) 0.6 βˆSRS 1.005 0.083 0.084 1.014 0.148 0.152 βˆODS 0.990 0.100 0.103 1.013 0.198 0.198 βˆP DS1 1.091 0.078 0.075 1.024 0.152 0.140 βˆP DS2 1.078 0.073 0.072 1.078 0.142 0.138 0.8 βˆSRS 1.004 0.116 0.122 1.029 0.204 0.205 βˆODS 0.983 0.106 0.105 1.020 0.202 0.203 βˆP DS1 1.083 0.102 0.099 1.050 0.193 0.178 βˆP DS2 1.068 0.098 0.097 1.081 0.186 0.184 ESE, the average of the estimates of standard errors; SSD, sample standard deviation.
Table 4.2: Results are based on the model log T = β
1X + β
2Z, where X ∼ N (0, 1) and Z ∼ B(0.5), and the cutpoints for the design were 0.3 and 0.7 sample quantiles.
β1 = 1 β2 = 1
(n0, n1, n3) Censoring Mean ESE SSD Mean ESE SSD (100,20,10) 0.6 βˆSRS 1.010 0.131 0.134 1.007 0.237 0.241
βˆODS 1.030 0.105 0.106 1.066 0.220 0.221 βˆP DS1 1.114 0.118 0.123 1.037 0.243 0.226 βˆP DS2 1.115 0.114 0.124 1.078 0.234 0.227 0.8 βˆSRS 1.022 0.184 0.195 1.030 0.329 0.345 βˆODS 1.036 0.119 0.120 1.076 0.243 0.256 βˆP DS1 1.147 0.141 0.148 1.018 0.284 0.251 βˆP DS2 1.140 0.140 0.155 1.049 0.280 0.260 (200,20,10) 0.6 βˆSRS 1.007 0.097 0.098 1.009 0.175 0.180 βˆODS 1.027 0.093 0.096 1.062 0.194 0.195 βˆP DS1 1.116 0.089 0.091 1.035 0.178 0.161 βˆP DS2 1.094 0.084 0.087 1.083 0.168 0.163 0.8 βˆSRS 1.014 0.137 0.149 1.025 0.242 0.247 βˆODS 1.033 0.102 0.100 1.086 0.207 0.211 βˆP DS1 1.137 0.111 0.114 1.052 0.218 0.194 βˆP DS2 1.120 0.109 0.111 1.070 0.211 0.194 (300,20,10) 0.6 βˆSRS 1.003 0.081 0.080 1.006 0.146 0.149 βˆODS 1.024 0.088 0.089 1.056 0.182 0.185 βˆP DS1 1.101 0.075 0.076 1.031 0.147 0.138 βˆP DS2 1.103 0.070 0.070 1.074 0.139 0.134 0.8 βˆSRS 1.011 0.115 0.122 1.016 0.200 0.212 βˆODS 1.033 0.093 0.093 1.082 0.190 0.195 βˆP DS1 1.123 0.095 0.093 1.047 0.183 0.166 βˆP DS2 1.124 0.090 0.090 1.064 0.173 0.157 ESE, the average of the estimates of standard errors; SSD, sample standard deviation.
Table 4.3: Results are based on the model log T = β
1X +β
2Z, where X ∼ N (0, 1) and Z ∼ B(0.5), the size of supplemental is selected by the fixed proportions, and the cutpoints for the design were 0.3 and 0.7 sample quantiles.
β1 = 1 β2 = 1
n1+ n3 Proportions Censoring Mean ESE SSD Mean ESE SSD 35 (0.5,0.5) 0.6 βˆSRS 1.018 0.130 0.132 1.014 0.234 0.238
Table 4.4: Results are based on the model log T = β
1X +β
2Z, where X ∼ N (0, 1) and Z ∼ B(0.5), the size of supplemental is selected by the fixed proportions, and the cutpoints for the design were 0.3 and 0.7 sample quantiles.
β1 = 1 β2 = 1
n1+ n3 Proportions Censoring Mean ESE SSD Mean ESE SSD 25 (0.5,0.5) 0.6 βˆSRS 1.004 0.099 0.099 1.002 0.178 0.177
Table 4.5: Results are based on the model log T = β
1X +β
2Z, where X ∼ N (0, 1) and Z ∼ B(0.5), the size of supplemental is selected by the fixed proportions, and the cutpoints for the design were 0.3 and 0.7 sample quantiles.
β1 = 1 β2 = 1
n1+ n3 Proportions Censoring Mean ESE SSD Mean ESE SSD 20 (0.5,0.5) 0.6 βˆSRS 1.002 0.083 0.084 1.006 0.148 0.148
Table 4.6: Results are based on the model log T = β
1X + β
2Z, where X ∼ N (0, 1) and Z ∼ B(0.5), and the supplemental sample include all cases.
β1 = 1 β2 = 1
(n0, n1, n3) Cutpoint Censoring Mean ESE SSD Mean ESE SSD (100,44,24) (0.3,0.7) 0.6 βˆSRS 1.014 0.117 0.119 1.015 0.211 0.220
βˆP DS1 1.105 0.124 0.131 1.019 0.249 0.227
(100,29,12) βˆSRS 1.008 0.127 0.133 1.005 0.229 0.237
βˆP DS2 1.113 0.122 0.129 1.069 0.245 0.230 (100,27,24) 0.8 βˆSRS 1.018 0.173 0.189 1.028 0.309 0.341 βˆP DS1 1.091 0.158 0.177 1.024 0.309 0.286
(100,29,11) βˆSRS 1.012 0.126 0.132 1.001 0.228 0.226
βˆP DS2 1.108 0.122 0.129 1.075 0.244 0.224 (100,18,5) (0.1,0.9) 0.6 βˆSRS 1.007 0.134 0.138 1.015 0.243 0.249 βˆP DS1 1.024 0.118 0.117 1.131 0.238 0.241
(100,15,4) βˆSRS 1.007 0.135 0.134 1.016 0.246 0.249
βˆP DS2 1.021 0.114 0.115 1.156 0.231 0.238 (100,17,10) 0.8 βˆSRS 1.024 0.184 0.205 1.036 0.332 0.365 βˆP DS1 1.047 0.162 0.160 1.089 0.321 0.309
(100,16,10) βˆSRS 1.019 0.186 0.204 1.035 0.336 0.364
βˆP DS2 1.042 0.168 0.176 1.098 0.328 0.319 (200,34,13) (0.3,0.7) 0.6 βˆSRS 1.006 0.095 0.094 1.004 0.170 0.169 βˆP DS1 1.103 0.093 0.098 1.028 0.181 0.160
(200,22,5) βˆSRS 1.010 0.098 0.099 1.007 0.177 0.176
βˆP DS2 1.097 0.090 0.096 1.076 0.175 0.168 (200,19,12) 0.8 βˆSRS 1.009 0.138 0.150 1.017 0.241 0.251 βˆP DS1 1.096 0.123 0.130 1.050 0.232 0.210
(200,22,5) βˆSRS 1.010 0.098 0.097 1.009 0.176 0.180
βˆP DS2 1.094 0.090 0.090 1.080 0.175 0.164 (300,30,9) (0.3,0.7) 0.6 βˆSRS 1.002 0.080 0.080 1.006 0.144 0.147 βˆP DS1 1.097 0.077 0.081 1.027 0.149 0.135
(300,19,4) βˆSRS 1.002 0.081 0.083 1.007 0.147 0.146
βˆP DS2 1.088 0.075 0.077 1.066 0.144 0.138 (300,15,7) 0.8 βˆSRS 1.002 0.117 0.122 1.013 0.204 0.206 βˆP DS1 1.097 0.105 0.109 1.053 0.196 0.174
(300,19,3) βˆSRS 1.007 0.082 0.084 1.015 0.148 0.152
βˆ 1.089 0.075 0.079 1.069 0.144 0.137
Table 4.7: Results are based on the model log T = β
1X + β
2Z, where X ∼ N (0, 1) and Z ∼ B(0.5), and the cutpoints for the design were 0.1 and 0.7 sample quantiles.
β1 = 1 β2 = 1
(n0, n1, n3) Censoring Mean ESE SSD Mean ESE SSD (100,10,10) 0.6 βˆSRS 1.018 0.141 0.145 1.020 0.254 0.254
βˆODS 1.109 0.117 0.110 1.158 0.239 0.244 βˆP DS1 1.102 0.122 0.123 1.022 0.248 0.245 βˆP DS2 1.066 0.117 0.125 1.035 0.233 0.240 0.8 βˆSRS 1.031 0.190 0.206 1.021 0.340 0.389 βˆODS 1.106 0.145 0.135 1.172 0.284 0.306 βˆP DS1 1.104 0.153 0.156 1.044 0.305 0.303 βˆP DS2 1.087 0.154 0.159 1.053 0.300 0.304 (200,10,10) 0.6 βˆSRS 1.015 0.103 0.104 1.017 0.183 0.181 βˆODS 1.110 0.101 0.093 1.156 0.203 0.204 βˆP DS1 1.087 0.093 0.092 1.019 0.178 0.175 βˆP DS2 1.048 0.086 0.088 1.037 0.167 0.174 0.8 βˆSRS 1.018 0.142 0.149 1.023 0.248 0.258 βˆODS 1.096 0.123 0.112 1.149 0.233 0.238 βˆP DS1 1.075 0.119 0.119 1.037 0.223 0.226 βˆP DS2 1.057 0.118 0.118 1.059 0.219 0.229 (300,10,10) 0.6 βˆSRS 1.011 0.086 0.085 1.007 0.152 0.155 βˆODS 1.115 0.096 0.088 1.121 0.190 0.189 βˆP DS1 1.074 0.078 0.078 1.007 0.147 0.145 0.8 βˆSRS 1.011 0.118 0.122 1.011 0.205 0.206 βˆODS 1.097 0.111 0.101 1.128 0.210 0.217 βˆP DS1 1.069 0.102 0.094 1.041 0.188 0.189 ESE, the average of the estimates of standard errors; SSD, sample standard deviation.
Table 4.8: Results are based on the model log T = β
1X + β
2Z, where X ∼ N (0, 1), Z ∼ B(0.5) and ∼ Gumble(0, 1), and the cutpoints for the design were 0.3 and 0.7 sample quantiles.
β1 = 1 β2 = 1
(n0, n1, n3) Censoring Mean ESE SSD Mean ESE SSD (100,10,10) 0.6 βˆSRS 1.010 0.127 0.134 1.011 0.231 0.228
βˆODS 0.999 0.106 0.104 1.031 0.210 0.209 βˆP DS1 1.100 0.109 0.106 0.999 0.233 0.208 βˆP DS2 1.099 0.108 0.108 1.044 0.230 0.212 0.8 βˆSRS 1.014 0.161 0.172 1.015 0.285 0.312 βˆODS 0.983 0.112 0.108 1.021 0.213 0.218 βˆP DS1 1.085 0.124 0.129 1.022 0.257 0.227 βˆP DS2 1.079 0.124 0.127 1.050 0.258 0.244 (200,10,10) 0.6 βˆSRS 1.004 0.093 0.092 1.014 0.170 0.175 βˆODS 0.991 0.093 0.089 1.022 0.185 0.184 βˆP DS1 1.089 0.083 0.082 1.000 0.171 0.157 βˆP DS2 1.076 0.080 0.078 1.037 0.165 0.157 0.8 βˆSRS 1.007 0.117 0.123 1.022 0.206 0.214 βˆODS 0.982 0.095 0.093 1.016 0.180 0.186 βˆP DS1 1.080 0.099 0.096 1.022 0.195 0.176 βˆP DS2 1.067 0.094 0.094 1.052 0.190 0.181 (300,10,10) 0.6 βˆSRS 1.002 0.077 0.078 1.016 0.140 0.137 βˆODS 0.995 0.088 0.089 1.026 0.173 0.170 βˆP DS1 1.082 0.071 0.066 0.998 0.141 0.130 βˆP DS2 1.071 0.068 0.067 1.039 0.136 0.128 0.8 βˆSRS 1.008 0.098 0.103 1.014 0.170 0.174 βˆODS 0.985 0.089 0.085 1.012 0.164 0.168 βˆP DS1 1.072 0.084 0.081 1.019 0.162 0.149 βˆP DS2 1.060 0.081 0.078 1.041 0.157 0.151 ESE, the average of the estimates of standard errors; SSD, sample standard deviation.