A Tutorial on Multilevel Survival Analysis: Methods, Models and Applications

Abstract

Data that have a multilevel structure occur frequently across a range of disciplines, including epidemiology, health services research, public health, education and sociology. We describe three families of regression models for the analysis of multilevel survival data. First, Cox proportional hazards models with mixed effects incorporate cluster-specific random effects that modify the baseline hazard function. Second, piecewise exponential survival models partition the duration of follow-up into mutually exclusive intervals and fit a model that assumes that the hazard function is constant within each interval. This is equivalent to a Poisson regression model that incorporates the duration of exposure within each interval. By incorporating cluster-specific random effects, generalised linear mixed models can be used to analyse these data. Third, after partitioning the duration of follow-up into mutually exclusive intervals, one can use discrete time survival models that use a complementary log-log generalised linear model to model the occurrence of the outcome of interest within each interval. Random effects can be incorporated to account for within-cluster homogeneity in outcomes. We illustrate the application of these methods using data consisting of patients hospitalised with a heart attack. We illustrate the application of these methods using three statistical programming languages (R, SAS and Stata).

Keywords: Cox proportional hazards model; Multilevel models; clustered data; event history models; frailty models; health services research; hierarchical regression model; statistical software; survival analysis.

Figure 1

Variation in hospital-specific hazards and survival ( frailty model). SD, standard deviation. [Colour figure can be viewed at wileyonlinelibrary.com]

Figure 2

Variation in hazard functions across hospitals (piecewise exponential model). SD, standard deviation. [Colour figure can be viewed at wileyonlinelibrary.com]

Statistical software output 1

SAS output for Cox frailty survival model (log-normal frailty distribution)

Statistical software output 2

Stata output for PWE survival model

Statistical software output 3

R output for discrete time mixed effects survival model

Statistical software output 4

SAS output for discrete time mixed effects survival model with random intercept and random effect for cardiogenic shock