parmsurv: a sas macro for flexible parametric survival

Parmsurv: a SAS Macro for Flexible Parametric Survival Analysis with Long-Term Predictions

Han Fu1*, Shahrul Mt-Isa2, Richard Baumgartner3, William Malbecq2

1Division of Biostatistics, The Ohio State University2Biostatistics and Research Decision Sciences (BARDS), MSD

3BARDS, Merck & Co., Inc.

* Author for correspondence: fu.607@osu.edu

Motivation and Background• Health economic evaluations require long-term predictions of survival beyond the follow-up period (e.g. 5 or 10 years)

• Fully parametric survival models• Convenient for long-term predictions• Suitable for non-proportional hazards, in contrast to the Cox model

• Generalized gamma (GG) and generalized F (GF) distributions• Extensive families• Contain well known distributions with various hazard shapes and complexity• Standard proportional hazards (PH) or accelerated failure time (AFT) models are often used• Only the location parameter may depend on covariates

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Existing Software

• PROC LIFEREG in SAS /STAT®: AFT models in the GG family• Does not include the GF model or user-defined distributions• Does not support regression on ancillary parameters

• streg in Stata®: most parametric models with ancillary regression• Does not include the gamma or GF • Does not support censoring types other than right censoring

• flexsurv R package: flexible parametric distributions including custom models• Long-term predictions not directly provided• Stratification or robust inference not supported

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ObjectiveDevelop a SAS macro that allows general parametric survival analysis

• Various distributions

• GG and GF distributions, their special cases including the exponential, Weibull, log normal, log logistic, etc., and Gompertz distribution• Proper custom survival distributions

•Modeling features

• Regression on the location parameter and/or ancillary parameters• Weighted regression, stratified regression and robust inference • Long-term predictions of survival and hazard rates

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Generalized Gamma (GG) Distribution

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4 Types of Hazard Shapes in GG Distribution

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Generalized F (GF) Distribution

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Other Built-in Distributions

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Adjusting for Covariates

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Log-Likelihood Function

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Statistical Inference

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Stratification

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Data Format

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Data Type Representation

Right censored

Left censored

Interval censored

Event observed

Data Type Representation

Right censored

Event observed

Computation

• PROC IML (Matrix language in SAS)

• Estimation: Non-linear programming (NLP) tool in SAS/IML®

• Optimization algorithm can be specified by user, default is Newton-Raphson• Other algorithms include quasi-Newton, trust region, Nelder-Mead, etc.

• Inference:• Approximate derivatives by finite differences: NLPFDD subroutine in SAS/IML®

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SAS Macro ParametersFunctional Area Parameters

Dataset data=

Response t1=, t2=, censor=, censval=0

Covariates covars=, anc=, class_cov=, refgrp=

Distribution dist=

Optimization optim_method=nlpnra, init=, lower = {. . . . . . . . . .}, upper = {. . . . . . . . . .},

Inference alpha=0.05, robust=F

Custom distribution density=, survival=, hazard=, custom_prep=, nanc=, location=beta, param_anc=, param_anc_transf=, log_transf_index=, log_density=, log_survival=

Prediction pred=, pred_max_time=, pred_plot_cl=T

Others weight=, strata=, noprint=F

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Primary Features of the Macro• Estimation and inference of parameters from built-in or custom distributions• Using data with different format and different censoring types• Allowing continuous/categorical covariates associated with primary / ancillary parameters, with / without case weights, with / without stratification• Using different optimization method, providing automatic initial values for optimization• Using regular / robust estimator for standard errors• Prediction in survival and hazard for certain covariates and time• Predicted survival and hazard curves for different covariates and/or stratified variables (with / without confidence bands)

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Simulation Settings

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Example 1: Fitting Standard Models

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Fit an exponential model%paramsurv(data=data, t1=time, censor=delta, covars= age sex, dist=exp)

Fitting Results: Weibull

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Fit a Weibull model%paramsurv(data=data, t1=time, censor=delta, covars= age sex, dist=Weibull)

Model Comparison

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Distribution Log-likelihood AIC BIC

Exponential -57.656 121.312 129.128

Weibull -56.043 120.086 130.507

Gamma -56.204 120.408 130.829

Log normal -60.688 129.375 139.796

Gompertz -57.249 122.499 132.920

Log logistic -58.903 125.806 136.227

Generalized Gamma -55.998 121.996 135.022

Generalized F -55.998 123.997 139.628

Example 2: Covariates on Ancillary Parameters

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Fitting Results

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Example 3: Custom Distribution

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Fitting Results

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Example 4: Prediction%paramsurv(data=data, t1=time, censor=delta, covars=age sex, dist=gengamma,

init={0 -0.5 1 0.5 0.5}, pred=pred, pred_max_time=5)

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Prediction

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Observations in the preddataset for prediction

Predicted survival and hazard (and S.E.) for specific covariates and time points

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Predicted Survival & Hazard Curves

Summary

Developed a SAS macro for general parametric survival analysis

• Allows parametric distributions of arbitrary complexity

• Accommodates fixed continuous / classification covariates

• Supports covariates associated with primary / ancillary parameters

• Allows case weights, stratification and robust inference

• Supports long-term predictions of survival and hazard rates

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References• Cox, Christopher, et al. "Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution." Statistics in medicine 26.23 (2007): 4352-4374.

• Cox, Christopher. "The generalized F distribution: an umbrella for parametric survival analysis." Statistics in medicine 27.21 (2008): 4301-4312.

• Jackson, Christopher H. "flexsurv: a platform for parametric survival modeling in R." Journal of statistical software 70 (2016).

• Lin, Danyu Y., and Lee-Jen Wei. "The robust inference for the Cox proportional hazards model." Journal of the American statistical Association 84.408 (1989): 1074-1078.

• Xu, Rengyi, Devan V. Mehrotra, and Pamela A. Shaw. "Hazard ratio inference in stratified clinical trials with time-to-event endpoints and limited sample size." Pharmaceutical statistics 18.3 (2019): 366-376.

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