The accelerated failure time (AFT) model is a linear model under the log-transformation of survival time that has been introduced as a useful alternative to the proportional hazards (PH) model. In this paper we propose variable-selection procedures of fixed effects in a parametric AFT model using penalized likelihood approaches. We use three popular penalty functions, least absolute shrinkage and selection operator (LASSO), adaptive LASSO and smoothly clipped absolute deviation (SCAD). With these procedures we can select important variables and estimate the fixed effects at the same time. The performance of the proposed method is evaluated using simulation studies, including the investigation of impact of misspecifying the assumed distribution. The proposed method is illustrated with a primary biliary cirrhosis (PBC) data set.
In survival analysis, accelerated failure time (AFT) model has been introduced as a useful alternative to proportional hazards (PH) model (Lawless, 1982). The PH model is modelled by fixed effects (e.g., regression coefficients) acting multiplicatively on the hazard rate of individual survival time. However, in the AFT model the fixed effects act linearly on the individual survival time, thus making the interpretation of the fixed effects easier than in the PH model. AFT model is robust against the misspecification of the assumed model due to its log-linear transformation (Hutton and Monaghan, 2002; Ha
Various penalized variable-selection methods in the semiparametric AFT model with an unspecified distribution have been studied (Huang
In this paper, we develop variable-selection procedures of fixed effects in parametric AFT model using a penalized likelihood approach. Here we consider two useful parametric distributions, lognormal and Weibull distributions, for survival analysis. For the variable selection, we use three popular penalty functions, least absolute shrinkage and selection operator (LASSO) (Tibshirani, 1996), adaptive LASSO (ALASSO) (Zou, 2006), and smoothly clipped absolute deviation (SCAD) (Fan and Li, 2001). We also show how to derive the penalized likelihood procedure. The performance of the proposed method is evaluated using simulation studies. In particular, the simulation shows that the proposed variable-selection method is somewhat robust against the misspecification of the assumed model. The proposed method is illustrated with a primary biliary cirrhosis (PBC) (Tibshirani, 1997) data set which is well known in the literature.
This paper is organized as follows. In Section 2, we briefly review the AFT model, and propose a penalized variable-selection method using AFT model, including the derivations of the estimation procedures. In Section 3, the results of simulation studies are presented to evaluate the validity of the proposed method. The proposed method is illustrated with the PBC data in Section 4. Discussion is given in Section 5. Finally, technical details are given in the
Let
where
For the distribution of
In this paper, if
Based on these two assumptions, we make inferences as shown below.
Now, we present how to derive a variable selection procedure using a penalized likelihood. In survival analysis with random censoring, observable random variables are given by
Let
where
For variable selection of fixed effects
where
LASSO (Tibshirani, 1996):
ALASSO (Zou, 2006):
where
SCAD (Fan and Li, 2001):
where
Figure 1 displays the shapes of LASSO and SCAD functions under
For the variable selection, we want to find the estimators
We call the resulting estimators penalized maximum likelihood estimators (PMLEs). The PMLEs are obtained by solving the following estimating equations:
Here we use
where
with a linear predictor
where
since
where
Wang
where df = tr[(
In summary, an outline of the proposed variable-selection algorithm is described as follows.
Step 1. Find initial values of
Step 2. In the inner loop, we maximize
Step 3. In the outer loop, we find
After convergence, we compute the estimated standard errors for
where
Simulation studies, based upon 100 replications of simulated data, are presented to evaluate the performance of the proposed variable-selection procedure for AFT models. Here, we compare the performances of the variable-selection methods using LASSO, ALASSO, and SCAD. Below we consider the two distributions (LN, Weibull) for this purpose. Following the simulation scheme of Fan and Li (2001), we generate the data from the AFT model (
Here, the corresponding covariates
For the LN case we consider
We also investigated the robustness of the proposed method when the true distribution of
In addition, we investigated the robustness of Weibull AFT model against mis-specifying distribution. Here, Weibull AFT model is fitted when the distribution of
For the illustration of the proposed method in Section 2, we consider the PBC data of the liver (Tibshirani, 1997). A total of 424 PBC patients met eligibility criteria for the randomized placebo controlled trial of the drug D-penicillamine. Here we consider 312 patients who participated in the randomized trial. Censoring rate due to survival was 59.8%. Table 6 summarizes the variables used in the analysis. For the analyses, all covariates (i.e., all variables except for Id, Futime and Status in Table 6) are standardized.
As presented above, we consider the two AFT models (i.e., LN and Weibull cases) with covariates in Table 7. First, we use two standard criteria of model selection: Akaike information criterion (AIC) and BIC, given by AIC= −2
From Table 7, we select the LN AFT model because the values of AIC and BIC in the LN are all smaller than those of the Weibull. Here we checked the adequacy of the lognormal assumption of survival time. This can be checked by a normal hazard plot (Klein and Moeschberger, 2003, p.410), i.e., we plot Φ^{−1}(1 –
Table 8 shows the estimated coefficients and SEs for the PBC in the LN case. As the result of the penalized variable selection, the values of the tuning parameters
Through penalized likelihood approach, we have shown the procedures that select important variables in the AFT model. We have demonstrated via simulation studies and illustration that the proposed variable-selection methods generally work well. Here we have found that the SCAD method performs better than the LASSO and ALASSO methods. The results confirm those in semi-parametric frailty hazard models by Ha
The AFT model has some advantages over Cox’s PH model as follows (Ha
We have also demonstrated via a simulation study that the proposed method is somewhat robust against misspecification of the assumed distribution. It would be also interested to investigate the robustness of the LN or Weibull AFT model against a further mis-specifying distribution, for example, when the true distribution of survival time
We have developed the variable-selection methods in AFT models with low-dimensional covariates (
Furthermore, the proposed method can be extended to AFT models allowing for random effects that can be useful for analyzing correlated survival data (Ha
Simulation results under LN AFT model (
Method | C | IC | PT | MSE | |
---|---|---|---|---|---|
100 | LASSO | 2.62 | 0.00 | 0.02 | 0.132 |
ALASSO | 4.18 | 0.00 | 0.41 | 0.079 | |
SCAD | 4.37 | 0.01 | 0.59 | 0.077 | |
300 | LASSO | 2.42 | 0.00 | 0.00 | 0.052 |
ALASSO | 4.39 | 0.00 | 0.45 | 0.021 | |
SCAD | 4.46 | 0.00 | 0.61 | 0.017 | |
500 | LASSO | 2.68 | 0.00 | 0.03 | 0.032 |
ALASSO | 4.50 | 0.00 | 0.59 | 0.015 | |
SCAD | 4.71 | 0.00 | 0.78 | 0.014 |
LN= lognormal; AFT = accelerated failure time; MSE= mean squared error; LASSO = least absolute shrinkage and selection operator; ALASSO = adaptive LASSO; SCAD = smoothly clipped absolute deviation.
Simulation results under Weibull AFT model (
Method | C | IC | PT | MSE | |
---|---|---|---|---|---|
100 | LASSO | 2.18 | 0 | 0.00 | 0.569 |
ALASSO | 4.29 | 0 | 0.51 | 0.027 | |
SCAD | 4.01 | 0 | 0.36 | 0.027 | |
300 | LASSO | 2.43 | 0 | 0.02 | 0.017 |
ALASSO | 4.53 | 0 | 0.63 | 0.008 | |
SCAD | 4.49 | 0 | 0.69 | 0.008 | |
500 | LASSO | 2.57 | 0 | 0.02 | 0.011 |
ALASSO | 4.66 | 0 | 0.72 | 0.005 | |
SCAD | 4.68 | 0 | 0.75 | 0.004 |
AFT = accelerated failure time; MSE= mean squared error; LASSO = least absolute shrinkage and selection operator; ALASSO = adaptive LASSO; SCAD = smoothly clipped absolute deviation.
Simulation results under Weibull AFT model (
Method | C | IC | PT | MSE | |
---|---|---|---|---|---|
100 | LASSO | 2.41 | 0 | 0.02 | 0.203 |
ALASSO | 4.02 | 0 | 0.35 | 0.112 | |
SCAD | 4.44 | 0 | 0.61 | 0.100 | |
300 | LASSO | 2.59 | 0 | 0.03 | 0.074 |
ALASSO | 4.47 | 0 | 0.52 | 0.034 | |
SCAD | 4.64 | 0 | 0.72 | 0.029 | |
500 | LASSO | 2.51 | 0 | 0.04 | 0.052 |
ALASSO | 4.52 | 0 | 0.58 | 0.018 | |
SCAD | 4.65 | 0 | 0.75 | 0.015 |
AFT = accelerated failure time; MSE= mean squared error; LASSO = least absolute shrinkage and selection operator; ALASSO = adaptive LASSO; SCAD = smoothly clipped absolute deviation.
Simulation results under Weibull AFT model (
Method | C | IC | PT | MSE | |
---|---|---|---|---|---|
100 | LASSO | 2.95 | 0.12 | 0.05 | 0.746 |
ALASSO | 4.05 | 0.32 | 0.30 | 0.596 | |
SCAD | 4.63 | 0.63 | 0.44 | 0.625 | |
300 | LASSO | 2.84 | 0.00 | 0.06 | 0.268 |
ALASSO | 4.42 | 0.01 | 0.50 | 0.166 | |
SCAD | 4.92 | 0.04 | 0.91 | 0.102 | |
500 | LASSO | 2.92 | 0.00 | 0.02 | 0.212 |
ALASSO | 4.48 | 0.00 | 0.55 | 0.091 | |
SCAD | 4.94 | 0.00 | 0.94 | 0.058 |
AFT = accelerated failure time; MSE= mean squared error; LASSO = least absolute shrinkage and selection operator; ALASSO = adaptive LASSO; SCAD = smoothly clipped absolute deviation.
Simulation results of fitting LN AFT model (
Error | Censoring | Method | C | IC | PT | MSE |
---|---|---|---|---|---|---|
45% | LASSO | 2.42 | 0.00 | 0.00 | 0.052 | |
ALASSO | 4.39 | 0.00 | 0.45 | 0.021 | ||
SCAD | 4.46 | 0.00 | 0.61 | 0.017 | ||
70% | LASSO | 2.40 | 0.00 | 0.02 | 0.128 | |
ALASSO | 4.38 | 0.00 | 0.45 | 0.045 | ||
SCAD | 4.57 | 0.00 | 0.63 | 0.032 | ||
45% | LASSO | 2.80 | 0.00 | 0.00 | 0.080 | |
ALASSO | 4.51 | 0.00 | 0.56 | 0.046 | ||
SCAD | 4.69 | 0.00 | 0.71 | 0.052 | ||
70% | LASSO | 2.61 | 0.00 | 0.00 | 0.176 | |
ALASSO | 4.32 | 0.01 | 0.43 | 0.109 | ||
SCAD | 4.90 | 0.03 | 0.89 | 0.172 | ||
Mix | 45% | LASSO | 2.90 | 0.00 | 0.04 | 0.139 |
ALASSO | 4.58 | 0.01 | 0.65 | 0.097 | ||
SCAD | 4.89 | 0.02 | 0.88 | 0.087 | ||
70% | LASSO | 2.88 | 0.00 | 0.02 | 0.252 | |
ALASSO | 4.42 | 0.02 | 0.52 | 0.192 | ||
SCAD | 4.91 | 0.11 | 0.84 | 0.227 |
Note:
LN= lognormal; AFT = accelerated failure time; MSE= mean squared error; LASSO = least absolute shrinkage and selection operator; ALASSO = adaptive LASSO; SCAD = smoothly clipped absolute deviation.
Explanation of variables for primary biliary cirrhosis data
Variable | Explanation |
---|---|
Id | Case number |
Futime | Number of days from registration to death |
Status | Status at endpoint (0: survival (59.8 %), 1: death) |
Drug | Types of drugs (1: D-penicillmain, 2: placebo) |
Age | In years |
Sex | Sex (0: male, 1: female) |
Ascites | Presence of ascites (0: no, 1: yes) |
Hepato | Presence of hepatomegaly or enlarged liver (0: no, 1: yes) |
Spiders | Blood vessel malformations in the skin (0: no, 1: yes) |
Edema | Presence of edema (0: no edema, 0.5: untreated or successfully treated, 1: edema despite diuretic therapy) |
Bili | Serum bilirunbin (mg/dl) |
Chol | Serum cholesterol (mg/dl) |
Albumin | Serum albumin (g/dl) |
Copper | Urine copper (ug/day) |
Alk_phos | Alkaline phosphotase (U/liter) |
Sgot | SGOT (U/ml) |
Trig | Triglycerides (mg/dl) |
Platelet | Platelets per cubic (ml/1000) |
Protime | Prothrombin time |
Stage | Histologic stage of disease |
Model selection for AFT model with primary biliary cirrhosis data
ℓ | AIC | BIC | |
---|---|---|---|
LN | −195.41 | 426.82 | 492.00 |
Weibull | −197.91 | 431.82 | 496.99 |
AFT = accelerated failure time; AIC = Akaike information criterion; BIC = Bayesian information criterion; LN = lognormal.
Variable selection using LN AFT model for primary biliary cirrhosis data
Variable | No penalty | LASSO | LASSO† | ALASSO | SCAD |
---|---|---|---|---|---|
Intercept | 8.073(0.086) | 7.885(0.060) | - | 7.994(0.065) | 7.989(0.066) |
Drug | −0.002(0.069) | 0.000(0.000) | 0.00(0.00) | 0.000(0.000) | 0.000(0.000) |
Age | −0.221(0.080) | −0.139(0.039) | 0.17(0.09) | −0.179(0.047) | −0.099(0.028) |
Sex | 0.091(0.068) | 0.016(0.011) | −0.01(0.03) | 0.000(0.000) | 0.000(0.000) |
Ascites | −0.112(0.076) | −0.092(0.032) | 0.04(0.07) | −0.023(0.009) | 0.000(0.000) |
Hepato | −0.005(0.080) | 0.000(0.000) | 0.00(0.00) | 0.000(0.000) | 0.000(0.000) |
Spiders | −0.116(0.072) | −0.051(0.024) | 0.02(0.05) | 0.000(0.000) | 0.000(0.000) |
Edema | −0.185(0.081) | −0.191(0.042) | 0.18(0.11) | −0.246(0.046) | −0.304(0.053) |
Bili | −0.202(0.086) | −0.204(0.043) | 0.35(0.12) | −0.244(0.047) | −0.306(0.053) |
Chol | −0.048(0.074) | 0.000(0.000) | 0.00(0.00) | 0.000(0.000) | 0.000(0.000) |
Albumin | 0.106(0.077) | 0.100(0.034) | −0.22(0.10) | 0.029(0.011) | 0.051(0.018) |
Copper | −0.148(0.073) | −0.152(0.040) | 0.21(0.11) | −0.143(0.037) | −0.116(0.031) |
Alk_phos | −0.040(0.061) | 0.000(0.000) | 0.00(0.00)) | 0.000(0.000) | 0.000(0.000) |
Sgot | −0.187(0.075) | −0.103(0.035) | 0.09(0.08) | −0.118(0.038) | −0.030(0.012) |
Trig | 0.022(0.072) | 0.000(0.000) | 0.00(0.00) | 0.000(0.000) | 0.000(0.000) |
Platelet | 0.004(0.072) | 0.000(0.000) | 0.00(0.00) | 0.000(0.000) | 0.000(0.000) |
Protime | −0.167(0.073) | −0.123(0.038) | 0.09(0.09) | −0.133(0.038) | −0.080(0.024) |
Stage | −0.244(0.091) | −0.181(0.044) | 0.21(0.09) | −0.259(0.055) | −0.275(0.057) |
^{†}indicates the results of variable selection from Cox’s PH model by Tibshirani (1997).
LN = lognormal; AFT = accelerated failure time; LASSO = least absolute shrinkage and selection operator; ALASSO = adaptive LASSO; SCAD = smoothly clipped absolute deviation.