Additive Risk Models for Survival Data with High‐Dimensional Covariates

Goodness-of-fit tests for Cox's proportional hazards model and Aalen's additive risk model, in which each model is compared on an equal footing with the best fitting fully nonparametric model, are developed. The goodness-of-fit statistics are based on differences each model, with a fully nonparametric estimator of d recently introduced by the authors. Here A( , z) denotes the conditional hazard function of the survival time of an individual with covariate vector z. Comparison of the results of the tests makes it possible to decide whether Cox's proportional hazards or Aalen's additive risk model gives a better fit to the data. In addition, a goodness-of-fit test for Cox's model within the family of all proportional hazards models A(t, z) = AO(t)r(z), where AO is a baseline hazard function and r is a general relative risk function, is developed. Additive hazards and proportional hazards regression models used in the analysis of censored survival data can give substantially different results. For instance, in connection with a study of cancer mortality among Japanese atomic bomb survivors, Muirhead & Darby (1987) have noted that the two models give substantially different estimates of the age-spe- cific probability that an individual will develop radiation induced cancer. Muirhead & Darby (see also Aranda-Ordaz, 1983) introduced a generalized parametric model which contains parametric additive hazards and proportional hazards models as special cases. The goodness- of-fit of each model is then obtained by comparing with the best fitting model within the generalized family, allowing the two special models to be treated on an equal footing. Beyond the parametric setting, much effort has been devoted to the development of goodness-of-fit tests for Cox's (1972) proportional hazards model

Simulation program for estimating statistical power of Cox's proportional hazards model assuming no specific distribution for the survival time

A Bayesian network interpretation of the Cox's proportional hazard model

We study a weighted least squares estimator for Aalen's additive risk model with right-censored survival data which allows for a very flexible handling of covariates. We divide the follow-up period into intervals and assume a constant hazard rate in each interval. The model is motivated as a piecewise approximation of a hazard function composed of three parts: arbitrary nonparametric functions for some covariate effects, smoothly varying functions for others, and known (or constant) functions for yet others. The proposed estimator is an extension of the grouped data version of the Huffer and McKeague (1991) estimator. For our model, since the number of parameters is finite (although large), conventional approaches (such as maximum likelihood) are easy to formulate and implement. The approach is illustrated by simulations, and is compared to the previous studies. The method is also applied to the Framingham study data.

SUMMARY The additive risk model assumes that the hazard function associated with a set of covariates is the sum of the baseline hazard function and the regression function of covariates. We propose two different test procedures for checking the adequacy of two-sample additive risk models for randomly censored observations. One is based on the martingale residuals and the other on the difference between weighted estimators of the excess risk. The test statistics are shown to be asymptotically normal under appropriate regularity conditions and consistent under any model misspecifications. Finally, two real examples are provided, along with results of a simulation study.

Optimal Duration of Imatinib Treatment / Deep Molecular Response for Treatment-Free Remission after Imatinib Discontinuation from a Canadian Tyrosine Kinase Inhibitor Discontinuation Trial

Background: The Cox proportional hazard (Cox PH) model is the most common and widely used method for analyzing time-to-event data. This approach, however, has been criticized for predicting patient outcomes since it assumes that the outcome is a linear combination of covariates and a constant hazard ratio over time, which is invalid in many cases and may not always be fulfilled, and thus Random Survival Forest (RSF) for time-to-event data has been proposed as an alternative to Cox PH. By considering non-linearity and interrelated covariates, the RSF is recognized as a way to correct bias in Cox proportional hazard models. Methods: The Cox PH model was compared to the Random Survival Forest by a relaxed proportional hazard (PH) assumption in this retrospective cohort study, which included 189 patients who underwent a new algorithmic treatment of an advanced human immunodeficiency virus clinical trial proposed by the World Health Organization from January to December 2018 at Kibong'oto infectious diseases hospital. Results: The study results show that the Random Survival Forest outperforms the Cox PH model in time-to-event data analysis by integrating covariates that do not meet proportional hazard assumptions based on predictive error curves, concordance indices, Brier scores, and integrated Brier scores as predictive indicators. Despite the fact that the PH assumption was relaxed for all covariates in the dataset, the Cox PH model selected only a few, and all deselected covariates were highly significant in the RSF, implying that the Cox PH model was inbuilt with the proportional hazard assumption. Conclusions: As a result, only covariates that satisfied the PH inbuilt assumption were chosen, regardless of whether the assumption was tested or not, rendering it ineffective when compared to a random survival forest. Therefore, having a dataset free from non-linearity and interrelated variables is impossible in the real world; hence, the RSF model outperforms the Cox PH model in predictive accuracy and covariate selection performance.

Gamma frailty model for survival risk estimation: an application to cancer data

: One may estimate a conditional hazard function from grouped (and possibly censored) survival data by the time and covariate specific occurrence/exposure rate. Asymptotic results for cumulative versions of this estimator are developed, utilizing the general frame-work of counting processes. In particular, a grouped data based goodness of fit test for Cox's proportional hazard model is given. Various constraints on the asymptotic behavior of the widths of the calendar periods and covariate strata employed in grouping the data are needed to prove the results. Actual performance of the estimators and test statistics is evaluated by Monte Carlo methods.

Preface. On the Entropies and Mutual Information of Some Bivariate Continuous Distributions Used in Reliability M. Ahsanullah, M. Habibullah. Initial Conditions Problem in Event History Analysis: An Indirect Inference Procedure M.Y. An. Analysis of Survival Data Under Competing Risks with Missing Cause of Death Information: Application and Implications for Study Design J.W. Andersen, et al. Conditional Proportional Hazards Models B.C. Arnold, Y.H. Kim. A Comparison of Conditional and Unconditional Inference Relating to Log-Gamma Distribution N. Balakrishnan, P.S. Chan. Universal Formulas for Treatment Effects from Noncompliance Data A.A. Balke, J. Pearl. Fundational Issues Concerning the Analysis of Censored Data R.E. Barlow, P. Tsai. Burn-in at the Component and System Level H.W. Block, et al. Identificationof Dependent Competing Risks Models K. Carling, T. Jacobson. Reliability Modelling for Optical Amplified Communication Systems M. Chown, G.A. Whitmore. A Parametric Approach to Measurement Errors in Receiver Operating Characteristic Studies M. Coffin, S. Sukhatme. Models for Degradation Processes and Event Times Based on Gaussian Processes K. Doksum, S.-L. Normand. A Non-Parametric Two-Sample Survival Test Based on a Single Occupancy Fermi-Dirac Model for the Discrete Range Distribution J.T. Efird, F. Pardo. Probability Approximations and Inequalities for Sequential Tests J. Glaz, J.R. Kenyon. Cure Mixture Models in Breast Cancer Survival Studies N.H. Gordon. A General Approach to Derive Chi-Square Type of Goodness-of-Fit Tests for Lifetime Data S. Hawala, J.-L. Wang. Prediction in Survival Analysis: Model or Medic? R. Henderson, M. Jones. Dynamic Reliability Models M. Hollander, E.A. Pena. Generalizations of Current Status Data with Applications N.P. Jewell, M. van der Laan. Trend Analysis of Multiple Counting Processes T. Kamakura. Historical Controls and Model Survival Analysis N. Keiding. A Random Effects Model for Multivariate Life Data A.C. Kimber. Statistical Challenges in Comparing Chemotherapy and Bone Marrow Transplantation as a Treatment for Leukemia J.P. Klein, M.-J. Zhang. Maximum Likelihood Estimation and the Multivariate Bernoulli Distribution: An Application to Reliability P.H. Kvam. Intermediate Clinical Events, Surrogate Markers and Survival M. Lefkopoulou, M. Zelen. A Semiparametric Bootstrap for Proportional Hazards Models T.M. Loughin, K.J. Koehler. Application of Cox Regression with a Change Point in Clinical Studies X. Luo, et al. Burn-in with Age Replacement A.W. Marshall. Fitting Cox's Proportional Hazards Model Using Grouped Survival Data I.W. McKeague, M.-J. Zhang. Statistical Methods for Dependent Competing Risks M.L. Moeschberger, J.P. Klein. Orthogonal Functions of Inverse Gaussian Disbrituions R. Nishii. Statistical Models for Quality of Life Measures Y. Palesch, A.J. Groos. Nonparametric Estimation of Regression Parameters from Censored Data with Two Discrete Covariates M.H. Rahbar. Locally Efficient Median Regressionwith Random Censoring and Surrogate Markers J.M. Robins. Survival Models for Heterogeneity Using the Non-Central Chi-Squared Distribution with Zero Degrees of Freedom C.S. Rocha. Efficiently Weighted Estimating Equations with Application to Proportional Excess Hazards P. Sasieni. Sequential Multi-Hypothesis Testing in Software Reliability J.S. Shieh, Y.L. Tong. Assessing Gamma Frailty Models for Clustered Failure Time Data J.H. Shih,

For Cox's proportional hazards model, test procedures are described for the comparison of randomly right-censored observed and expected (standard population) survival data. It is observed that the class of linear rank tests proposed by Prentice (1978, Biometrika 65, 167-179) may be employed to test regression hypotheses in this one-sample setting. In order to compare globally the observed and expected survival data the one-sample limit of Mantel's logrank test is formally derived. The statistic derived is identical to one proposed from a diflerent viewpoint by Breslow (1975, Intemational Statistical Review 43, 45-58). Throughout it is assumed that the standard population survival function is a continuous function of time and that the survival or censoring time for each study-sample individual is exact and is not grouped into categories.

Background and Objectives The purpose of this study was to analyze the survival data of salivary gland cancer (SGCs) patients to construct machine learning and deep learning models that can predict survival and use them to stratify SGC patients according to risk estimate.Subjects and Method We retrospectively analyzed the clinicopathologic data from 460 patients with SGCs from 2006 to 2018.Results In Cox proportional hazard (CPH) model, pM, stage, lymphovascular invasion, lymph node ratio, and age exhibited significant correlation with patient’s survival. In the CPH model, the c-index value for the training set was 0.85, and that for the test set was 0.81. In the Random Survival Forest model, the c-index value for the training set was 0.86, and that for the test set was 0.82. Stage and age exhibited high importance in both the Random Survival Forest and CPH models. In the deep learning-based model, the c-index value was 0.72 for the training set and 0.72 for the test set. Among the three models mentioned above, the Random Survival Forest model exhibited the highest performance in predicting the survival of SGC patients.Conclusion A survival prediction model using machine learning techniques showed acceptable performance in predicting the survival of SGC patients. Although large-scale clinical and multicenter studies should be conducted to establish more powerful predictive model, we expect that individualized treatment can be realized according to risk stratification made by the machine learning model.

Various prognostic models based on survival analysis methods have beenproposed to predict the risk of preeclampsia (PE). To develop a more accurate yetinterpretable prediction approach, we utilized clinical data from pregnant womencollected at a hospital in Jakarta and applied the XGBoost-Cox ProportionalHazard Model (XGB-Cox). This model integrates the predictive power of theXGBoost machine learning algorithm with the Cox Proportional Hazard (Cox-PH)model, which estimates the effect of covariates on event time. Our results showthat the XGB-Cox model outperforms the traditional Cox-PH model based on fourevaluation metrics: log-likelihood, log-rank test, concordance index (C-index), andBrier score. The XGB-Cox model achieved a higher C-index of 0.8908 comparedto 0.7548 for Cox-PH, indicating improved risk discrimination.Kaplan-Meier curves suggest that XGB-Cox provides better separation across risk quartiles.While XGB-Cox generally yields lower Brier Scores, its performance declines atlater gestational weeks. The Cox-PH model remains superior in interpretability,offering clear hazard ratios, while XGB-Cox enhances model fitness and stillprovides meaningful insights into feature importance. Additionally, sensitivity analysis underscores the need to carefully determine the proportion of censored data, as excessive censoring affects model stability. These findings suggest that XGB-Cox provides a robust predictive framework for early PE risk assessment, supporting its potential application in clinical decision-making for maternalhealthcare.

Validation of Recent Response Criteria (ELN-22, IWG-23 and PB-CR) in 1634 MDS/CMML/AML Patients Treated with HMA or HMA-Ven Using CPH Models and a CPH Deep Neural Network - Can or Should Response Criteria be Harmonized for Similarly Treated Patients?

Lenalidomide Treatment Is Not Related to AML Progression Risk but Is Associated with a Survival Benefit in RBC Transfusion-Dependent Patients with IPSS Low- or Int-1-Risk MDS with del5q: Results From a Comparative Study