Abstract <p>The exponential function is commonly used in modeling Cox proportional hazard model. When the lifetime data are correlated and exhibit a clustered structure, the more appropriate statistical methods are studied by combining the methods of multivariate shared frailty model, survival tree, and Bayesian approach. This is demonstrated with an extensive simulation study for time to tooth loss data coming from elderly patients with the aim to identify unknown risk groups. The distribution of frailty term may affect the modeling process in generating Bayesian survival tree. Then, two frailty distributions of gamma and log-normal were focused in multivariate shared exponential survival trees, using Bayesian approach. A total of 45 simulated clustered survival datasets were generated based on number of clusters, cluster size, and right censoring rate. Each dataset contained correlated failure times, and up to 50 covariates at both the cluster and unit levels depending on the selected variables from the stepwise procedure in the Cox proportional hazard model. Next, each dataset was resampled 1000 times, and for each resampled dataset 70<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8268_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\%\)</EquationSource>  </InlineEquation> of the clusters were used for training and the remaining 30<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8268_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\%\)</EquationSource>  </InlineEquation> for testing. Across all models, it was found that the classification accuracy increased when the cluster size and number of clusters went up, but consistently decayed when the right-censoring rate increased. Moreover, the Bayesian multivariate survival tree approach incorporating the shared log-normal frailty with an exponential baseline hazard achieved the highest classification accuracy. In conclusion, this model is recommended for its superior classification accuracy across varying conditions. The better performance of the log-normal model highlights the importance of the frailty distribution</p>

Importance of Frailty Distributions for Bayesian Multivariate Exponential Survival Trees to Identify Unknown Risk Groups

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发表日期：

2025年4月1日

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Patcharaporn Porndumnernsawat,
Till D. Frank,
Lily Ingsrisawang

摘要

Abstract

The exponential function is commonly used in modeling Cox proportional hazard model. When the lifetime data are correlated and exhibit a clustered structure, the more appropriate statistical methods are studied by combining the methods of multivariate shared frailty model, survival tree, and Bayesian approach. This is demonstrated with an extensive simulation study for time to tooth loss data coming from elderly patients with the aim to identify unknown risk groups. The distribution of frailty term may affect the modeling process in generating Bayesian survival tree. Then, two frailty distributions of gamma and log-normal were focused in multivariate shared exponential survival trees, using Bayesian approach. A total of 45 simulated clustered survival datasets were generated based on number of clusters, cluster size, and right censoring rate. Each dataset contained correlated failure times, and up to 50 covariates at both the cluster and unit levels depending on the selected variables from the stepwise procedure in the Cox proportional hazard model. Next, each dataset was resampled 1000 times, and for each resampled dataset 70 \(\%\) of the clusters were used for training and the remaining 30 \(\%\) for testing. Across all models, it was found that the classification accuracy increased when the cluster size and number of clusters went up, but consistently decayed when the right-censoring rate increased. Moreover, the Bayesian multivariate survival tree approach incorporating the shared log-normal frailty with an exponential baseline hazard achieved the highest classification accuracy. In conclusion, this model is recommended for its superior classification accuracy across varying conditions. The better performance of the log-normal model highlights the importance of the frailty distribution