Maximum Partial Likelihood Estimator Research Articles

Fitting the Cox proportional hazards model to big data.

The semiparametric Cox proportional hazards model, together with the partial likelihood principle, has been widely used to study the effects of potentially time-dependent covariates on a possibly censored event time. We propose a computationally efficient method for fitting the Cox model to big data involving millions of study subjects. Specifically, we perform maximum partial likelihood estimation on a small subset of the whole data and improve the initial estimator by incorporating the remaining data through one-step estimation with estimated efficient score functions. We show that the final estimator has the same asymptotic distribution as the conventional maximum partial likelihood estimator using the whole dataset but requires only a small fraction of computation time. We demonstrate the usefulness of the proposed method through extensive simulation studies and an application to the UK Biobank data.

Time series regression models for zero-inflated proportions

Time series of proportions are often encountered in applications such as ecology, environmental science and public health. Strategies for such data include linear regression after logistic transformation. Though easy to fit, the transformation approach renders covariate effects uninterpretable on the scale on which they were observed owing to Jensen's inequality. An alternative to the transformation approach has been to directly model the response via the beta distribution. In this paper, we extend zero-inflated beta regression models for independent proportions to time series data that is bounded over the unit interval and that may take on zero values. Estimation is within the partial-likelihood framework and is computationally feasible to implement. We outline the asymptotic theory of our maximum partial likelihood estimators under mild regularity conditions and investigate their bias and variability using simulation studies. The utility of our method is illustrated using two real data examples.

Debiased lasso for stratified Cox models with application to the national kidney transplant data.

The Scientific Registry of Transplant Recipients (SRTR) system has become a rich resource for understanding the complex mechanisms of graft failure after kidney transplant, a crucial step for allocating organs effectively and implementing appropriate care. As transplant centers that treated patients might strongly confound graft failures, Cox models stratified by centers can eliminate their confounding effects. Also, since recipient age is a proven non-modifiable risk factor, a common practice is to fit models separately by recipient age groups. The moderate sample sizes, relative to the number of covariates, in some age groups may lead to biased maximum stratified partial likelihood estimates and unreliable confidence intervals even when samples still outnumber covariates. To draw reliable inference on a comprehensive list of risk factors measured from both donors and recipients in SRTR, we propose a de-biased lasso approach via quadratic programming for fitting stratified Cox models. We establish asymptotic properties and verify via simulations that our method produces consistent estimates and confidence intervals with nominal coverage probabilities. Accounting for nearly 100 confounders in SRTR, the de-biased method detects that the graft failure hazard nonlinearly increases with donor's age among all recipient age groups, and that organs from older donors more adversely impact the younger recipients. Our method also delineates the associations between graft failure and many risk factors such as recipients' primary diagnoses (e.g. polycystic disease, glomerular disease, and diabetes) and donor-recipient mismatches for human leukocyte antigen loci across recipient age groups. These results may inform the refinement of donor-recipient matching criteria for stakeholders.

Predictive value of intravascular ultrasound for the function of intermediate coronary lesions

BackgroundIntravascular ultrasound (IVUS) can provide detailed coronary anatomic parameters. The purpose of our study was to evaluate the parameters measured by IVUS for the prediction of intermediate coronary lesions function by referencing quantitative fraction ratio (QFR) ≤ 0.80 (vs. > 0.80).MethodsEighty four cases with 92 intermediate coronary lesions in vessels with a diameter ≥ 2.50 mm were enrolled. Paired assessment of IVUS and cQFR was available, and vessels with cQFR ≤ 0.8 were considered the positive reference standard. Logistic regression was used to select model variables by a maximum partial likelihood estimation test and receiver operating characteristic curve (ROC) analysis to evaluate the diagnostic value of different indices.ResultsPlaque burden (PB) and lesion length (LL) of IVUS were independent risk factors for the function of coronary lesions. The predictive probability P was derived from the combined PB and LL model. The area under the curve (AUC) of PB, (minimum lumen area) MLA, and LL and the predicted probability P are 0.789,0.732,0731, and 0.863, respectively (P < 0.01). The AUC of the predicted probability P was the biggest among them; the prediction accuracy of cQFR ≤ 0.8 was 84.8%, and the sensitivity of the diagnostic model was 0.826, specificity was 0. 725, and P < 0.01.ConclusionPB and LL of IVUS were independent risk factors influencing the function of intermediate coronary lesions. The model combining the PB and LL may predict coronary artery function better than any other single parameter.

Pemodelan Waktu Survival Pasien Tuberkulosis menggunakan Regresi Cox Proportional Hazard dengan Data Tersensor

Cox proportional hazard regression is a type of survival analysis that can be applied to tuberculosis cases. This study aims to determine the Cox proportional hazard regression model and the factors that influence the survival time of tuberculosis patients at RSUP Dr. M. Djamil Padang. The survival period used is the time when the patient is taking TB treatment at RSUP Dr. M. Djamil Padang in 2021 until the patient is declared dead. The method used in the Cox Proportional Hazard Regression analysis is the Maximum Partial Likelihood Estimation Method. By using the cox proportional hazard regression model, the factors that influence the survival time of tuberculosis patients at RSUP Dr. M. Djamil's is BMI (X3) , leukocytes (X5) , fever (X9) , shortness of breath (X11) , and decreased appetite (X12) . The Cox Proportional Hazard Regression Model obtained is hi(t) = h0(t) exp(1,315X3 + 1,224X5 + 1,138X9 +1,623X11 + 1,251X12).

Approximating Partial Likelihood Estimators via Optimal Subsampling

With the growing availability of large-scale biomedical data, it is often time-consuming or infeasible to directly perform traditional statistical analysis with relatively limited computing resources at hand. We propose a fast subsampling method to effectively approximate the full data maximum partial likelihood estimator in Cox’s model, which largely reduces the computational burden when analyzing massive survival data. We establish consistency and asymptotic normality of a general subsample-based estimator. The optimal subsampling probabilities with explicit expressions are determined via minimizing the trace of the asymptotic variance-covariance matrix for a linearly transformed parameter estimator. We propose a two-step subsampling algorithm for practical implementation, which has a significant reduction in computing time compared to the full data method. The asymptotic properties of the resulting two-step subsample-based estimator is also established. Extensive numerical experiments and a real-world example are provided to assess our subsampling strategy. Supplemental materials for this article are available online.

Extending the Liu estimator for the Cox proportional hazards regression model with multicollinearity

In this article, we present the Liu estimator for the Cox proportional hazards (PH) model. The maximum partial likelihood estimator (MPLE) is commonly used for estimation of the coefficients of the Cox PH model. The MPLE performs well if the covariates are uncorrelated. However, in many situations, covariates become seriously correlated, and then the MPLE is inept to produce stable estimates for the unknown coefficients. To overcome this situation, the literature suggests using the ridge estimator as an alternative to the MPLE for the Cox PH model in the presence of multicollinearity. In the present article, we extend the Liu estimator, a popular superseder of the ridge estimator, for the Cox PH model and discuss its properties. The performance of the proposed estimator has been compared with the available estimators using the scalar mean squared error criterion through the Monte Carlo simulations. A numerical example has also been provided for the illustration.

Inference in the presence of likelihood monotonicity for proportional hazards regression

Proportional hazards are often used to model event time data subject to censoring. Samples involving discrete covariates with strong effects can lead to infinite maximum partial likelihood estimates. A methodology is presented for eliminating nuisance parameters estimated at infinity using approximate conditional inference. Of primary interest is testing in cases in which the parameter of primary interest has a finite estimate, but in which other parameters are estimated at infinity.

Analysis and asymptotic theory for nested case-control designs under highly stratified proportional hazards models.

Nested case-control sampled event time data under a highly stratified proportional hazards model, in which the number of strata increases proportional to sample size, is described and analyzed. The data can be characterized as stratified sampling from the event time risk sets and the analysis approach of Borgan et al.(Ann Stat 23:1749-1778, 1995) is adapted to accommodate both the stratification and case-control sampling from the stratified risk sets. Conditions for the consistency and asymptotic normality of the maximum partial likelihood estimator are provided and the results are used to compare the efficiency of the stratified analysis to an unstratified analysis when the baseline hazards can be semi-parametrically modeled in two special cases. Using the stratified sampling representation of the stratified analysis, methods for absolute risk estimation described by Borgan et al.(1995) for nested case-control data are used to develop methods for absolute risk estimation under the stratified model. The methods are illustrated by a year of birth stratified analysis of radon exposure and lung cancer mortality in a cohort of uranium miners from the Colorado Plateau.

Active-set algorithm-based statistical inference for shape-restricted generalized additive Cox regression models

Recently the shape-restricted inference has gained popularity in statistical and econometric literature to relax the linear or quadratic covariate effect in regression analyses. The typical shape-restricted covariate effect includes monotone increasing, decreasing, convexity or concavity. In this paper, we introduce the shape-restricted inference to the celebrated Cox regression model (SR-Cox), in which the covariate response is modelled as shape-restricted additive functions. The SR-Cox regression approximates the shape-restricted functions using a spline basis expansion with data-driven choice of knots. The underlying minimization of negative log-likelihood function is formulated as a convex optimization problem, which is solved with an active-set optimization algorithm. The highlight of this algorithm is that it eliminates the superfluous knots automatically. When covariate effects include combinations of convex or concave terms with unknown forms and linear terms, the most interesting finding is that SR-Cox produces accurate linear covariate effect estimates which are comparable to the maximum partial likelihood estimates if indeed the forms are known. We conclude that concave or convex SR-Cox models could significantly improve nonlinear covariate response recovery and model goodness of fit.

Generalized autoregressive moving average models: an efficient estimation approach

In this paper, we propose an efficient estimation approach, so-called the pretest and shrinkage approach in estimating the regression parameters of the generalized autoregressive moving average (GARMA) model, which are pervasive for modelling binary and count time series data. This model accommodates a set of covariates in addition to the ARMA parameters. We want to estimate regression and ARMA parameters when some of the regression parameters may belong to a subspace. We apply the maximum partial likelihood method to obtain the unrestricted maximum partial likelihood estimator(UMPLE) and also the restricted maximum partial likelihood estimator (RMPLE) for the model with parameter restriction and then present the improved pretest and shrinkage estimators. We establish the asymptotic distributional biases and risks of the proposed estimators and evaluate their relative performance with respect to the UMPLE. The performance of the proposed estimators is investigated using simulation studies. A real data example is provided to illustrate the practical usefulness of the estimators.

Dimension reduction for integrative survival analysis.

We propose a constrained maximum partial likelihood estimator for dimension reduction in integrative (e.g., pan-cancer) survival analysis with high-dimensional predictors. We assume that for each population in the study, the hazard function follows a distinct Cox proportional hazards model. To borrow information across populations, we assume that each of the hazard functions depend only on a small number of linear combinations of the predictors (i.e., "factors"). We estimate these linear combinations using an algorithm based on "distance-to-set" penalties. This allows us to impose both low-rankness and sparsity on the regression coefficient matrix estimator. We derive asymptotic results that reveal that our estimator is more efficient than fitting a separate proportional hazards model for each population. Numerical experiments suggest that our method outperforms competitors under various data generating models. We use our method to perform a pan-cancer survival analysis relating protein expression to survival across 18 distinct cancer types. Our approach identifies six linear combinations, depending on only 20 proteins, which explain survival across the cancer types. Finally, to validate our fitted model, we show that our estimated factors can lead to better prediction than competitors on four externaldatasets.

Deep learning for the partially linear Cox model

While deep learning approaches to survival data have demonstrated empirical success in applications, most of these methods are difficult to interpret and mathematical understanding of them is lacking. This paper studies the partially linear Cox model, where the nonlinear component of the model is implemented using a deep neural network. The proposed approach is flexible and able to circumvent the curse of dimensionality, yet it facilitates interpretability of the effects of treatment covariates on survival. We establish asymptotic theories of maximum partial likelihood estimators and show that our nonparametric deep neural network estimator achieves the minimax optimal rate of convergence (up to a polylogarithmic factor). Moreover, we prove that the corresponding finite-dimensional estimator for treatment covariate effects is n-consistent, asymptotically normal and attains semiparametric efficiency. Extensive simulation studies and analyses of two real survival data sets show the proposed estimator produces confidence intervals with superior coverage as well as survival time predictions with superior concordance to actual survival times.

Distributed Cox proportional hazards regression using summary-level information.

Individual-level data sharing across multiple sites can be infeasible due to privacy and logistical concerns. This article proposes a general distributed methodology to fit Cox proportional hazards models without sharing individual-level data in multi-site studies. We make inferences on the log hazard ratios based on an approximated partial likelihood score function that uses only summary-level statistics. This approach can be applied to both stratified and unstratified models, accommodate both discrete and continuous exposure variables, and permit the adjustment of multiple covariates. In particular, the fitting of stratified Cox models can be carried out with only one file transfer of summary-level information. We derive the asymptotic properties of the proposed estimators and compare the proposed estimators with the maximum partial likelihood estimators using pooled individual-level data and meta-analysis methods through simulation studies. We apply the proposed method to a real-world data set to examine the effect of sleeve gastrectomy versus Roux-en-Y gastric bypass on the time to first postoperative readmission.

Efficient estimation method for generalized ARFIMA models

This paper focuses on pretest and shrinkage estimation strategies for generalized autoregressive fractionally integrated moving average (GARFIMA) models when some of the regression parameters are possible to restrict to a subspace. These estimation strategies are constructed on the assumption that some covariates are not statistically significant for the response. To define the pretest and shrinkage estimators, we fit two models: one includes all the covariates and the others are subject to linear constraint based on the auxiliary information of the insignificant covariates. The unrestricted and restricted estimators are then combined optimally to get the pretest and shrinkage estimators. We enlighten the statistical properties of the shrinkage and pretest estimators in terms of asymptotic bias and risk. We examine the comparative performance of pretest and shrinkage estimators with respect to unrestricted maximum partial likelihood estimator (UMPLE). We show that the shrinkage estimators have a lower relative mean squared error as compared to the UMPLE when the number of significant covariates exceeds two. Monte Carlo simulations are conducted to examine the relative performance of the proposed estimators to the UMPLE. An empirical application is used for the usefulness of our proposed estimation strategies.

Survival Analysis Competing Risk using Fine-Gray Subdistribution Model with the Maximum Partial Likelihood Estimation Method

Survival analysis is a statistical method which the variable of concern is the time until the event occurs. In survival analysis, there is a situation where an individual can experience more than one type of event and the occurrence of these events prevents the occurrence of other types of events. This situation is called competing risk. The situation causes the Kaplan Meier method, which is a method of estimating the survival function, cannot be used. The Cumulative Incidence Function method is proposed as a solution to solve competing risk events uses the probability of each type of event. The Cox regression model is also modified to allow for competing risk is called the Fine-Gray subdistribution model using the Maximum Partial Likelihood Estimation. This study examines the estimation of parameter Fine-Gray subdistribution model and applies it to melanoma case. Melanoma is a type skin cancer that can spread to other organs in the body. In the case of death, melanoma patients who experience death with other causes are called a competing risks and death with melanoma are considered as the event of interest. Study showed that factors that influence melanoma mortality are age, sex, tumor thickness, and cultured epidermal autografts. Patients death with melanoma was 1.400 times higher risk than death with other causes.

A unified approach for synthesizing population-level covariate effect information in semiparametric estimation with survival data.

There has been a growing interest in developing methodologies to combine information from public domains to improve efficiency in the analysis of relatively small-scale studies that collect more detailed patient-level information. The auxiliary information is usually given in the form of summary statistics or regression coefficients. Thus, the question arises as to how to incorporate the summary information in the model estimation procedure. In this article, we consider statistical analysis of right-censored survival data when additional information about the covariate effects evaluated in a reduced Cox model is available. Recognizing that such external information can be summarized using population moments, we present a unified framework by employing the generalized method of moments to combine information from different sources for the analysis of survival data. The proposed estimator can be shown to be consistent and asymptotically normal; moreover, it is more efficient than the maximum partial likelihood estimator. We also consider incorporating uncertainty of the external information in the inference procedure. Simulation studies show that, by incorporating the additional summary information, the proposed estimators enjoy a substantial gain in efficiency over the conventional approach. A data analysis of a pancreatic cancer cohort study is presented to illustrate the methods and theory.

Semiparametric Inference for the Functional Cox Model

This article studies penalized semiparametric maximum partial likelihood estimation and hypothesis testing for the functional Cox model in analyzing right-censored data with both functional and scalar predictors. Deriving the asymptotic joint distribution of finite-dimensional and infinite-dimensional estimators is a very challenging theoretical problem due to the complexity of semiparametric models. For the problem, we construct the Sobolev space equipped with a special inner product and discover a new joint Bahadur representation of estimators of the unknown slope function and coefficients. Using this key tool, we establish the asymptotic joint normality of the proposed estimators and the weak convergence of the estimated slope function, and then construct local and global confidence intervals for an unknown slope function. Furthermore, we study a penalized partial likelihood ratio test, show that the test statistic enjoys the Wilks phenomenon, and also verify the optimality of the test. The theoretical results are examined through simulation studies, and a right-censored data example from the Improving Care of Acute Lung Injury Patients study is provided for illustration. Supplementary materials for this article are available online.

Network Imputation for a Spatial Autoregression Model with Incomplete Data

Researchers typically encounter missing data in practice and have developed various imputation methods. However, the existing methods are mainly developed for independent data and the assumption of independence disregards the connections of units through various social relationships (e.g., friendship, follower-followee relationship). In fact, the observed responses from connected friends should provide valuable information for missing responses. This factor motivates us to conduct imputation in this paper by borrowing information from connected friends using a network structure. With the missing at random assumption and using observed information only, we propose a partial likelihood approach and develop the corresponding maximum partial likelihood estimator (MPLE). The estimator’s consistency and asymptotic normality are established. Using the MPLE, we then develop a novel regression imputation method. The method utilizes both auxiliary information and connected complete units (i.e., network information); using the imputed data, we can compute the sample mean of the responses. We show this method to be consistent and asymptotically normal. Compared with the imputation method using auxiliary information only (i.e., ignoring network information), the proposed estimator is statistically more efficient. Extensive simulation studies are conducted to demonstrate its finite sample performance. We then analyze a real example about QQ in mainland China for illustration.

A matlab code to compute prediction of survival trends in patients with DHF

This study aims to create matlab code in estimating the parameters of cox regression model. The written matlab code consists of two algorithms. the first algorithm is to obtain the parameter estimation at the baseline hazard function derived from the weibull distribution while the second is on the cox model.The parameter estimation used in the first algorithm uses maximum likelihood estimation (MLE) and the second algorithm uses the maximum partial likelihood estimation (MPLE). The written matlab code is simulated to predict the survival time of DHF patients. The results is that age and thrombocyte are significant covariates affecting survival time of patients with DHF.