Improved AIC Selection Strategy for Survival Analysis

SUMMARY

In survival analysis, it is of interest to appropriately select significant predictors. In this paper, we extend the AIC_C selection procedure of Hurvich and Tsai to survival models to improve the traditional AIC for small sample sizes. A theoretical verification under a special case of the exponential distribution is provided. Simulation studies illustrate that the proposed method substantially outperforms its counterpart: AIC, in small samples, and competes it in moderate and large samples. Two real data sets are also analyzed.

1 Introduction

In clinical trials, biological and biomedical applications, many variables may be available for the initial analysis, and spurious covariates may increase prediction error. Deciding which covariates to be kept in the statistical model has always been a tricky task for data analysis. Conventional variable selection techniques, such as AIC (Akaike, 1974), BIC (Schwarz, 1978), and C_p (Mallows, 1973), have widely been used to select an appropriate model. These criteria work well and are implemented in the most well-developed statistical software such as R and SAS. Their deficiency in small samples was pointed out by Sugiura (1978) and emphasized by Hurvich and Tsai (1989). The latter authors showed that AIC may be drastically biased for the linear model, and developed a modified version, AIC_C, which is nearly unbiased for estimating Kullback-Leibler information and provides better model choices than AIC in small samples. Tsai and his colleagues generalized Hurvich and Tsai's criterion to diverse situations like the extended quasi-likelihood model (Hurvich and Tsai, 1995), the nonparametric regression (Hurvich et al., 1998), and the semiparametric regression (Hurvich and Tsai, 1999).

Traditional variable selection criteria such as AIC and BIC have been extended to survival analysis. Faraggi and Simon (1998) proposed a Bayesian variable selection method, which is an extension of Lindley's (1968) variable selection criterion for the linear model, for censored data based on the sufficiency and asymptotic normality of the maximum partial likelihood estimator. Volinsky and Raftery (2000) extended the BIC to the Cox model. They proposed a modification of the penalty term in the BIC so that it is defined in terms of the number of uncensored events instead of the number of observations. Tibshirani (1997) extended his LASSO variable selection procedure to the Cox model. More recently, Fan and Li (2002) derived a nonconcave penalized partial likelihood for the Cox model and the Cox frailty model. Although all of these approaches have been demonstrated to be promising, they may not be accepted in practice because (i) the computation of some methods is not simple and sometimes has a requirement of determining prior information, and (ii) few existing computation packages (to the best of our knowledge, there is a package glmpath for the LASSO algorithm) have been developed for practitioners' use. The aim of this paper is to fill this gap. We propose here an improved AIC variable selection method for survival analysis. This work is motivated by Hurvich and Tsai (1989), whose focus is on linear models. We extend Hurvich and Tsai's approach to survival models and numerically justify the superiority of the proposed criterion over other traditional criteria in small sample sizes. The proposed method can be implemented in the existing software, such as R/Splus and SAS. This availability may make the method easily implement in practice.

The rest of the paper is organized as follows. In Section 2 we propose an improved AIC selection procedure for survival models. A particular case of the exponential distribution for the survival time is considered, which serves as a theoretical justification of the proposed criterion. Section 3 gives the results of extensive simulation experiments to illustrate the proposed method, and compare it with its competitors. Two real examples are examined in Section 4. We conclude the paper with some discussions in Section 5. Technical details are given in the Appendix.

2 Improved AIC for survival analysis data

Let T , C, and x be the survival time, censoring time, and the associated p × 1 covariates respectively. Let Z = min(T, C) be the observed time and δ = I(T ≤ C) be the censoring indicator. Let h(t|x) and S(t|x) be the conditional hazard and survival functions of T given x, respectively. The complete likelihood of the data is given by

$$L=\prod _{u}h(Z_i\mid {\mathbf{x}}_i)\prod _{i=1}^{n}S(Z_i\mid {\mathbf{x}}_i),$$ (2.1)

where n is the total number of observations, and the subscript u denotes the product over the uncensored data. In this paper, we focus on the accelerated life time (ALT) model, one of the most useful parametric life models, of form

$$\log \left(T\right)=\alpha +{\mathbf{x}}^{\mathrm{T}}\beta +\sigma \epsilon .$$ (2.2)

Let S₀(t) denote the survival function of T when x = 0, and h₀(t) be the hazard risk of S₀(t). It follows that

$$\begin{array}{cc}\hfill S(t\mid \mathbf{x})& =S_0\left\{t\exp (-{\mathbf{x}}^{\mathrm{T}}\beta )\right\},\hfill \\ \hfill h(t\mid \mathbf{x})& =h_0\left\{t\exp (-{\mathbf{x}}^{\mathrm{T}}\beta )\right\}\exp (-{\mathbf{x}}^{\mathrm{T}}\beta ).\hfill \end{array}$$

In a consequence, we obtain the log-likelihood of the observed data {(x_i, Z_i, δ_i), i = 1, . . . , n}

$$l(\mathbf{x},Z,\beta )=\sum _u(-{\mathbf{x}}_i^{\mathrm{T}}\beta +\log \left[h_0\left\{Z_i\exp (-{\mathbf{x}}_i^{\mathrm{T}}\beta )\right\}\right])+\sum _{i=1}^n\log \left[S_0\left\{Z_i\exp (-{\mathbf{x}}_i^{\mathrm{T}}\beta )\right\}\right]$$ (2.3)

Collett (1994) suggested that the AIC for survival models should be

$$\mathrm{AIC}=-2\log \left(\text{likelihood}\right)+2(p+2+k),$$

where k = 0 for the exponential model, k = 1 for the Weibull, log-logistic and log-normal models, and k = 2 for the generalized gamma model. Following Hurvich and Tsai (1989), we propose an improved AIC as follows

$${\mathrm{AIC}}_{\mathrm{SUR}}=\mathrm{AIC}+\frac{2(p+2)(p+3)}{n-p-3}.$$ (2.4)

Different choices for the error distribution of ε yield different regression models, and then different log-likelihood functions given in (2.3). The routines to finish the calculations on the log-likelihood functions and AICs (and then AIC_SURs) are available in the most statistical packages like R/Splus and SAS.

A commonly used criterion of measuring the difference between the candidate model and the true model is the Kullback-Leibler information Δ = E₀(−2 log L), where E₀ denotes the expectation with respect to the true model, and L is the likelihood function under the candidate model. In the remainder of this section, we use this measure to derive a more precise model selection criterion for the special case of the exponential distribution to demonstrate the rationality of the proposed AIC_SUR given in (2.4).

Consider the ALT model (2.2) with σ = 1 and ε following an extreme value distribution whose density function is exp (v − e^v). Then the survival time T has the exponential distribution with the density function λe^−λt, where λ = exp{−(α + x^Tβ)}. If we denote ${\lambda }_i=\exp \{-(\alpha +{\mathbf{x}}_i^{\mathrm{T}}\beta )\}$, then h(Z_i | x_i) = λ_i, and S(Z_i | x_i) = exp(−λ_iZ_i). So the log-likelihood function from (2.1) is given by

$$\log L=\sum _u\log {\lambda }_i+\sum _{i=1}^n(-{\lambda }_i{Z}_i).$$

From this, we see that the Kullback-Leibler information is

$$\begin{array}{cc}\hfill \Delta (\alpha ,\beta )& =-2\sum _u\log {\lambda }_i+2\sum _{i=1}^n{\lambda }_i{E}_0\left(Z_i\right)\hfill \\ \hfill & =-2\sum _u\log {\lambda }_i+2\sum _{i=1}^n\frac{{\lambda }_i}{{\lambda }_{i0}}(1-e^{-{\lambda }_{i0}C}),\hfill \end{array}$$

where the censoring time is assumed to be a constant, ${\lambda }_{i0}=\exp \{-({\alpha }_0+{\mathbf{x}}_{0i}^{\mathrm{T}}{\beta }_0)\}$, and α₀ and β₀ are the parameters in the true model.

Following Akaike (1974) and Hurvich and Tsai (1989) (see also Burnham and Anderson, 1998), a reasonable measure representing the discrepancy between the candidate and true models would be $E_0\Delta (\widehat{\alpha },\widehat{\beta })$, where $\widehat{\alpha }$ and $\widehat{\beta }$ are the estimators of α and β under the candidate model. That is, we would choose those candidate models which minimize $E_0\Delta (\widehat{\alpha },\widehat{\beta })$. In the Appendix we derive an (approximately) unbiased estimator, AIC_exp given in (A.4), of $E_0\Delta (\widehat{\alpha },\widehat{\beta })$ and this can be used to obtain a feasible model selection criterion.

We now numerically demonstrate the rationality of the proposed AIC_SUR in (2.4) by comparing it with AIC_exp in (A.4) under the exponential distribution which is regarded as more precise.

Generate data from the model y = x^Tβ + ε, where β = (1, 2, 3, 4)^T, x follows a 4-dimensional normal distribution with the mean zero and covariance matrix I_4×4, and ε follows an extreme value distribution with the density function exp (v − e^v). We consider the combinations of n = 20, 30, 40, 50 and the censoring variable C = 5, 10, 15, 20, 25, 30, and repeat 500 simulations for each combination. Table 1 presents the means and standard errors of AIC_SUR and AIC_exp. It is seen from the table that the values of AIC_SUR and AIC_exp are very close, suggesting that the difference between the two model selection criteria would often be quite minor. This implies the rationality of AIC_SUR from one aspect. Of course, the above demonstration is based on a special distribution-exponential distribution. So in the following section, we conduct some simulations to study the behavior in selecting true models of AIC_SUR under various distributions.

Table 1.

The comparison of AIC_SUR and AIC_exp under the exponential distribution based on 100 replications

n	C	AICSUR	AICSUR(se)	AICexp	AICexp(se)
20	5	−49.05	5.52	−48.85	5.53
	10	−45.69	5.6	−44.53	5.57
	15	−40.72	5.93	−40.69	5.88
	20	−42	5.76	−41.65	5.79
	25	−20.09	5.43	−20.7	5.45
	30	−36.67	5.47	−36.85	5.53
30	5	−74.96	5.94	−76.26	5.94
	10	−70.99	6.38	−73.16	6.37
	15	−52.18	6.29	−53.6	6.25
	20	−46.12	6.67	−47.07	6.68
	25	−50.02	6.28	−52.12	6.31
	30	−50.76	6.26	−52.56	6.3
40	5	−102.21	6.39	−104.51	6.39
	10	−83.2	6.52	−84.97	6.52
	15	−84.94	6.64	−87.27	6.63
	20	−79.45	7.33	−81.7	7.34
	25	−68.97	6.9	−71.82	6.89
	30	−77	6.6	−79.54	6.59
50	5	−116.25	6.69	−119.95	6.69
	10	−123.37	7	−125.84	7.02
	15	−104.48	7.3	−107.31	7.29
	20	−95.02	6.58	−97.96	6.59
	25	−77.79	7.03	−80.54	7.02
	30	−76.77	6.72	−79.96	6.74

3 Simulation study

In this section, we investigate the finite sample performance of the proposed procedure AIC_SUR by Monte Carlo simulations, and illustrate the proposed methodology by analyzing two real data sets in next section.

Example 1

Generate data from the model

$$y={\mathbf{x}}^{\mathrm{T}}\beta +\sigma \epsilon ,$$

where β = (1, 2, 3, 4, 0, 0, 0, 0)^T, x follows an 8-dimensional normal distribution with the mean zero and covariance matrix I_8×8. We consider three scenarios: (i) ε follows a logistic distribution, (ii) ε follows a log-normal distribution, and (iii) ε follows an extreme distribution. We take the location and scale parameters 0 and 1, respectively. The value of the censoring random variable C is generated by the uniform distribution U(0, 10) for each observation. For each scenario, we take n = 12, 20, 30, and σ² = 0.1, 0.5, 1. At each of 27 configurations, 500 independent data sets are generated. Similar to Hurvich and Tsai (1989), our candidate models are those whose predictors are sequential columns of X; i.e., consist of columns 1, · · · , r of X. The true model consists of the first 4 columns of X. We use three criteria: AIC, BIC, and AIC_SUR to select a value of r for each configuration, respectively. Tables 2-4 summarize the frequencies of the order selected by the specified criterion for scenarios 1−3, respectively. It is observed that AIC_SUR consistently provides the best selection of r = 4 among the three criteria studied, regardless of sample sizes and variances. Even when n = 12, AIC_SUR generally selects at least 250 times of the correct model, while AIC selects only around 200 times of the correct model. When n = 20, the number of the correct times selected by AIC_SUR is double to that of AIC. Usually, the best model can be identified more frequently when n = 30. However, AIC_SUR still substantially outperforms AIC. It is also seen from Tables 2 -4 that BIC is usually better than AIC but substantially inferior to AIC_SUR for the cases considered here.

Table 2.

Scenario 1-Frequency of order selected using different criteria in 500 replications of model fitting with the true order r₀ = 4 under the logistic distribution^*

			Selected model order r
n	${\sigma }_{\epsilon }^2$	Criterion	3	4	5	6	7	8
12	0.1	AIC	112	198	91	57	34	8
		BIC	104	201	94	59	35	7
		AICSUR	206	240	40	13	1	0
	0.5	AIC	127	183	100	51	26	13
		BIC	124	180	105	51	26	14
		AICSUR	194	251	42	11	0	2
	1	AIC	118	162	108	69	34	9
		BIC	110	161	112	71	35	11
		AICSUR	203	233	46	14	3	1
20	0.1	AIC	8	108	63	69	107	145
		BIC	9	141	66	72	97	115
		AICSUR	17	289	69	52	43	30
	0.5	AIC	8	133	62	66	86	145
		BIC	8	167	65	69	74	117
		AICSUR	15	312	59	47	35	32
	1	AIC	4	130	57	79	98	132
		BIC	4	162	60	76	94	104
		AICSUR	10	300	54	49	54	33
30	0.1	AIC	2	204	52	62	79	101
		BIC	2	270	51	54	58	65
		AICSUR	3	316	57	48	36	40
	0.5	AIC	2	211	50	58	66	113
		BIC	3	288	51	48	45	65
		AICSUR	3	330	57	39	33	38
	1	AIC	0	232	53	30	69	116
		BIC	0	309	48	21	47	75
		AICSUR	1	350	59	25	27	38

^*The censoring variable C is generated by the uniform distribution U (0, 10).

Table 4.

Scenario 3-Frequency of order selected using different criteria in 500 replications of model fitting with the true order r₀ = 4 under the extreme value distribution^*

			Selected model order r
n	${\sigma }_{\epsilon }^2$	Criterion	3	4	5	6	7	8
12	0.1	AIC	103	222	94	59	13	9
		BIC	96	223	97	61	13	10
		AICSUR	137	299	44	17	2	1
	0.5	AIC	124	191	82	66	28	9
		BIC	119	191	83	69	28	10
		AICSUR	181	263	42	11	3	0
	1	AIC	132	164	95	59	40	10
		BIC	121	169	102	57	40	11
		AICSUR	244	208	38	9	1	0
20	0.1	AIC	4	113	57	87	104	135
		BIC	4	152	63	84	88	109
		AICSUR	4	268	74	71	47	36
	0.5	AIC	6	136	69	67	101	121
		BIC	5	166	72	63	91	103
		AICSUR	9	302	76	42	41	30
	1	AIC	17	122	57	74	95	135
		BIC	21	144	61	72	87	115
		AICSUR	44	263	64	46	51	32
30	0.1	AIC	3	197	54	64	61	121
		BIC	3	267	56	51	49	74
		AICSUR	3	324	53	40	34	46
	0.5	AIC	0	218	50	68	59	105
		BIC	0	284	53	48	52	63
		AICSUR	1	327	57	41	34	40
	1	AIC	6	211	61	61	53	108
		BIC	8	284	62	44	34	68
		AICSUR	11	339	55	34	28	33

^*The censoring variable C is generated by the uniform distribution U (0, 10).

4 Real Data Analysis

Example 2

We fit the motor data set, which was obtained by Nelson and Hahn (1972) and studied by Kalbfleisch and Prentice (1980), using the exponential, Weibull, log-logistic, and log-normal models. The response variable and covariate are the hour to the failure of motorette and operating temperature, respectively. Nelson and Hahn (1972) used the log-normal model, while Kalbfleisch and Prentice (1980) used the Weibull model for the analysis of this dataset because the latter authors thought that the Weibull model generates a larger likelihood value. On the basis of our simulations (data not shown), this evidence may be not enough to be convinced. We therefore apply the proposed method to the analysis of this dataset. To compare with the results of Nelson and Hahn (1972) and of Kalbfleisch and Prentice (1980), we make the transformation x = 1000/(273.2 + temperature) and exclude the ten observations at the temperature level of 150% because the experiment was an accelerated process to speed up the failure time. A total of 30 observations are used in our analysis. We fit the four models and present the AIC and AIC_SUR values in Table 5. The results of the two criteria uniformly indicate that the Weibull model is most appropriate. This confirmation convinces that Kalbfleisch and Prentice's choice is appropriate. The estimates and their related quantities for the four models are presented in Table 6. Shown in Table 6 are the estimates, their corresponding standard deviations, the z-ratios, and the p-values obtained by testing the null hypothesis that the corresponding parameter is zero.

Table 5.

Values of AIC and AIC_SUR for the Motor data set under various models

	AIC	AICSUR
Weibull	294.69	296.29
Exponential	309.61	311.21
Log-logistic	295.68	297.28
Log-normal	297.73	299.33

Table 6.

Estimated results of parameters for the Motor data set under various models

model		Value	Std. Error	z	p-value
Weibull	(Intercept)	−11.89	1.97	−6.05	0
	temp	9.04	0.91	9.98	0
	Log(scale)	−1.02	0.22	−4.63	0
Exponential	(Intercept)	−8.99	5.5	−1.63	0.1
	temp	7.83	2.54	3.08	0
Log-logistic	(Intercept)	−11.11	2.21	−5.02	0
	temp	8.61	1.03	8.38	0
	Log(scale)	−1.19	0.22	−5.46	0
Log-normal	(Intercept)	−10.47	2.77	−3.78	0
	temp	8.32	1.28	6.48	0
	Log(scale)	−0.5	0.18	−2.75	0.01

Example 3

In this example, we apply the proposed method to analyze the data set from a study of the bone marrow transplantation (BMT) for leukemia. This study was designed in 1984 as a single institution (Ohio State University Hospitals, OSU) study and was modified in 1987 to include the five institutions known to be using this preparative regimen in all patients with the acute myelocytic leukemia (AML). All patients who underwent the marrow transplantation for the AML using this preparative regimen at the participating institutions were reported. One hundred twenty-seven patients were with the AML aged 7 to 55 (median 30) who were treated from March 1, 1984 through June 30, 1989 at the five separate centers with the allogeneic BMT following preparation with Bu and Cy. Fifty-five of them underwent their transplantation at Ohio State University Hospitals (OSU; Columbus), 23 at Wilford Hall at Lackland Air Force Base (San Antonio, TX), 22 at Hahnemann University (Philadelphia, PA), 17 at St Vincent's Hospital (Sydney, Australia), and 10 at Alfred Hospital (Melbourne, Australia). More details of the study are referred to Copelan et al. (1991).

Our response variable is the disease free survival time, T , and the disease free survival indicator (1-dead or relapsed, 0-alive and disease free), δ . The potential covariates in this study include the following variables:

• X₁: patient age in year;

• X₂: donor age in year;

• X₃: patient sex (1-male, 0-female);

• X₄: donor sex (1-male, 0-female);

• X₅: patient cytomegalovirus (CMV) immune status (1-CMV positive, 0-CMV negative);

• X₆: donor CMV status (1-CMV positive, 0-CMV negative);

• X₇: waiting time to transplant in day;

• X₈: French-American-British (FAB, 1-FAB grade 4 or 5 and AML, 0-otherwise);

• X₉: hospital (1-Ohio State University, 2-Alferd , 3-St. Vincent, 4-Hahnemann);

• X₁₀: methotrexate (MTX) used as a graft-versus-host-prophylactic (1-yes, 0-no).

For an illustration, we consider only the observations of ALL the patients. The X₈ values are all zeros and therefore excluded in our analysis. A total of 79 combinations of the covariates is considered. We fit the exponential, Weibull, log-logistic, and log-normal models to the data for each combination, and select the corresponding best model by AIC and AIC_SUR.

Using the Weibull and exponential models, AIC and AIC_SUR select the model with the covariates (X₁, X₂, X₆, X₉) as the best one, and AIC = 358.19 (Weibull) and 358.21 (exponential) and AIC_SUR = 360.07 (Weibull) and 360.08 (exponential). It is clear that the values of both AIC and AIC_SUR under the Weibull and exponential models are very close. The corresponding estimates under these two setups are given in Table 7. Although both AIC and AIC_SUR suggest the Weibull model, the p-value of log(scale) indicates that the exponential model is appropriate. For the log-logistic model, AIC and AIC_SUR select the model with the covariates (X₁, X₂, X₆, X₁₀) as the best one, and AIC = 361.70 and AIC_SUR = 363.55. The p-value of the log(scale) indicates that the scale is not significantly different from 1. For the log-normal model, AIC selects the model with the covariates (X₁, X₂, X₆, X₁₀) as the best one with AIC = 363.99, while AIC_SUR selects the model with the covariates (X₁, X₆, X₁₀) as the best one with AIC_SUR = 363.15. Seeing the p-value of testing the parameters in the model selected by AIC, one may notice that X₂ and X₁₀ are not statistically significant, and the model selected by AIC_SUR seems more reasonable. In summary, we recommend to use the exponential model to fit the data with the covariates (X₁, X₂, X₆, X₉) on the basis of the above analysis. AIC_SUR makes us confident to this selection.

Table 7.

Results of variable selection using AIC and AIC_SUR and the corresponding estimated results for the BMT data set under various models

Model	Criterion		Value	Std. Error	z	p-value
Weibull	AIC/AICSUR	Intercept	9.679	0.986	9.813	0
		X1	−0.227	0.051	−4.448	0
		X2	0.114	0.035	3.274	0.001
		X6	1.663	0.512	3.249	0.001
		X9	−0.686	0.254	−2.703	0.007
		Log(scale)	0.022	0.179	0.126	0.9
Exponential	AIC/AICSUR	Intercept	9.657	0.952	10.146	0
		X1	−0.226	0.049	−4.653	0
		X2	0.114	0.034	3.384	0.001
		X6	1.651	0.492	3.359	0.001
		X9	−0.683	0.247	−2.761	0.006
Log-logistic	AIC/AICSUR	Intercept	8.312	0.94	8.844	0
		X1	−0.176	0.051	−3.436	0.001
		X2	0.082	0.037	2.224	0.026
		X6	1.352	0.594	2.277	0.023
		X10	−1.093	0.489	−2.235	0.025
		Log(scale)	−0.182	0.18	−1.009	0.313
Log-normal	AIC	Intercept	8.611	1.001	8.604	0
		X1	−0.167	0.054	−3.077	0.002
		X2	0.06	0.042	1.433	0.152
		X6	1.297	0.6	2.163	0.031
		X10	−1.059	0.558	−1.897	0.058
		Log(scale)	0.452	0.155	2.911	0.004
	AICSUR	Intercept	9.025	0.982	9.194	0
		X1	−0.116	0.04	−2.897	0.004
		X6	1.183	0.595	1.989	0.047
		X10	−1.111	0.561	−1.982	0.047
		Log(scale)	0.465	0.156	2.99	0.003

5 Discussion

To select an appropriate model for survival analysis, we generalized Hurvich and Tsai's (1989) approach and developed an improved AIC selection procedure, AIC_SUR. The proposed method was shown to be superior to the traditional AIC and BIC through simulation studies. It is interesting to observe from our simulations that when the sample size is not small (n = 20 and 30), the efficiency of AIC_SUR can be greatly increased if we use the total number of uncensored observations instead of the total number of observations n in the extra penalty term of AIC_SUR (data not shown). Our method was also applied to analyze two real data sets.

The proposed AIC_SUR is a general criterion of selecting survival models. It can be applied to the exponential, Weibull, log-logistic, log-normal and generalized gamma models etc. As a theoretical verification for AIC_SUR, we derived a more precise model selection criterion AIC_exp for the particular scenario of the exponential distribution for the survival time with constant censoring. The calculation results showed that the discrepancy between the two model selection criteria is quite minor under the exponential distribution. Of course, the further justification is necessary under the more general cases of the distributions for the survival time and this warrants our future research.

Unlike other advanced selection procedures, the proposed method is very easy to implement and computationally efficient. These features make the method promising in practice. The efficient R/Splus computation codes were developed and are available from the authors upon request. Extension of the idea to the goodness-of-fit and semiparametric survival models would be possible and will also be studied in our future work.

Table 3.

Scenario 2-Frequency of order selected using different criteria in 500 replications of model fitting with the true order r₀ = 4 under the log-normal distribution^*

			Selected model order r
n	${\sigma }_{\epsilon }^2$	Criterion	3	4	5	6	7	8
12	0.1	AIC	96	223	96	59	21	5
		BIC	89	224	98	60	24	5
		AICSUR	129	302	49	18	2	0
	0.5	AIC	111	219	97	42	21	10
		BIC	105	222	99	40	23	11
		AICSUR	137	302	51	10	0	0
	1	AIC	103	202	91	68	27	9
		BIC	98	207	91	67	26	11
		AICSUR	161	279	44	15	1	0
20	0.1	AIC	5	154	63	76	92	110
		BIC	5	176	69	71	86	93
		AICSUR	7	296	70	55	41	31
	0.5	AIC	2	143	63	89	91	112
		BIC	2	169	68	80	82	99
		AICSUR	4	303	58	63	46	26
	1	AIC	7	156	53	71	85	128
		BIC	7	184	51	71	78	109
		AICSUR	12	343	44	46	34	21
30	0.1	AIC	1	204	59	57	65	114
		BIC	1	282	51	45	50	71
		AICSUR	1	335	52	45	32	35
	0.5	AIC	1	213	62	43	72	109
		BIC	1	291	51	39	51	67
		AICSUR	1	346	47	37	35	34
	1	AIC	0	199	79	56	59	107
		BIC	0	281	76	43	41	59
		AICSUR	0	325	78	36	23	38

^*The censoring variable C is generated by the uniform distribution U (0, 10).

APPENDIX

An estimator of $E_0\Delta (\widehat{\alpha },\widehat{\beta })$ under the exponential distribution

First note that

$$\begin{array}{cc}\hfill E_0\left\{\frac{{\widehat{\lambda }}_i}{{\lambda }_{i0}}\left(1-e^{-{\lambda }_{i0}C}\right)\right\}& =E_{Z_{-i}}{E}_{Z_i}\left\{\frac{{\widehat{\lambda }}_i(Z_i,Z_{-i})}{{\lambda }_{i0}}\left(1-e^{-{\lambda }_{i0}C}\right)\right\}\hfill \\ \hfill & =E_{Z_{-i}}\left[{\int }_0^{\infty }\left\{\frac{{\widehat{\lambda }}_i(\min (t_i,C),Z_{-i})}{{\lambda }_{i0}}\left(1-e^{-{\lambda }_{i0}C}\right)\cdot {\lambda }_{i0}{e}^{-{\lambda }_{i0}{t}_i}\right\}d{t}_i\right]\hfill \\ \hfill & =E_{Z_{-i}}[{\int }_0^C\left\{\frac{{\widehat{\lambda }}_i(t_i,Z_{-i})}{{\lambda }_{i0}}\left(1-e^{-{\lambda }_{i0}C}\right)\cdot {\lambda }_{i0}{e}^{-{\lambda }_{i0}{t}_i}\right\}d{t}_i\phantom{]}\hfill \\ \hfill & +\phantom{[}{\int }_C^{\infty }\left\{\frac{{\widehat{\lambda }}_i(C_i,Z_{-i})}{{\lambda }_{i0}}\left(1-e^{-{\lambda }_{i0}C}\right)\cdot {\lambda }_{i0}{e}^{-{\lambda }_{i0}{t}_i}\right\}d{t}_i]\hfill \\ \hfill & \equiv E_{Z_{-i}}(U+V),\hfill \end{array}$$

where ${\widehat{\lambda }}_i(\mathrm{a},\mathrm{b})$ denotes the estimated value of λ_i based on the data (a, b) and is assumed to be continuous, and Z_−i means the vector consisting of Z₁, ..., Z_i−1, Z_i+1, ..., Z_n.

It is readily seen that

$$\begin{array}{cc}\hfill V& =\frac{1-e^{-{\lambda }_{i0}C}}{{\lambda }_{i0}}\cdot {\widehat{\lambda }}_i(C,Z_{-i})e^{-{\lambda }_{i0}C}\hfill \\ \hfill & =e^{-{\lambda }_{i0}C}\cdot E_{Z_i}\left[Z_i{\widehat{\lambda }}_i(C,Z_{-i})\right].\hfill \end{array}$$ (A.1)

On the other hand, it can be shown that

$$\begin{array}{cc}\hfill U& =\left(1-e^{-{\lambda }_{i0}C}\right)\cdot {\int }_0^C{\widehat{\lambda }}_i(t_i,Z_{-i})e^{-{\lambda }_{i0}{t}_i}d{t}_i\hfill \\ \hfill & =\left(1-e^{-{\lambda }_{i0}C}\right)\cdot {\int }_0^C{e}^{-{\lambda }_{i0}{t}_i}d\left\{{\int }_0^{t_i}{\widehat{\lambda }}_i(w,Z_{-i})dw\right\}\hfill \\ \hfill & =\left(1-e^{-{\lambda }_{i0}C}\right)[e^{-{\lambda }_{i0}C}\cdot {\int }_0^C{\widehat{\lambda }}_i(w,Z_{-i})dw\phantom{]}\hfill \\ \hfill & +\phantom{[}{\int }_0^C\left\{{\int }_0^{t_i}{\widehat{\lambda }}_i(w,Z_{-i})dw\right\}\cdot {\lambda }_{i0}{e}^{-{\lambda }_{i0}{t}_i}d{t}_i]\hfill \\ \hfill & =\left(1-e^{-{\lambda }_{i0}C}\right)[{\int }_0^{\infty }\left\{{\int }_0^C{\widehat{\lambda }}_i(w,Z_{-i})dw\right\}\cdot {\lambda }_{i0}{e}^{-{\lambda }_{i0}{t}_i}d{t}_i\phantom{]}\hfill \\ \hfill & +\phantom{[}{\int }_0^C\left\{{\int }_0^{t_i}{\widehat{\lambda }}_i(w,Z_{-i})dw\right\}\cdot {\lambda }_{i0}{e}^{-{\lambda }_{i0}{t}_i}d{t}_i]\hfill \\ \hfill & =\left(1-e^{-{\lambda }_{i0}C}\right)\cdot {\int }_0^{\infty }\left\{{\int }_0^{\min (t_i,C)}{\widehat{\lambda }}_i(w,Z_{-i})dw\right\}{\lambda }_{i0}{e}^{-{\lambda }_{i0}{t}_i}d{t}_i\hfill \\ \hfill & =\left(1-e^{-{\lambda }_{i0}C}\right)\cdot E_{Z_i}\left\{{\int }_0^{Z_i}{\widehat{\lambda }}_i(w,Z_{-i})dw\right\}.\hfill \end{array}$$ (A.2)

From formulas (A.1) and (A.2), we obtain

$$\begin{array}{cc}\hfill & E_0\left\{\frac{{\widehat{\lambda }}_i}{{\lambda }_{i0}}\left(1-e^{-{\lambda }_{i0}C}\right)\right\}\hfill \\ \hfill & =E_{Z_{-i}}\left[\left(1-e^{-{\lambda }_{i0}C}\right)\cdot E_{Z_i}\left\{{\int }_0^{Z_i}{\widehat{\lambda }}_i(w,Z_{-i)}dw\right\}+e^{-{\lambda }_{i0}C}\cdot E_{Z_i}\left\{Z_i{\widehat{\lambda }}_i(C,Z_{-i})\right\}\right]\hfill \\ \hfill & =\left(1-e^{-{\lambda }_{i0}C}\right)\cdot E_0\left\{{\int }_0^{Z_i}{\widehat{\lambda }}_i(w,Z_{-i})dw\right\}+e^{-{\lambda }_{i0}C}\cdot E_0\left\{Z_i{\widehat{\lambda }}_i(C,Z_{-i})\right\}.\hfill \end{array}$$ (A.3)

Therefore,

$$\begin{array}{cc}\hfill E_0\Delta (\widehat{\alpha },\widehat{\beta })& =E_0\left\{-2\sum _u\log {\widehat{\lambda }}_i\right\}=2\sum _{i=1}^n{E}_0\left\{\frac{{\widehat{\lambda }}_i}{{\lambda }_{i0}}\left(1-e^{-{\lambda }_{i0}C}\right)\right\}\hfill \\ \hfill & =E_0\left[-2\sum _u\log {\widehat{\lambda }}_i+2\sum _{i=1}^n\left\{\left(1-e^{-{\lambda }_{i0}C}\right)\cdot {\int }_0^{Z_i}{\widehat{\lambda }}_i(w,Z_{-i})dw+e^{-{\lambda }_{i0}C}\cdot Z_i{\widehat{\lambda }}_i(C,Z_{-i})\right\}\right]\hfill \\ \hfill & +E_0(\{-2\sum _u\log {\widehat{\lambda }}_i+2\sum _{i=1}^n{Z}_i{\widehat{\lambda }}_i\}+2\sum _{i=1}^n[\left(1-e^{-{\lambda }_{i0}C}\right)\cdot \left\{{\int }_0^{Z_i}{\widehat{\lambda }}_i(w,Z_{-i})dw-Z_i{\widehat{\lambda }}_i\right\}\phantom{]}\phantom{)}\hfill \\ \hfill & \phantom{(}\phantom{[}+e^{-{\lambda }_{i0}C}\cdot \left\{Z_i{\widehat{\lambda }}_i(C,Z_{-i})-Z_i{\widehat{\lambda }}_i\right\}]).\hfill \end{array}$$

Thus, we propose to select the best ALT model which minimizes the following Kullback-Leibler information:

$$\begin{array}{cc}\hfill {\mathrm{AIC}}_{\exp }=& -2\log \left(\text{likelihood}\right)+2\sum _{i=1}^n[\left(1-\psi \left({\widehat{\lambda }}_{i0}\right)\right)\cdot \left\{{\int }_0^{Z_i}{\widehat{\lambda }}_i(w,Z_{-i})dw-Z_i{\widehat{\lambda }}_i\right\}\phantom{]}\hfill \\ \hfill & +\psi \left({\widehat{\lambda }}_{i0}\right)\cdot \phantom{[}\left\{Z_i{\widehat{\lambda }}_i(C,Z_{-i})-Z_i{\widehat{\lambda }}_i\right\}],\hfill \end{array}$$ (A.4)

where $\psi \left({\widehat{\lambda }}_{i0}\right)$ is an estimator of exp(−λ_i0C), for which we provide some estimation methods below. Noting that Z_i, Z_−i and ${\widehat{\lambda }}_i$ are all known, the calculation on the integral in AIC_exp is easy.

Observing that

$$E_0\left(Z_i\right)=\{1-\exp (-{\lambda }_{i0}C)\}∕{\lambda }_{i0},$$ (A.5)

and

$$E_0\left(Z_i^2\right)=-\frac{2}{{\lambda }_{i0}}\left\{C{e}^{-{\lambda }_{i0}C}-\frac{1}{{\lambda }_{i0}}\left(1-e^{-{\lambda }_{i0}C}\right)\right\},$$

we have

$$E_0\left(\frac{2Z_i-{\lambda }_{i0}{Z}_i^2}{2C}\right)=e^{-{\lambda }_{i0}C}.$$ (A.6)

Therefore, combining (A.5) and (A.6), we can obtain an estimator of λ_i0 as

$${\widehat{\lambda }}_{i0}=\frac{2(C-Z_i)}{Z_i(2C-Z_i)}.$$ (A.7)

Thus, a natural estimator of exp(−λ_i0C) is $\exp (-{\widehat{\lambda }}_{i0}C)$ with ${\widehat{\lambda }}_{i0}$ given in (A.7). On the other hand, by virtue of (A.6) and (A.7), we can get another estimator of exp(−λ_i0C) as

$$\frac{2Z_i-{\widehat{\lambda }}_{i0}{Z}_i^2}{2C}=\frac{Z_i}{2C-Z_i}.$$ (A.8)