Summary
We propose a computationally and statistically efficient divide-and-conquer (DAC) algorithm to fit sparse Cox regression to massive datasets where the sample size 
is exceedingly large and the covariate dimension 
is not small but 
. The proposed algorithm achieves computational efficiency through a one-step linear approximation followed by a least square approximation to the partial likelihood (PL). These sequences of linearization enable us to maximize the PL with only a small subset and perform penalized estimation via a fast approximation to the PL. The algorithm is applicable for the analysis of both time-independent and time-dependent survival data. Simulations suggest that the proposed DAC algorithm substantially outperforms the full sample-based estimators and the existing DAC algorithm with respect to the computational speed, while it achieves similar statistical efficiency as the full sample-based estimators. The proposed algorithm was applied to extraordinarily large survival datasets for the prediction of heart failure-specific readmission within 30 days among Medicare heart failure patients.
Keywords: Cox proportional hazards model, Distributed learning, Divide-and-conquer, Least square approximation, Shrinkage estimation, Variable selection
1. Introduction
Large datasets derived from health insurance claims and electronic health records are increasingly available for healthcare and medical research. These datasets are valuable sources for the development of risk prediction models, which are the key components of precision medicine. Fitting risk prediction models to a dataset with a massive sample size (
), however, is computationally challenging, especially when the number of candidate predictors (
) is also large and yet only a small subset of the predictors is informative. While it is statistically feasible to fit a full model with all 
variables, deriving parsimonious risk prediction models has the advantage of being more clinically interpretable and easier to implement in practice. In such a setting, it is highly desirable to fit a sparse regression model to simultaneously remove non-informative predictors and estimate the effects of the informative predictors. When the outcome of interest is time-to-event and is subject to censoring, one may obtain a sparse risk prediction model by fitting a regularized Cox proportional hazards model (Cox, 1972) with penalty functions such as the adaptive least absolute shrinkage and selection operator (LASSO) penalty (Zhang and Lu, 2007).
When 
is extraordinarily large, directly fitting an adaptive LASSO (aLASSO) penalized Cox model to such a dataset is not computationally feasible. To overcome the computational difficulty, one may employ the divide-and-conquer (DAC) strategy, which typically divides the full sample into 
subsets, solves the optimization problem using each subset, and combines the subset-specific estimates into a combined estimate. Various DAC algorithms have been proposed to fit penalized regression models. For example, Chen and Xie (2014) proposed a DAC algorithm to fit penalized generalized linear models (GLMs). The algorithm obtains a sparse GLM estimate for each subset and then combine subset-specific estimates by majority voting and averaging. Tang and others (2016) proposed an alternative DAC algorithm to fit GLM with an extremely large 
and a large 
by combining de-biased LASSO estimates from each subset. While both algorithms are effective in reducing the computation burden compared to fitting a penalized regression model to the full data, they remain computationally intensive as 
penalized estimation procedures will be required. In addition, the DAC strategy has not been extended to the survival data analysis.
In this article, we propose a novel DAC algorithm using a sequence of linearizations, denoted by 
, to fit aLASSO penalized Cox proportional hazards models, which can further reduce the computation burden compared to the existing DAC algorithms. 
starts with obtaining an estimator that maximizes the partial likelihood (PL) of a subset of the full data, which is then updated using all subsets via one-step approximations. The updated estimator serves as a 
-consistent initial estimator for the aLASSO problem and approximates the full sample-based maximum PL estimator. Subsequently, we obtain the final aLASSO estimator based on an objective function applying the least square approximation (LSA) to the PL as in Wang and Leng (2007). The LSA allows us to fit the aLASSO using a pseudo-likelihood based on a sample of size 
. The penalized regression is only fit once to a 
dimensional psuedo data in the proposed 
algorithm and the improvement in computation cost is substantial if 
. Our proposed 
algorithm can also accommodate time-dependent covariates.
The rest of the article is organized as follows. We detail the 
algorithm in Section 2. In Section 3, we present simulation results demonstrating the superiority of 
compared to the existing methods when covariates are time-independent and when some covariates are time-dependent. In Section 4, we employ the 
algorithm to develop risk prediction models for 30-day readmission after an index heart failure hospitalization with data from over 10 million Medicare patients by fitting regularized Cox models with (i) 
time-independent covariates and (ii) 
time-independent covariates and 
time-dependent environmental covariates. We conclude with some discussions in Section 5.
2. Methods
2.1. Notation and settings
Let 
denote the survival time and 
denote the 
vector of bounded and potentially time-dependent covariates. Due to censoring, for 
, we only observe 
, where 
, 
, and 
is the censoring time assumed to be independent of 
given 
. Suppose the data for analysis consist of 
subjects with independent realizations of 
, denoted by 
, where we assume that 
.
We denote the index set for the full data by 
. For all DAC algorithms discussed in this article, we randomly partition 
into 
subsets with the 
-th subset denoted by 
. Without loss of generality, we assume that 
is an integer and that the index set for the subset 
is 
. For any index set 
, we denote the size of 
by 
with 
if 
and 
if 
. Throughout we assume that 
such that 
and 
.
We aim to predict 
based on 
via a Cox model with conditional hazard function
 |
(2.1) |
where 
is the baseline hazard function. Our goal is to develop a computationally and statistically efficient procedure to estimate 
using 
under the assumption that 
is sparse with the size of the active set 
much smaller than 
. When 
is not extraordinarily large, we may obtain an efficient estimate, denoted by 
, based on the aLASSO penalized PL likelihood estimator as proposed in Zhang and Lu (2007). Specifically,
 |
(2.2) |
where for any index set 
,
 |
(2.3) |
is an initial 
-consistent estimator of model (2.1), 
is a tuning parameter, and 
. A simple choice of 
is 
, where for any set 
,
 |
Following the arguments given in Zhang and Lu (2007), when 
and 
, we can show that 
achieves the variable selection consistency, i.e. the estimated active set 
satisfies 
and that the oracle property holds, i.e.
 |
where 
if 
is a vector and 
if 
is a matrix,
 |
, 
, and for any vector 
, 
, 
, 
.
When 
is not too large, multiple algorithms are available to solve (2.2) with time-independent covariates, including a coordinate gradient descent algorithm (Simon and others, 2011), a least angle regression-like algorithm (Park and Hastie, 2007), a combination of gradient descent-Newton Raphson method (Goeman, 2010), and a modified shooting algorithm (Zhang and Lu, 2007). Unfortunately, when 
is extraordinarily large, existing algorithms for fitting (2.2) are highly computationally intensive and subject to memory constraints. These algorithms may even be infeasible to compute in the presence of time-dependent covariates as each subject contribute multiple observations in the fitting.
2.2. The 
algorithm
The goal of this article is to develop an estimator that achieves the same asymptotic efficiency as 
but can be computed very efficiently.
Our proposed algorithm, 
, for attaining such a property is motivated by the LSA proposed in Wang and Leng (2007), with the LSA applied to the full sample-based PL. Specifically, it is not difficult to show that 
is asymptotically equivalent to 
, where
 |
That is, 
will also achieve the variable selection consistency as 
and 
has the same limiting distribution as that of 
. This suggests that an estimator can recover the distribution of 
if we can construct accurate DAC approximations to 
and 
. To this end, we propose a linearization-based DAC estimator, denoted by 
, which requires three main steps: (i) obtaining an estimator for the unpenalized problem 
based on a subset, say 
; (ii) obtaining updated estimators for the unpenalized problem through one-step approximations using all 
subsets; and (iii) constructing an aLASSO penalized estimator based on LSA. The procedure also brings a 
that well approximates 
.
Specifically, in step (i), we use subset 
to obtain a standard maximum PL estimator,
 |
In step (ii), we obtain a DAC one-step approximation to 
,
 |
where
 |
(2.4) |
Let 
be our DAC approximation to 
. In practice, we find that it suffices to let 
. Finally, we apply the LSA to the PL and approximate 
using 
, where
 |
The optimization problem in step (iii) is equivalent to
 |
(2.5) |
where 
is a 
vector and 
is a 
matrix. The linearization in step (iii) is exactly the same as that in Zhang and Lu (2007), which allows us to solve the penalized regression step using a pseudo likelihood based on a sample of size 
. The computation cost of this step compared to solving (2.2) reduces substantially when 
. In the Appendix of the supplementary material available at Biostatistics online, we show that 
. It then follows from the similar arguments given in Wang and Leng (2007) that if 
, 
, the estimated active set using 
achieves the variable selection consistency, i.e. 
and the oracle property holds, i.e. 
and 
have the same limiting distribution.
2.3. Tuning and standard error calculation
The tuning parameter 
is chosen by minimizing the Bayesian information criteria (BIC) of the fitted model. Volinsky and Raftery (2000) showed that the exact Bayes factor can be better approximated for the Cox model if the number of uncensored cases, 
, is used to penalize the degrees of freedom in the BIC. Specifically, for any given tuning parameter 
with its corresponding estimate of 
, 
, the BIC suggested by Volinsky and Raftery (2000) is defined as
 |
(2.6) |
where 
. With the LSA, we may further approximate 
by
 |
(2.7) |
For the estimation of 
, we chose a 
such that 
is minimized. The oracle property is expected to hold in the setting where 
and 
is extraordinarily large. We may thus estimate the variance-covariance matrix for 
using 
. For 
, a 
confidence interval for 
can be calculated accordingly.
3. Simulations
3.1. Simulation settings
We performed two sets of simulations to evaluate the performance of 
for the fitting of sparse Cox models, one with only time-independent covariates and the other with time-dependent covariates. For both scenarios, we focused primarily on 
and 
. We consider the number of iterations 
and 
to examine the impact of 
on the proposed estimator.
3.1.1. Time-independent covariates
We conducted extensive simulations to evaluate the performance of the proposed estimator 
relative to (a) the performance of the full sample-based aLASSO estimator for the Cox model 
and (b) a majority voting-based DAC method for the Cox model, denoted by 
also with 
, penalized by a minimax concave penalty (MCP), which extends the majority voting-based DAC scheme for GLM proposed by Chen and Xie (2014). The reason of choosing 
as a comparison is that there is no other DAC method available for the Cox model and only Chen and Xie (2014) considered a similar majority voting-based DAC method for the penalized GLM with non-adaptive penalties. We set a priori that 
sets the estimate of a coefficient at zero, if at least 50% of the subset-specific estimates have a zero estimate for that coefficient. In addition, we compared the performance of the DAC estimator 
relative to the full sample maximum PL estimator 
.
For the penalized procedures, we selected the tuning parameter based on the BIC discussed in Section 2.3. The aLASSO procedures were fit using the 
function in R package 
(Friedman and others, 2010; Simon and others, 2011; R Core Team, 2017) with 
; the MCP procedures were fit using the 
function in R package 
(Breheny and Huang, 2011; R Core Team, 2017). When there are ties among failure times, we used the Efron’s method within each data subset (Efron, 1977).
For the covariates, we considered 
and 
. We generated 
from a multivariate normal distribution with mean 
and variance-covariance matrix 
, where 
denotes a 
vector with all elements being 
and we considered 
, 
, and 
to represent weak, moderate, and strong correlations among the covariates. For a given 
, we generated 
from a Weibull distribution with a shape parameter of 2 and a scale parameter of 
, where we considered three choices of 
to reflect different degrees of sparsity and signal strength:
 |
For censoring, we generated 
from an exponential distribution with a rate parameter of 
, resulting in 
of censoring across different configurations.
We additionally considered 
, 
, 
and 
to evaluate how different choices of 
impact the relative performance of different procedures.
3.1.2. Time-dependent covariates
We also conducted simulations for the settings where time-dependent covariates are present to evaluate the performance of 
. Since neither 
nor 
allows time-dependent survival data, we used 
as a benchmark to compare 
with. In addition, we compared the performance of 
relative to 
.
We considered 
consisting of 
time-independent covariates and 
time-dependent covariates. The simulation of the survival data with time-dependent covariates extended the simulation scheme of Austin (2012) from dichotomous time-dependent covariates to continuous time-dependent covariates. We considered four time intervals 
, 
, 
, and 
, where the time-dependent covariates are constant within each interval but can vary between intervals. We generated 
from a multivariate normal distribution with mean 
and variance-covariance matrix 
, where 
are the time-independent covariates and 
are the time-dependent for 
. We similarly considered 
to represent weak, moderate, and strong correlations.
We generated 
from a Weibull distribution with a shape parameter of 2 and a scale parameter of 
, where 
,
 |
We considered an administrative censoring with 
, leading to 
censoring under the three scenarios represented by weak, moderate, and strong correlations of the design matrix.
3.1.3. Measures of performance
For any 
, we report (i) the average computation time for 
; (ii) the global mean squared error (GMSE), defined as 
; (iii) empirical probability of 
; (iv) the bias of each individual coefficient; and (v) mean squared error (MSE) of each individual coefficient. For 
and 
, we also report the empirical coverage level of the 95% normal confidence interval with standard error estimated as described in Section 2.3. For any 
, we report (i) the average computation time for 
; (ii) the global mean squared error (GMSE), defined as 
.
When only computing time is of interest, we calculate the average computation time for each configuration by averaging over 10 simulated datasets performed on Intel® Xeon® CPU E5-2697 v3 @ 2.60GHz. The average computation time for each configuration in the time-dependent settings is based on simulations using 50 simulated datasets performed on Intel® Xeon® E5-2620 v3 @2.40GHz. The statistical performance is evaluated based on 1000 simulated datasets for each configuration. Although the DAC algorithms can be more efficiently implemented via parallel computing, we report the computational time for DAC algorithms carried out without parallelization for fair comparisons to other algorithms, and for each simulation, we run a single-core job including all the methods under comparison.
3.2. Simulation results
We first show in Table 1, the average computation time and GMSE of unpenalized estimators 
and 
. The results suggest that 
with two iterations (
) attains a GMSE comparable to the full sample-based estimator 
and reduced the computation time by more than 50%. The DAC estimator 
with two iterations (
) has a similar GMSE to 
. Across all settings, the results of 
are nearly identical with 
or 
and hence we summarize below the results for 
only for 
unless noted otherwise.
Table 1.
Comparisons of 
and 
with respect to average computation time in seconds and global mean squared error (GMSE 
) for the estimation of 
, 
, 
, and 
| (a) Time-independent | ||||||||
|---|---|---|---|---|---|---|---|---|
|
|
|
||||||
|
Estimator | Time | GMSE | Time | GMSE | Time | GMSE | |
|
||||||||
| 0.20 |
|
5.2 | 19.3 | 4.5 | 21.1 | 4.3 | 21.2 | |
|
|
10.1 | 19.3 | 8.7 | 21.1 | 8.4 | 21.2 | |
|
15.1 | 19.3 | 12.9 | 21.1 | 12.4 | 21.2 | ||
|
26.1 | 19.2 | 24.0 | 21.0 | 24.2 | 21.1 | ||
| 0.50 |
|
6.3 | 18.1 | 5.8 | 19.5 | 5.5 | 20.4 | |
|
|
10.3 | 18.1 | 9.8 | 19.5 | 9.3 | 20.4 | |
|
14.3 | 18.1 | 13.7 | 19.5 | 13.2 | 20.4 | ||
|
32.9 | 18.0 | 31.3 | 19.4 | 30.3 | 20.4 | ||
| 0.80 |
|
5.3 | 17.9 | 5.2 | 18.7 | 5.1 | 19.7 | |
|
|
9.2 | 17.8 | 9.0 | 18.7 | 8.9 | 19.7 | |
|
13.1 | 17.8 | 12.9 | 18.7 | 12.8 | 19.7 | ||
|
30.3 | 17.7 | 29.6 | 18.6 | 29.2 | 19.7 | ||
|
||||||||
| 0.20 |
|
10.2 | 74.9 | 13.9 | 83.8 | 17.1 | 85.0 | |
|
|
19.0 | 74.5 | 26.4 | 83.4 | 32.7 | 84.6 | |
|
27.8 | 74.5 | 38.8 | 83.4 | 48.2 | 84.6 | ||
|
62.0 | 74.3 | 83.1 | 83.2 | 102 | 84.4 | ||
| 0.50 |
|
20.5 | 69.0 | 22.8 | 76.5 | 24.7 | 80.6 | |
|
|
39.3 | 68.3 | 43.9 | 76.1 | 47.7 | 80.3 | |
|
58.0 | 68.3 | 64.8 | 76.1 | 70.5 | 80.3 | ||
|
126 | 68.1 | 142 | 75.9 | 151 | 80.1 | ||
| 0.80 |
|
26.9 | 66.0 | 28.6 | 72.0 | 30.0 | 77.3 | |
|
|
51.9 | 65.0 | 55.2 | 71.5 | 57.9 | 76.9 | |
|
76.8 | 65.0 | 81.7 | 71.5 | 85.7 | 76.9 | ||
|
169 | 64.9 | 180 | 71.3 | 190 | 76.7 | ||
| (b) Time-dependent | ||||
|
||||
|
Estimator | Time | GMSE | |
| 0.2 |
|
62.0 | 17.9 | |
|
|
121 | 17.9 | |
|
179 | 17.9 | ||
|
262 | 18.0 | ||
| 0.5 |
|
58.0 | 17.9 | |
|
|
112 | 17.9 | |
|
166 | 17.9 | ||
|
263 | 17.9 | ||
| 0.8 |
|
58.3 | 18.0 | |
|
|
114 | 18.0 | |
|
169 | 18.0 | ||
|
254 | 17.9 | ||
3.2.1. Computation time
The computation time is summarized in Tables 2 and 3 for 
and 
, and Tables S1 and S2 of supplementary material available at Biostatistics online for 
for time-independent survival data. There are substantial differences in computation time across methods. Across different settings, the average computation time of 
ranges from 
to 
s for 
and from 
to 
s for 
, with virtually all time spent on the computation of the unpenalized estimator 
. On the contrary, 
requires a substantially longer computation time with average time ranging from 
to 
s for 
and from 
to 
s for 
. This suggests that the computation time of 
is about 
of the full sample estimator when 
and about 
when 
. On the other hand, 
has a substantially longer average computation time than 
. This is because the MCP procedure, requiring more computational time than the aLASSO, needs to be fitted 
times. As shown in Table S3 of supplementary material available at Biostatistics online, when the sample size varies from 
to 
, the computation time increases for all methods but the increase is more drastic for 
and 
. The full sample estimator 
can only be calculated when requesting a very large memory (150GB) when 
while 
can be computed with a much smaller memory.
Table 2.
Comparisons of 
, 
, and 
for estimating 
with respect to average computation time in seconds, GMSE (
), coefficient-specific empirical probability (%) of 
, bias (
), MSE (
), and empirical coverage probability (%) of the confidence intervals
|
|
|
||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
|
|
|
|
|
|
|
|
||||||||
|
|
|
|
|
|
|
|
|
|
|||||||
|
||||||||||||||||
| TIME | 5.20 | 10.2 | 15.1 | 443 | 1451 | 6.30 | 10.3 | 14.3 | 499 | 1642 | 5.30 | 9.20 | 13.1 | 500 | 1652 | |
| GMSE | 4.32 | 4.27 | 4.27 | 4.24 | 5.61 | 4.53 | 4.42 | 4.42 | 4.41 | 7.40 | 5.17 | 5.01 | 5.01 | 4.97 | 9.91 | |
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | –0.74 | 0.49 | 0.49 | 0.63 | 12.9 | –1.64 | –0.24 | –0.24 | –0.19 | 12.3 | –0.27 | 1.16 | 1.16 | 1.16 | 14.7 | |
| MSE | 0.55 | 0.55 | 0.55 | 0.55 | 0.72 | 0.70 | 0.70 | 0.70 | 0.70 | 0.86 | 1.50 | 1.50 | 1.50 | 1.50 | 1.76 | |
| CovP | 94.4 | 94.7 | 94.7 | — | — | 94.4 | 94.7 | 94.7 | — | — | 94.6 | 94.7 | 94.7 | — | — | |
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | –1.47 | –0.85 | –0.85 | –0.72 | 5.61 | –0.63 | 0.07 | 0.07 | 0.03 | 6.55 | –0.02 | 0.72 | 0.72 | 0.73 | 8.81 | |
| MSE | 0.49 | 0.48 | 0.48 | 0.48 | 0.52 | 0.55 | 0.55 | 0.55 | 0.55 | 0.60 | 1.48 | 1.48 | 1.48 | 1.46 | 1.60 | |
| CovP | 93.0 | 93.3 | 93.3 | — | — | 94.9 | 95.1 | 95.1 | — | — | 95.0 | 95.1 | 95.1 | — | — | |
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | –1.39 | –1.09 | –1.09 | –1.18 | 2.75 | –0.82 | –0.49 | –0.49 | –0.44 | 3.88 | –3.37 | –3.02 | –3.02 | –2.87 | –3.18 | |
| MSE | 0.44 | 0.44 | 0.44 | 0.44 | 0.46 | 0.63 | 0.63 | 0.63 | 0.63 | 0.67 | 1.49 | 1.48 | 1.48 | 1.48 | 1.76 | |
| CovP | 94.5 | 94.4 | 94.4 | — | — | 94.4 | 94.3 | 94.3 | — | — | 94.4 | 94.4 | 94.4 | — | — | |
|
%zero | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 |
| Bias | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| MSE | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
|
||||||||||||||||
| TIME | 10.2 | 19.1 | 27.8 | 607 | 2246 | 20.6 | 39.4 | 58.1 | 859 | 3316 | 27.0 | 52.0 | 76.9 | 1016 | 3641 | |
| GMSE | 4.62 | 4.10 | 4.10 | 4.08 | 5.50 | 5.83 | 4.61 | 4.61 | 4.61 | 9.50 | 6.91 | 5.05 | 5.05 | 5.06 | 17.0 | |
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | –5.46 | 0.31 | 0.31 | 0.27 | 13.2 | –6.75 | 0.25 | 0.25 | 0.33 | 16.0 | –6.38 | 0.94 | 0.94 | 0.84 | 20.7 | |
| MSE | 0.58 | 0.53 | 0.53 | 0.53 | 0.71 | 0.77 | 0.70 | 0.70 | 0.70 | 0.97 | 1.49 | 1.45 | 1.45 | 1.45 | 1.96 | |
| CovP | 94.4 | 94.7 | 94.7 | — | — | 94.2 | 94.7 | 94.7 | — | — | 94.3 | 94.8 | 94.8 | — | — | |
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | –2.09 | 0.80 | 0.80 | 0.81 | 7.88 | –3.11 | 0.29 | 0.29 | 0.43 | 8.89 | –3.47 | 0.24 | 0.24 | 0.57 | 11.4 | |
| MSE | 0.45 | 0.44 | 0.44 | 0.44 | 0.51 | 0.68 | 0.66 | 0.66 | 0.66 | 0.76 | 1.42 | 1.42 | 1.42 | 1.42 | 1.62 | |
| CovP | 94.9 | 95.0 | 95.0 | — | — | 94.7 | 94.8 | 94.8 | — | — | 94.8 | 94.9 | 94.9 | — | — | |
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | –4.15 | –2.68 | –2.68 | –2.69 | 0.40 | –4.15 | –2.44 | –2.44 | –2.52 | 3.63 | –2.02 | –0.15 | –0.15 | –0.09 | 1.09 | |
| MSE | 0.44 | 0.43 | 0.43 | 0.43 | 0.45 | 0.61 | 0.60 | 0.60 | 0.60 | 0.61 | 1.34 | 1.33 | 1.33 | 1.33 | 1.62 | |
| CovP | 94.5 | 94.6 | 94.6 | — | — | 94.5 | 94.6 | 94.6 | — | — | 94.7 | 94.9 | 94.9 | — | — | |
|
%zero | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 |
| Bias | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| MSE | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
Table 3.
Comparisons of 
, 
, and 
for estimating 
with respect to average computation time in seconds, GMSE (
), coefficient-specific empirical probability (%) of 
, bias (
), MSE (
), and empirical coverage probability (%) of the confidence intervals
|
|
|
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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
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|
|
|
|
|
|
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|
|
|
|
|
|
|
|
|
|
|||||||
|
||||||||||||||||
| TIME | 4.5 | 8.7 | 12.9 | 461 | 1554 | 5.9 | 9.8 | 13.7 | 497 | 1628 | 5.2 | 9.1 | 12.9 | 499 | 1657 | |
| GMSE | 7.27 | 7.24 | 7.24 | 7.22 | 870 | 6.92 | 6.86 | 6.86 | 6.84 | 1176 | 7.21 | 7.12 | 7.12 | 7.17 | 4452 | |
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | 1.14 | 1.59 | 1.59 | 1.60 | 97.8 | 1.47 | 2.07 | 2.07 | 2.11 | 110 | 4.82 | 5.45 | 5.45 | 1.39 | 219 | |
| MSE | 0.53 | 0.53 | 0.53 | 0.53 | 10.1 | 0.71 | 0.71 | 0.71 | 0.71 | 12.8 | 1.70 | 1.70 | 1.70 | 1.68 | 49.6 | |
| CovP | 94.4 | 94.8 | 94.8 | — | — | 94.5 | 94.9 | 94.9 | — | — | 94.5 | 94.8 | 94.8 | — | — | |
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | 0.72 | 0.94 | 0.94 | 1.14 | 98.3 | 1.61 | 1.90 | 1.90 | 2.02 | 108 | 3.63 | 3.95 | 3.95 | 1.06 | 206 | |
| MSE | 0.50 | 0.50 | 0.50 | 0.50 | 10.2 | 0.76 | 0.76 | 0.76 | 0.75 | 12.4 | 1.65 | 1.65 | 1.65 | 1.66 | 44.2 | |
| CovP | 95.0 | 95.1 | 95.1 | — | — | 95.0 | 95.1 | 95.1 | — | — | 95.0 | 95.1 | 95.1 | — | — | |
|
%zero | 0 | 0 | 0 | 0 | 61.6 | 0 | 0 | 0 | 0 | 100 | 0 | 0 | 0 | 0 | 100 |
| Bias | –4.21 | –4.16 | –4.16 | –4.21 |
 407 |
–5.65 | –5.55 | –5.55 | –5.52 |
 500 |
–11.4 | –11.3 | –11.3 | –7.22 |
 500 |
|
| MSE | 0.50 | 0.50 | 0.50 | 0.49 | 180 | 0.71 | 0.71 | 0.71 | 0.70 | 250 | 2.08 | 2.07 | 2.07 | 2.05 | 250 | |
| CovP | 95.2 | 95.2 | 95.2 | — | — | 94.8 | 94.9 | 94.9 | — | — | 91.5 | 91.7 | 91.7 | — | — | |
|
%zero | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 99.9 | 100 | 100 | 100 | 100 |
| Bias | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.09 | 0 | 0 | 0 | 0 | |
| MSE | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.01 | 0 | 0 | 0 | 0 | |
|
||||||||||||||||
| TIME | 14.0 | 26.5 | 38.9 | 696 | 2755 | 22.9 | 43.9 | 64.9 | 920 | 3465 | 28.7 | 55.3 | 81.8 | 1051 | 3764 | |
| GMSE | 7.53 | 7.29 | 7.29 | 7.28 | 1496 | 7.58 | 7.05 | 7.05 | 7.03 | 1300 | 8.38 | 7.46 | 7.46 | 7.67 | 12696 | |
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | –0.02 | 2.00 | 2.00 | 2.07 | 139 | 0.84 | 3.59 | 3.59 | 3.58 | 125.2 | 5.70 | 8.81 | 8.81 | 0.58 | 172 | |
| MSE | 0.54 | 0.54 | 0.54 | 0.54 | 19.8 | 0.80 | 0.79 | 0.79 | 0.79 | 16.5 | 1.62 | 1.67 | 1.67 | 1.62 | 31.2 | |
| CovP | 95.4 | 94.9 | 94.9 | — | — | 94.8 | 94.9 | 94.9 | — | — | 96.3 | 96.2 | 96.2 | — | — | |
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | –0.02 | 0.99 | 0.99 | 1.08 | 142 | 0.15 | 1.59 | 1.59 | 1.77 | 122 | 3.94 | 5.37 | 5.37 | 5.56 | 151 | |
| MSE | 0.49 | 0.49 | 0.49 | 0.49 | 20.8 | 0.75 | 0.75 | 0.75 | 0.74 | 15.8 | 1.72 | 1.73 | 1.73 | 1.79 | 25.1 | |
| CovP | 96.4 | 96.4 | 96.4 | — | — | 95.4 | 95.2 | 95.2 | — | — | 95.1 | 95.3 | 95.3 | — | — | |
|
%zero | 0 | 0 | 0 | 0 | 99.1 | 0 | 0 | 0 | 0 | 100 | 0 | 0 | 0 | 0 | 100 |
| Bias | –6.39 | –6.07 | –6.07 | –6.08 |
 498 |
–8.32 | –7.92 | –7.92 | –7.83 |
 500 |
–18.5 | –18.1 | –18.1 | –15.4 |
 500 |
|
| MSE | 0.59 | 0.57 | 0.57 | 0.57 | 249 | 0.84 | 0.82 | 0.82 | 0.82 | 250 | 2.38 | 2.38 | 2.38 | 2.43 | 250 | |
| CovP | 93.6 | 93.2 | 93.2 | — | — | 93.1 | 93.5 | 93.5 | — | — | 90.5 | 91.0 | 91.0 | — | — | |
|
%zero | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 |
| Bias | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| MSE | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
In the presence of time-dependent covariates, Table 4 shows that 
has an average computation time of 112–121 s for 
and 
; 
has an average computation time of 254–264 s. Virtually all computation time for 
and 
is spent on the computation of the unpenalized initial estimator 
, which has more observations and requires substantially more computation time compared to the setting with time-independent covariates given the same 
and 
.
Table 4.
Performance of 
and 
for estimating 
with respect to average computation time in seconds, GMSE (
), coefficient-specific empirical probability (%) of 
, bias (
), MSE (
), and empirical coverage probability (%) of the confidence intervals
|
|
|
|||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
|
|
|
|
|
||||||||
|
|
|
|
|
|
|
|
|
|
||||
| Time | 62.0 | 121 | 179 | 262 | 58.0 | 112 | 166 | 264 | 58.3 | 114 | 169 | 254 | |
| GMSE | 3.55 | 3.55 | 3.55 | 3.55 | 3.68 | 3.68 | 3.68 | 3.69 | 4.44 | 4.44 | 4.44 | 4.44 | |
| Time-independent covariates | |||||||||||||
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | 1.45 | 1.46 | 1.46 | 1.46 | 4.62 | 4.63 | 4.63 | 4.65 | 11.9 | 11.9 | 11.9 | 12.0 | |
| MSE | 0.22 | 0.22 | 0.22 | 0.22 | 0.36 | 0.36 | 0.36 | 0.36 | 1.02 | 1.02 | 1.02 | 1.03 | |
| CovP | 95.7 | 95.7 | 95.7 | 95.8 | 94.6 | 94.6 | 94.6 | 94.7 | 93.9 | 93.9 | 93.9 | 93.5 | |
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | –0.19 | –0.19 | –0.19 | –0.14 | 0.53 | 0.54 | 0.54 | 0.61 | 3.45 | 3.47 | 3.47 | 3.45 | |
| MSE | 0.22 | 0.22 | 0.22 | 0.22 | 0.33 | 0.33 | 0.33 | 0.33 | 0.93 | 0.93 | 0.93 | 0.93 | |
| CovP | 95.4 | 95.5 | 95.5 | 95.5 | 96.0 | 96.1 | 96.1 | 96.0 | 93.7 | 93.8 | 93.8 | 93.7 | |
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | –4.55 | –4.56 | –4.56 | –4.58 | –6.79 | –6.78 | –6.78 | –6.89 | –17.4 | –17.4 | –17.4 | –17.2 | |
| MSE | 0.25 | 0.25 | 0.25 | 0.25 | 0.45 | 0.45 | 0.45 | 0.45 | 1.59 | 1.59 | 1.59 | 1.57 | |
| CovP | 94.1 | 94.1 | 94.1 | 94.4 | 90.5 | 90.8 | 90.8 | 90.8 | 86.8 | 86.7 | 86.7 | 86.8 | |
|
%zero | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 |
| Bias | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| MSE | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| Time-independent covariates | |||||||||||||
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | 1.01 | 1.01 | 1.01 | 1.04 | 2.65 | 2.65 | 2.65 | 2.61 | 9.72 | 9.74 | 9.74 | 9.74 | |
| MSE | 0.21 | 0.21 | 0.21 | 0.21 | 0.37 | 0.37 | 0.37 | 0.38 | 0.93 | 0.93 | 0.93 | 0.95 | |
| CovP | 96.3 | 96.3 | 96.3 | 96.3 | 94.6 | 94.6 | 94.6 | 94.4 | 95.6 | 95.5 | 95.5 | 95.4 | |
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | –1.01 | –1.01 | –1.01 | –1.01 | 1.11 | 1.11 | 1.11 | 1.15 | 3.26 | 3.26 | 3.26 | 3.19 | |
| MSE | 0.21 | 0.21 | 0.21 | 0.21 | 0.33 | 0.33 | 0.33 | 0.33 | 0.93 | 0.93 | 0.93 | 0.94 | |
| CovP | 95.8 | 95.9 | 95.9 | 96.0 | 95.2 | 95.2 | 95.2 | 95.3 | 94.9 | 94.8 | 94.8 | 95.0 | |
|
%zero | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Bias | –3.81 | –3.81 | –3.81 | –3.81 | –6.28 | –6.26 | –6.26 | –6.30 | –16.5 | –16.5 | –16.5 | –16.6 | |
| MSE | 0.26 | 0.26 | 0.26 | 0.25 | 0.45 | 0.45 | 0.45 | 0.45 | 1.49 | 1.48 | 1.48 | 1.49 | |
| CovP | 93.8 | 93.8 | 93.8 | 93.5 | 91.2 | 91.1 | 91.1 | 91.4 | 86.2 | 86.1 | 86.1 | 86.0 | |
|
%zero | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 |
| Bias | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| MSE | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
3.2.2. Statistical performance
The results for the simulation scenarios with only time-independent covariates are summarized in Tables 2 and 3 for 
and 
, and Tables S1 and S2 of supplementary material available at Biostatistics online for 
. In general, 
is able to achieve a statistical performance comparable to 
, while 
generally has a worse performance, with respect to the GMSE and variable selection, bias, and MSE of individual coefficient. For example, as shown in Table 2, the GMSEs (
) for 
, 
, and 
are respectively 
, 
, and 
when 
and 
; 
, 
, 
when 
and 
. The relative performance of different procedures has similar patterns across different levels of correlation 
among the covariates. When the signals are relatively strong and sparse as for 
or 
, 
and 
have small biases and achieved perfect variable selection, while 
substantially excludes the 
signal when 
. For the more challenging case of 
where some of the signals are weak, the variable selection of 
and 
is also near perfect. Both penalized estimators for the weakest signal (0.035) exhibit a small amount of bias when 
and 
and an increased bias when 
. Such biases in the weak signals are expected for shrinkage estimators (Menelaos and others, 2016), especially in the presence of high correlation among covariates. However, it is important to note that 
and 
perform nearly identically, suggesting that our 
procedure incurs negligible additional approximation errors. On the other hand, 
has difficulty in detecting the 0.05 and 0.035 signals and tends to produce substantially higher MSE than 
.
The empirical coverage levels for the confidence intervals are close to the nominal level across all settings except for the very challenging setting with very weak signals when the correlation is 
. This again is due to the bias inherent in shrinkage estimators. The relative performance of 
, 
and 
remains similar across different sample sizes. When 
varies from 
to 
, 
remains the best performing estimator with precision comparable to 
while maintaining substantial advantage with respect to computational efficiency.
The results for the time-dependent survival are summarized in Table 4. We find that 
also generally has a good performance in estimating 
for both time-independent and time-dependent covariates. The variable selection consistency holds perfectly for all parameters of interest. The coverage of the confidence intervals also has similar patterns as the case with time-independent covariates.
The proposed DAC approach was implemented as an R software package divideconquer, which is available at https://github.com/michaelyanwang/divideconquer.
4. Application of the DAC procedure to Medicare data
We applied the proposed 
algorithm to develop risk prediction models for heart failure-specific readmission or death within 30 days of discharge among Medicare patients who were admitted due to heart failure. The Medicare inpatient claims were assembled for all Medicare fee-for-service beneficiaries during 
to identify the eligible study population. The index date was defined as the discharge date of the first heart failure admission of each patient. We restricted the study population to patients who were discharged alive from the first heart failure admission. The outcome of interest was time to heart failure-specific readmission or death after the first heart failure admission. Because readmission rates within 30 days were used to assess the quality of care at hospitals by the Centers for Medicare and Medicaid Services (CMS) (CMS, 2016), we censored the time to readmission at 30 days. For a patient who was readmitted or died on the same day as discharge (whose claim did not indicate discharge dead), the time-to-event was set at 0.5 days. Due to the large number of ICD-9 codes, we classified each discharge ICD-9 code into disease phenotypes indexed by phenotype codes according to Denny and others (2013). A heart failure admission or readmission was identified if the claim for that admission or readmission had a heart failure phenotype code at discharge.
We considered two sets of covariates: (i) time-independent covariates including baseline individual-level covariates collected at time of discharge from the index heart failure hospitalization, baseline area-level covariates at the residential ZIP code of each patient, and indicators for time trend including dummy variables for each year and dummy variables for each months, and (ii) time-dependent predictors that vary day-by-day. Baseline individual-level covariates included age, sex, race (white, black, others), calendar year, and month of the discharge, Charlson comorbidity index (CCI) (Quan and others, 2005) which described the degree of illness of a patient, and indicators for nonrare comorbidities (defined as prevalence 
among the study population). Baseline area-level covariates included socioeconomic status variables (percent black residents [ranging from 0 to 1], percent Hispanic residents [ranging from 0 to 1], median household income [per 10 000 increase], median home value [per 10 000 increase], percent below poverty [ranging from 0 to 1], percent below high school [ranging from 0 to 1], percent owned houses [ranging from 0 to 1]), population density (1000 per squared kilometer), and health status variables (percent taking hemoglobin A1C test [ranging from 0 to 1], average BMI, percent ambulance use [ranging from 0 to 1], percent having low-density lipoprotein test [ranging from 0 to 1], and smoke rate [ranging from 0 to 1]). The time-dependent covariates included daily fine particulate matter (PM
) predicted using a neural network algorithm (Di and others, 2016), daily temperature with its quadratic form, and daily dew point temperature with its quadratic form. There were 574 time-independent covariates and five time-dependent covariates.
There were 
eligible patients with a total of 
heart failure readmissions or deaths, among which 
were readmissions and 
were deaths. After expanding the dataset by accounting for time-dependent variables which vary day-by-day, the time-dependent dataset contained 
rows of records.
We fitted cause-specific Cox models for readmission due to heart failure or deaths as a composite outcome, considering two separate models: (i) a model containing only time-independent covariates and (ii) a model incorporating time-dependent covariates. In both cases, the datasets were too large for 
package to analyze as a whole, demonstrating the need for 
.
4.1. Time-independent covariates only
We applied 
with 
and paralleled 
on 25 Authentic AMD Little Endian @2.30GHz CPUs. Computing 
with 
took 1.1 h, including the time of reading datasets from hard drives during each iteration of the update of the one-step estimator. Figure 1a shows the hazard ratio of each covariate based on 
with 
predicting heart failure-specific readmission and death within 30 days.
Fig. 1.
Hazard ratios of each covariate in estimating hazard of heart failure readmissions or death within 30 days after the first admission using 
. (a) time-independent variables, (a) time-independent variables (continue), (b) time-dependent variables, and (b) time-dependent variables (continue).
Fig. 1.
Continued. a(3) and a(4) time-independent variables.
Fig. 1.
Continued. b(3) and b(4) time-dependent variables.
Multiple comorbidities are associated with an increased risk of 30-day readmission or death with the leading factors including renal failures, cancers, malnutrition, subdural or intracerebral hemorrhage, myocardial infarction, endocarditis, respiratory failure, and cardiac arrest. CCI is also associated with an increased hazard of the outcome. These findings are generally consistent with those reported in the literature. For example, Philbin and DiSalvo (1999) reported that ischemic heart disease, diabetes, renal diseases, and idiopathic cardiomyopathy were associated with an increased risk of heart failure-specific readmission within a year. Leading factors negatively associated with readmissions include virus infections, asthma, and chronic kidney disease in earlier stages. These negative association findings are reflective of both clinical practice patterns and the biological effects, as most of the negative predictors are generally less severe than the positive predictors.
Some socioeconomic status predictors are relatively less important in predicting the outcome after accounting for the phenotypes, where percentage of black people, median household income, and percent below poverty are dropped and dual eligibility, median home value, percent below high school has a small hazard ratio. By comparison, Philbin and others (2001) reported a decrease in readmission as neighborhood income increased. Foraker and others (2011) reported that given comorbidity measured by CCI, the readmission or death hazard was higher for low socioeconomic status patients. The present article considered more detailed phenotypes in addition to CCI suggesting a relatively smaller impact of socioeconomic status. The difference in results is possibly because comorbidity may be on the causal pathway between socioeconomic status and readmission or death. Adjusting for a detailed set of co-morbidities partially blocks the effect of socioeconomic status. Percent Hispanic residents is negatively associated with readmission or death. Percent occupied houses increase the risk of readmission or death, which is consistent with the strong positive prediction by population density. Most ecological health variables showed a small hazard ratio.
Black and other race groups have a lower hazard than white. Females have a lower hazard than males, which is consistent with Roger and others (2004) that females had a higher survival rate than males after heart failure. Age is associated with an increased hazard of readmission or death, as expected.
The coefficient by month suggests a higher risk of readmission or death in cold seasons than warm seasons, with a larger negative hazard ratio for summer indicators. The short-term readmission or death rate is decreasing over time, which is suggested by the negative hazard ratio of later years. The later calendar year being negatively associated with readmission risk may be an indication of improved follow-up care for patients discharged from heart failure. Consistently, (Roger and others, 2004) also suggested an improved heart failure survival rate over time.
4.2. Incorporating time-dependent covariates
The analysis in Section 4.2 has two goals. First, the covariates serve as the risk predictors of the hazard of heart failure-specific readmission. Second, all covariates other than PM
serve as the potential confounders of the association between PM
and readmission, particularly time trend and area-level covariates. The 
procedure is a variable selection technique to drop non-informative confounders given the high dimensionality of confounders. This goal aligns with Belloni and others (2014), which constructs separate penalized regressions for the propensity score model and outcome regression model to identify confounders. We, herein, focused on building a penalized regression for the outcome regression model.
We applied 
algorithm with 
to this time-dependent survival dataset. The procedure was paralleled on 10 Authentic AMD Little Endian @2.30GHz CPUs. The estimation of 
with 
took 36.5 h, including the time of loading the datasets into memory. The result suggests each 10 
increase in daily PM
was associated with 
increase of risk (95% confidence interval: 0.3–0.7) adjusting for individual-level, area-level covariates, and temperature. Because there is rare evidence on whether air pollution is associated with heart failure-specific readmission or death among heart failure patients and it is rare to estimate the health effect of daily air pollution using a time-dependent Cox model, this model provides a novel approach to address a new research question. While evidence is rare on the association between daily PM
and heart failure-specific readmission, some studies used case-crossover design to estimate the effect of short-term PM
on the incidence of heart failure admissions. Pope and others (2008) found that a 10 
increase in 14-day moving average PM
was associated with a 13.1 (1.3–26.2) increase in the incidence of heart failure admissions among elderly patients; Zanobetti and others (2009) reported that each 10 
increase in 2-day averaged PM
was associated with a 1.85 (1.19–2.51) increase in the incidence of congestive heart failure admission. There is also a large body of literature suggesting that short-term exposure to PM
is associated with an increased risk of death. For example, Di and others (2017) shows among the Medicare population during 2000–2012 that each 10 
increase in PM
was associated with an 1.05% (0.95–1.15) increase in mortality risk. In addition, Figure 1b shows the covariate-specific estimates of the hazard ratio for all the covariates, with the estimates consistent with the analysis of time-independent dataset.
Fig. 1.
Continued. b(1) and b(2) time-dependent variables.
5. Discussion
The proposed 
procedure for fitting aLASSO penalized Cox model reduces the computation cost, while it maintains the precision of estimation and accuracy in variable selection with an extraordinarily large 
and a numerically large 
. The use of 
makes it feasible to obtain the 
-consistent estimator required by the penalized step (e.g. when there is a constraint in RAM) and shortens the computation time of the initial estimator by 
. The improvement in the computation time was substantial in the regularized regression step. The LSA converted the fitting of regularized regression from using a dataset of size 
to a dataset of size 
.
The majority voting-based method 
with MCP (Chen and Xie, 2014) had a substantially longer computation time than 
. The difference primarily comes from (i) the fact that the Cox model with MCP is fitted 
times and (ii) the computational efficiency between 
algorithm which is more efficient than the MCP algorithm in 
(Breheny and Huang, 2011).
The difference in variable selection between 
and 
(Chen and Xie, 2014) is primarily due to the majority voting. The simulations in Chen and Xie (2014) have shown that an increase in the percentage for the majority voting decreased the sensitivity and increased the specificity of variable selection. Similarly in the simulations of the present study with a 
of the majority vote, Chen and Xie’s procedure showed a high specificity but the sensitivity was low for weaker signals as demonstrated in the simulation studies.
For non-weak signals, the oracle property appears to hold well as evidenced by the simulation results for 
and 
shown in Tables 2 and 3. For weak signals such as 0.035 in 
, the oracle property does not appear to hold even with 
and the bias in the shrinkage estimators results in confidence intervals with low coverage. This is consistent with the impossibility result shown in Potscher and Schneider (2009), which suggests difficulty in deriving precise interval estimators when aLASSO penalty is employed.
When parallel computing is available, a larger 
may be preferable for our algorithm as it can reduce the overall computational time provided that 
. When 
is not large relative to 
, we may increase the stability of the algorithm by replacing 
with 
for some 
such that 
. Potential approaches to further reduce the computation burden for large 
settings include employing a screening step or employing DAC in the second step of the algorithm. However, deriving the statistical properties of DAC algorithms in the large 
setting can be more involved and warrants further research.
Supplementary Material
Acknowledgments
Conflict of Interest: None declared.
Funding
USEPA (RD-83587201), an National Institutes of Health (U54 HG007963), and National Institute of Environmental Health Sciences (R01 ES024332 and P30 ES000002). Its contents are solely the responsibility of the grantee and do not necessarily represent the official views of the USEPA. Further, USEPA does not endorse the purchase of any commercial products or services mentioned in the publication. The analysis of the Medicare data was performed on the secured clusters of Research Computing, Faculty of Arts and Sciences at Harvard University.
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