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Proportional Hazards Modeling of Survival Data: With Distribution-Free Methods

Harold Bishop
Organization: Process Modeling Solutions, Inc.

2006 Wolfram Technology Conference
Conference location

Champaign IL

This presentation demonstrates the application of Mathematica to solve distribution-free problems in survival analysis using the proportional hazards model. Continuous-time survival data is modeled with right censoring and covariates. Getting the data formatted can be conveniently performed for handling in multiple ways while retaining identifiability. Mathematica provides the means to build a high-quality statistical analysis tool that can be quickly customized to fit the problem, not putting the burden on the practitioner to mold the problem into one that fits the tool.

In the proportional hazards model, the basic assumption is a where ais the baseline hazard function under a. x is a vector of covariates contributing to the survivability of individual items. A data type is defined to organize the items by covariates as well as risk sets for failure at specific times. This data type facilitates the calculations involved in applying the statistical method and makes procedures clear with regard to what mathematics is being performed. Model setup of variables includes automatic variable definitions, naming of coefficients (β), and provides a data pattern for transforming and recoding the data to fit continuous and nominal scale layouts. Three different data sets are used to demonstrate the versatility of this Mathematica approach. For solving the partial likelihood function NMaximize was used after testing other approaches with FindMaximum and a direct Newton method. NMaximize was fast and able to consistently reach a solution using simulated annealing for the data sets demonstrated. Once the a have been found, the survivor function can be estimated for the baseline and consequently for any covariate vector x within the range of the model. Testing the proportional hazards assumption, comparing independent variable effects through stratification, and computing exact survival contributions with ties in the data are easily performed and visible within the Mathematica environment producing textbook-quality output in tables and graphs.

*Mathematics > Probability and Statistics

ProportionalHazards.zip (335 KB) - ZIP archive [for Mathematica 5.2]