In the applications of COX regression models, we always encounter data sets t hat contain too many variables that only a few of them contribute to the model. Therefore, it will waste much more samples to estimate the “noneffective” variables in the inference. In this paper, we use a sequential procedure for constructing the fixed size confidence set for the “effective” parameters to the model based on an adaptive shrinkage estimate such that the “effective” coefficients can be efficiently identified with the minimum sample size. Fixed design is considered for numerical simulation. The strong consistency, asymptotic distributions and convergence rates of estimates under the fixed design are obtained. In addition, the sequential procedure is shown to be asymptotically optimal in the sense of Chow and Robbins (1965).
The COX proportional hazards model is a popular choice for the analysis of censored survival data with covariates, illustrated in [
For handling the problem mentioned above, we propose a sequential procedure for constructing the fixed size confidence set for effective parameters based on an adaptive shrinkage estimate (ASE) such that the effective coefficients can be efficiently identified with the minimum sample size. Suppose the conditional hazard rate of a survival time, T, given the regressor vector, X, is written as
h ( t | x ) = h 0 ( x ) exp ( β ′ X ) , t ≥ 0 (1)
In the paper, it will be studied under fixed design and the consistency and asymptotic properties of the proposed estimator will be obtained under this design. The rest of this paper is organized as follows. In Section 2, we will give the adaptive shrinkage estimate (ASE) based on the Maximum Partial Likelihood Estimate (MPLE) of COX regression models and their asymptotic properties. In section 3, sequential sampling strategy based on ASE and stopping rule as well as random size confident set is presented. In Section 4, an example with numerical simulation is given to illustrate the performance of the proposed method via sequential fixed size confidence estimation using synthesized data sets.
Let T i and C i be the potential failure time and censoring time of the i-th ( i ∈ N + ) subject from a random sample with n individuals, respectively, and X i = ( X i 1 , X i 2 , ⋯ , X i n ) T be a p-dimensional vector of covariates which assumed to be time-independent throughout this paper for the i-th individual. Assume that T i and C i are conditionally independent given X i . In practice, the failure time T i might not always be observed due to censoring because of the termination of study or early withdrawal from the study. What we can actually observe are Y i = min { T i , C i } , the smaller of the failure time and the censoring time, and δ i = I { T i ≤ C i } , the indicator that failure has been observed. The data then consist of the triplets ( Y i , δ i , X i ) , i = 1 , 2 , ⋯ , n . Suppose there is no tie among failure times. Let t 1 , t 2 , ⋯ , t n denote the N ordered times of observed failures and (j) be the label of the individual that fails at t j . Let R j be the risk set at time t j , i.e. R j = { i : Y i ≥ t j } . The partial likelihood of the model (1) is defined as
∏ j = 1 N exp ( β T X ( j ) ) ∑ i ∈ R j exp ( β T X ( i ) ) (2)
and the log partial likelihood is then,
L ( β ) = ∑ j = 1 N { β T X ( j ) − log [ ∑ i ∈ R j exp ( β T X i ) ] } (3)
The maximum partial likelihood estimate of β , β ˜ , is found by solving the score equation U ( β ) = 0 , where
U ( β ) = ∂ L ( β ) ∂ β = ∑ j = 1 N { X ( j ) − ∑ i ∈ R j X i exp ( β T X i ) ∑ i ∈ R j exp ( β T X i ) } (4)
Let κ = κ ( n ) be a non-random function of n such that for some 0 < δ < 1 / 2 and γ > 0 , n 1 / 2 κ → 0 and n 1 / 2 + γ δ κ → ∞ , as n → ∞ . In this paper, we need the following assumptions:
(A1) x i satisfies sup i ‖ x i ‖ < ∞ , and the residual term
ε i = Λ ^ ( Y i ) exp ( β ^ T X i )
has E | ε i | ζ < ∞ for some ζ > 2 , where Λ ^ is some cumulative baseline function.
(A2) lim n → ∞ I n ( β ) / n = Σ , where I n ( β ) is the information matrix of β and Σ is a positive matrix.
Then, Theorem 3.1 in [
κ n j = κ | β ˜ n j | − γ
with β ˜ n j being the j-th components of β ˜ n . From (16) and asymptotic property of β ˜ n we have n 1 / 2 κ n j → 0 × I ( β 0 j ≠ 0 ) + ∞ × I ( β 0 j = 0 ) almost surely as n → ∞ . Where I ( ⋅ ) denotes the indicator function and we presume that 0 × ∞ = 0 . Similar to Wang and Chang, define β ^ n = I n ( ε ) β ˜ n as an adaptive shrinkage estimate (ASE) of β 0 , where I n ( ε ) = d i a g { I n 1 ( ε ) , I n 2 ( ε ) , ⋯ , I n p ( ε ) } is a p × p diagonal matrix. So far, we get good statistical properties of the proposed ASE estimate under non-random sample size, but our goal is to determine a sample size under which the ASE attains the required accuracy. To this end, we will introduce the sequential sampling scheme based on the ASE below. It is known that construction of the confidence set for β 0 depends on the asymptotic distribution of β ^ n and sample size under sequential analysis is a random variable. So we need to study asymptotic properties of ASE under random sample size. Fortunately, property of uniform continuity in probability, see in [
Theorem 1. Suppose that the (A1) and (A2) are satisfied, and let N ( t ) be a positive integer-valued random variable such that N ( t ) / t converges to 1 in probability as t → ∞ . Then
N ( t ) ( β ^ N ( t ) − β 0 ) → N ( 0 , I 0 Σ I 0 − 1 )
in distribution as t → ∞ .
From Theorem 1, we can construct a confidence set of β 0 and a stopping rule on sequential sampling procedure to determine final sample size. Let { ( y i , x i ) : i = 1 , 2 , ⋯ , k } be the first k observations and denoted by C k . Define a stopping rule N d as
N = N d ≡ inf { k : d 2 a k 2 ≥ ν k , ∀ k ≥ n 0 } (5)
For sequential estimation procedure, one new observation is collected at a time until the stopping criterion is satisfied. When the stopping rule holds, based on N samples a confidence set of β 0 is constructed as follow,
R N = { Z ∈ R p : S N N ≤ d 2 ν N ; I N j ( ε ) = 0 → z j = 0 , 1 ≤ j ≤ p } (6)
where S N = ( Z N 1 − β ^ N 1 ) T Σ ˜ 11 ( Z N 1 − β ^ N 1 ) . Properties of the sequential procedure and the confidence set R N are summarized below.
Theorem 2. Assume that the (A1) and (A2) are satisfied, and let N be the
stopping time defined in Equation (5). Then 1) lim d → 0 d 2 N / a 2 ν = 1 almost surely; 2) lim d → 0 d 2 N / a 2 ν = 1 ; 3) lim d → 0 d 2 E ( N ) / a 2 ν = 1 ; 4) lim d → 0 p ^ 0 ( N ) = p 0 almost surely; 5) lim d → 0 E ( p ^ 0 ( N ) ) = p 0 where ν is the maximum eigen-value of matrix I 0 Σ − 1 I 0 .
We evaluate the performance of the proposed method via sequential fixed size confidence estimation using synthesized data sets. As mentioned previously, by the definition of the stopping rule, when sampling is stopped, the final confidence ellipsoid constructed will have the prescribed precision and coverage probability. Thus, we can compare the average stopping times of procedures based on MPLE and ASE. Since the proposed method ignores the non-effective variables, we expect the average stopping time to be significantly smaller than that of the procedure based on MPLE with no variable identification mechanism. If the p 0 variables are known in advance, then the most efficient procedureis, of course, to use only these p 0 variables. Therefore, we also construct a sequential procedure under such a situation, and the results of the cases with known p 0 can serve as the baseline, in which the smallest sample size is achieved, asymptotically.
The synthesized data sets for the model with fixed designs are generated as follows: the regressor x i are generated independently from a standard multivariate normal distribution with mean 0 and identity covariance matrix beforehand, and the error term e i is independently drawn from the standard normal distribution for each i ≥ 1 . The system error is assumed to follow the standard normal distribution. The response generated by model (1) with the arbitrary h 0 ( t ) = t 2 without loss of the generality and the true parameter β 0 = ( − 1.2 , 2.0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ) with 8 non-effective variables. Different precisions of confidence ellipsoid d ∈ { 0.3 , 0.4 , 0.5 , 0.6 } are chosed with coverage probability equal to 95% α = 0.05 in the simulation. We choose γ = 1 , δ = 0.45 and θ = 0.75 in analyzing simulated data. When applying the ASE method, the regularization parameter ε needs to be determined by some model selection criteria, as the AIC, BIC together with a GCV method. For convenience, we only use BIC to illustrate our method,
BIC = − 2 ( ∑ j = 1 N ( β T X ( j ) − log ( ∑ j ∈ R j exp ( β T X i ) ) ) ) + log ( n ) × d f / n ,
where df is the number of the non-zero components in β .
| β 0 = ( − 1.2 , 2.0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ) | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| MPLE p 0 | ASE | MPLE | ||||||||
| Design | d | N | κ * | CP | N | κ | CP | N | κ | CP |
| fixed | 0.6 | 95.740 (14.75)** | 1.028 | 0.95 | 107.62 (17.40) | 1.044 | 0.93 | 305.8 (23.194) | 1.01 | 0.94 |
| 0.5 | 130.44 (19.75) | 1.017 | 0.98 | 141.52 (19.13) | 1.034 | 0.93 | 419.84 (29.586) | 1.006 | 1 | |
| 0.4 | 198.18 (25.587) | 1.008 | 0.93 | 199.394 (25.311) | 1.021 | 0.928 | 632.936 (37.868) | 1.003 | 0.93 | |
| 0.3 | 359.68 (38.211) | 1.004 | 0.95 | 333.676 (37.087) | 1.017 | 0.94 | 1100.05 (44.707) | 1.002 | 0.97 | |
κ * = d 2 N / ( a 2 ν ) ; C P + is the empirical coverage probability of 95% confidence ellipsoid region R N ; **Empirical standard deviations are in parentheses.
| β 1 = − 1.2 , β 2 = 2.0 | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| ASE | MPLE | ||||||||
| Design | d | N i c ∗ | N c ∗ | β 1 | β 2 | N i c ∗ | N c ∗ | β 1 | β 2 |
| fixed | 0.6 | 0 | 7.876 | −1.263 (0.155) | 2.10 (0.210) | - | - | −1.220 (0.09) | 2.051 (0.01) |
| 0.5 | 0 | 7.916 | −1.243 (0.129) | 2.072 (0.173) | - | - | −1.214 (0.007) | 2.041 (0.096) | |
| 0.4 | 0 | 7.932 | −1.232 (0.105) | 2.053 (0.142) | - | - | −1.214 (0.064) | 2.011 (0.082) | |
| 0.3 | 0 | 7.956 | −1.220 (0.076) | 2.037 (0.107) | - | - | −1.202 (0.042) | 2.005 (0.057) | |
N i c ∗ and N c ∗ are the average number of zero components in β correctly identified and nonzero components incorrectly estimated as zero values, respectively.
Based on an ASE estimate of the parameter in COX regression model, a sequential sampling procedure is constructed to estimate the minimum sample size to identify the effective variables and simultaneously make estimate of parameters with required accuracy. We prove that the proposed sequential procedure is asymptotically optimal in the sense of Chow and Robbins [
This research was supported by Research projects of universities in Xinjiang Uygur Autonomous Region under Grant No. XJEDU2016I033 and Xinjiang Normal University postdoctoral research foundation under Grant No. XJNUBS1539.
The authors declare no conflicts of interest regarding the publication of this paper.
H.B., Lu, Zhou, J.L. and Dong, C.L. (2021) Sequential Shrinkage Estimate for COX Regression Models with Uncertain Number of Effective Variables. Modeling and Numerical Simulation of Material Science, 11, 47-53. https://doi.org/10.4236/mnsms.2021.113004