Chapter 2 Partially Linear Accelerated Failure Time Model
2.5 Numerical Studies
2.5.3 Simulation
We conduct Monte Carlo simulations to assess the performance of our proposed methodology. Similar settings as in Chen (2005) are adopted. The following simulation was done using software R, and quantreg package. We consider the AFT-PL model
log(Ti) = φ(Zi) + βTXi+ εi, (i = 1, ..., n).
The sample size n was set to be 100. The predictors Xi= (X1i, X2i, X3i)T were generated
from a I.I.D. 3-dimensional uniform distribution U (0, 5). The random variable Zi is
correlated with Xi through the relation Zi = 0.1(X1i+ ηi), where ηi follows a uniform
distribution U (0, 5) and independent of all other random variables. We let the true regression coefficient β = (3, 1.5, 1)T. We considered nonlinear effects, φ(Zi) = eZi+
√ Zi,
and quadratic effects, φ(Zi) = Zi2. And εi is independent of (Zi, Xi) and is set to
be N (0, 1) distributed. Censoring random variables were generated according to the formula, Ci = φ(Zi) + XTi β + Ui∗, where Ui∗ follows a uniform distribution U (0, c).
The observed survival time ˜Ti = TiV eCi. Here c was chosen such that the censoring
proportion is approximately 20% (varied from 30% to 10%).
When conducting the local gehan procedure, we chose the kernel function to be the Epanechnikov kernel, i.e., K(u) = 3/4(1 − u2)+. We used mean squared error M SE or
partial mean squared error P M SE (i.e. ( ˆβ − β)TE(XXT)( ˆβ − β)) as the criteria for choosing the bandwidth. The trend of M SE(z0, h) or P M SE(z0, h) with respect to h
2.5 Numerical Studies 37
curves of M SE(z0, h) and P M SE(z0, h), with z0 equal to 0.25, 0.5 and 0.75 respectively,
when choosing φ(Z) = eZ+√Z for example. It implies that h = n−0.2(≈ 0.4) is a good choice of bandwidth, which leads to relatively small value of M SE(z0, h) or P M SE(z0, h)
and balances the bias and variance. Another way is to choose bandwidth via cross validation.
We then compared our local Gehan estimator (Locz0 − AF T ), with z0 set to be
0.25, 0.5, and 0.75, the global Gehan estimator (Glob1 − AF T ) obtained by averaging
the local Gehan estimators at all points Zi for i = 1, ..., n , the global Gehan estimator
(Glob2−AF T ) based on profile Gehan loss function (2.9), and the global Gehan estimator
(Glob3− AF T ), with ng = 20, based on global Gehan loss function (2.10), with other
existing estimators in the literature, the stratified estimator in Chen, Shen and Ying (2005) (SK − AF T ) where K denotes the number of strata, the penalized regression
spline estimator (PSP-AFT) in Qi et al. (2010) with r knots (r=2,4, and 8), the AFT model with true φ plugged in (AF T − φ). We used 5-fold cross-validation to tune the regularization parameter in PSP-AFT method, with tuning parameter chosen to be γ = 10γ∗, where γ∗is a sequence of equal spaced constants. Table 2.1 and 2.2 summarize the mean bias (Bias), standard deviation (SD) and mean squared error (M SE) of ˆβ over 200 Monte Carlo data sets when φ(Z) = eZ +√Z and φ(Z) = Z2, respectively. The results for the estimated coefficients of the predictors X2 and X3, on which Z does
not depend, show that all the rank-based methods work equally well and that their performance does not differ much from that of the estimator with the true φ plugged in. As for the estimated coefficient of the predictor X1 on which Z depends, the proposed
local Gehan estimator outperforms the penalized regression splines estimator as well as the stratified estimator in terms of the Bias, indicating a better estimation accuracy and so does the global Gehan estimator obtained by averaging local Gehan estimators (Glob1− AF T ). Meanwhile, the global Gehan estimator obtained by minimizing (2.9)
(Glob2− AF T ) yields an SD closest to that of AF T − φ, which shows its advantage
over other estimators, although it obtains the largest Bias. Further, compared with the above two global Gehan estimators, the global Gehan estimator obtained by minimizing (2.10) (Glob3− AF T ) yields comparable Bias but largest SD, however, it has an obvious
advantage in shortening the operation time. The Bias of SK−AF T for ˆβ1decreases while
its SD increases obviously when the number of strata becomes large, thus the method for choosing K to reach the bias-variance balance in SK− AF T method remains to be
investigated, leading to a shortcoming of this method over our method. In the setting of our interest, P SP − AF T has the largest bias for ˆβ1compared to stratified method, local
Gehan method, and global Gehan methods (Glob1− AF T and Glob3− AF T ), especially
when the nonparametric component φ(Z) = eZ+√Z, which demonstrates its relatively bad performance for non-polynomial effect.
At the end, we evaluated the resampling scheme for approximating the standard errors of our proposed estimators. We randomly perturbed the local (or global) Gehan loss function 200 times; each time the random variables {Wi}ni=1 were generated from
the Gamma(0.25, 2) distribution. The results are almost the same with different choices of nonparametric function φ(·). Table 2.3 and 2.4 summarize the SD and SE(std(SE)) for the local Gehan estimator at three points z0 = 0.25, 0.50 and 0.75 and for the
2.5 Numerical Studies 39
global Gehan estimators, where SD denotes the standard deviation of 200 estimated ˆβj
(j = 1, 2, 3) and SE(std(SE)) denotes the mean (standard deviation) of 200 estimated standard errors for ˆβj (j = 1, 2, 3) from the resampling scheme. We observe that the
SD and SE are very close to each other, which justifies the accuracy of the resampling scheme for estimating the standard error.
Figure 2.1 The curves of M SE(z0, h) and P M SE(z0, h) when φ(Z) = eZ+
√ Z 0.1 0.2 0.3 0.4 0.5 0.00 0.05 0.10 0.15 0.20 z0=0.25 h MSE, PMEm 0.1 0.2 0.3 0.4 0.5 0.00 0.05 0.10 0.15 0.20 z0=0.5 h MSE, PMEm 0.1 0.2 0.3 0.4 0.5 0.00 0.05 0.10 0.15 0.20 z0=0.75 h MSE, PMEm
Table 2.1 Simulation results for parameter estimation ( ˆβ) when φ(Z) = eZ+√Z in AFT-PL model.
φ(Z) = eZ+√Z
ˆ
β1 βˆ2 βˆ3
Bias SD MSE Bias SD MSE Bias SD MSE
S2− AF T 0.115 0.092 0.022 0.004 0.079 0.006 0.004 0.079 0.006 S5− AF T 0.036 0.107 0.013 0.003 0.077 0.006 0.005 0.081 0.007 S20− AF T 0.003 0.134 0.018 0.009 0.094 0.009 0.007 0.092 0.008 S40− AF T -0.000 0.176 0.031 0.008 0.113 0.013 0.004 0.122 0.015 P SP − AF T (r = 2) 0.044 0.118 0.016 0.001 0.079 0.006 0.004 0.080 0.006 P SP − AF T (r = 4) 0.052 0.117 0.016 0.002 0.079 0.006 0.003 0.080 0.006 P SP − AF T (r = 8) 0.098 0.115 0.023 0.004 0.080 0.006 0.005 0.080 0.006 Loc0.25− AF T -0.003 0.107 0.011 0.003 0.076 0.006 0.004 0.080 0.006 Loc0.5− AF T -0.002 0.107 0.011 0.004 0.074 0.005 0.003 0.080 0.006 Loc0.75− AF T -0.001 0.109 0.012 0.004 0.079 0.006 0.002 0.079 0.006 Glob1− AF T -0.001 0.107 0.011 0.003 0.075 0.006 0.003 0.078 0.006 Glob2− AF T 0.161 0.087 0.034 0.000 0.081 0.007 0.007 0.083 0.007 Glob3− AF T -0.002 0.130 0.017 -0.002 0.089 0.008 0.006 0.098 0.010 AF T − φ -0.004 0.074 0.005 0.003 0.075 0.006 0.003 0.075 0.006 † Notes: (i)S
K − AF T , stratified AFT estimator with K strata; (ii)P SP − AF T , penalized
regression spline estimator with r knots; (iii)Locz0 − AF T , local Gehan estimator at z0; (iv)Glob1− AF T , global Gehan estimator obtained by averaging local Gehan estimators at
all Zi, for i = 1, 2, ..., n; (v)Glob2− AF T , global Gehan estimator based on profile Gehan loss
function (2.9); (vi)Glob3− AF T , global Gehan estimator based on global Gehan loss function
2.5 Numerical Studies 41
Table 2.2 Simulation results for parameter estimation ( ˆβ) when φ(Z) = Z2in AFT-PL model.
φ(Z) = Z2
ˆ
β1 βˆ2 βˆ3
Bias SD MSE Bias SD MSE Bias SD MSE
S2− AF T 0.041 0.090 0.010 0.004 0.076 0.006 0.004 0.077 0.006 S5− AF T 0.015 0.106 0.012 0.003 0.077 0.006 0.004 0.080 0.006 S20− AF T 0.001 0.135 0.018 0.008 0.094 0.009 0.008 0.093 0.009 S40− AF T -0.001 0.175 0.031 0.008 0.111 0.012 0.004 0.121 0.015 P SP − AF T (r = 2) 0.022 0.104 0.011 0.002 0.078 0.006 0.003 0.078 0.006 P SP − AF T (r = 4) 0.024 0.099 0.010 0.003 0.078 0.006 0.003 0.079 0.006 P SP − AF T (r = 8) 0.051 0.094 0.011 0.003 0.077 0.006 0.004 0.079 0.006 Loc0.25− AF T -0.003 0.107 0.012 0.003 0.075 0.006 0.005 0.080 0.006 Loc0.5− AF T -0.002 0.107 0.011 0.004 0.074 0.005 0.004 0.080 0.006 Loc0.75− AF T -0.001 0.110 0.012 0.004 0.079 0.006 0.002 0.078 0.006 Glob1− AF T -0.001 0.107 0.011 0.003 0.075 0.006 0.003 0.078 0.006 Glob2− AF T 0.069 0.080 0.011 0.002 0.076 0.006 0.005 0.077 0.006 Glob3− AF T -0.002 0.130 0.017 -0.002 0.089 0.008 0.006 0.098 0.010 AF T − φ -0.004 0.074 0.005 0.003 0.075 0.006 0.003 0.075 0.006 † Notes: (i)S
K − AF T , stratified AFT estimator with K strata; (ii)P SP − AF T , penalized
regression spline estimator with r knots; (iii)Locz0 − AF T , local Gehan estimator at z0; (iv)Glob1− AF T , global Gehan estimator obtained by averaging local Gehan estimators at
all Zi, for i = 1, 2, ..., n; (v)Glob2− AF T , global Gehan estimator based on profile Gehan loss
function (2.9); (vi)Glob3− AF T , global Gehan estimator based on global Gehan loss function
Table 2.3 Standard deviations of the local and global Gehan estimators when φ(Z) = eZ+√Z in AFT-PL model. φ(Z) = eZ+√Z, (W i+ Wj) Wi∼ Gamma(0.25, 2) ˆ β1 βˆ2 βˆ3
SD SE(std(SE)) SD SE(std(SE)) SD SE(std(SE))
Loc0.25− AF T 0.107 0.111(0.022) 0.075 0.084(0.015) 0.080 0.085(0.017) Loc0.5− AF T 0.107 0.098(0.018) 0.074 0.075(0.012) 0.076 0.074(0.012) Loc0.75− AF T 0.109 0.100(0.017) 0.079 0.076(0.010) 0.079 0.076(0.011) Glob1− AF T 0.107 0.103(0.018) 0.075 0.078(0.012) 0.078 0.078(0.011) Glob2− AF T 0.087 0.079(0.012) 0.081 0.078(0.012) 0.083 0.079(0.012) Glob3− AF T 0.130 0.125(0.022) 0.089 0.084(0.010) 0.098 0.092(0.015) † Notes: (i)Loc
z0− AF T , local Gehan estimator at z0; (ii)Glob1− AF T , global Gehan
estimator obtained by averaging local Gehan estimators at all Zi, for i = 1, 2, ..., n;
(iii)Glob2− AF T , global Gehan estimator based on profile Gehan loss function (2.9);
2.5 Numerical Studies 43
Table 2.4 Standard deviations of the local and global Gehan estimator when φ(Z) = Z2 in AFT-PL model.
φ(Z) = Z2, (W
i+ Wj) Wi∼ Gamma(0.25, 2)
ˆ
β1 βˆ2 βˆ3
SD SE(std(SE)) SD SE(std(SE)) SD SE(std(SE))
Loc0.25− AF T 0.107 0.111(0.022) 0.075 0.084(0.015) 0.080 0.085(0.017) Loc0.5− AF T 0.107 0.098(0.018) 0.074 0.075(0.012) 0.076 0.074(0.012) Loc0.75− AF T 0.110 0.100(0.017) 0.079 0.076(0.010) 0.079 0.076(0.011) Glob1− AF T 0.107 0.103(0.018) 0.075 0.078(0.012) 0.078 0.078(0.011) Glob2− AF T 0.080 0.072(0.011) 0.076 0.073(0.011) 0.077 0.073(0.011) Glob3− AF T 0.130 0.125(0.022) 0.089 0.084(0.010) 0.098 0.092(0.015) † Notes: (i)Loc
z0− AF T , local Gehan estimator at z0; (ii)Glob1− AF T , global Gehan
estimator obtained by averaging local Gehan estimators at all Zi, for i = 1, 2, ..., n;
(iii)Glob2− AF T , global Gehan estimator based on profile Gehan loss function (2.9);