Statistics for Maintenance Modelling: Proportional Hazards and Accelerated Life Models

# Model Checking¤

Return

Let us note the effect of y on the survival function in the form of the equation h(t; y) = h0(t)g(y) where g(y) = . Since S(t) = (from The Hazard Function screen in the Continuous Lifetime Distributions 1 session):

S(t; y) = = = = {S0(t)}g(y),

taking logs of S(t; y) gives:

ln S(t; y) = -g(y) h0(u)du.

Taking logs again of -ln S(t; y), we finally have:

ln(-lnS(t; y)) = ln(g(y)) + ln(-ln(S0(t)))

where S0(t) = is the baseline survival function, therefore ln S0(t) = - h0(u)du.

In other words, the ln(-ln) survival functions given y are parallel for different covariates. This suggests that if there are many observations for each y, a check on the proportional hazards assumption can be made by computing the Kaplan-Meier estimator of the survival function i(t) = (t; yi) from the data at each yi. Plots of ln(-lni(t)) versus t should be roughly parallel if a proportional hazards model is appropriate.