In lots of scientific trials, the result of curiosity could also be some type of time to occasion end result. This may very well be time till dying, time till hospitalization, or time till another (sometimes unfavourable) well being occasion.
One easy manner to model that is to use a Cox proportional hazard (a.okay.a. exponential) model. The Cox model is one of the most popular in both academic research and for HTA submissions. For the Cox evaluation to be legitimate, nevertheless, a few key assumptions should maintain. These embody: (i) hazard are proportional, (ii) linearity, (iii) additive, and (iv) examination of high-leverage factors. Let’s take a look at every of those in flip.
Take a look at for proportional hazard
Step one when analyzing whether or not the hazard is proportional is to do a visible inspection. One can plot the Kaplan-Meier curves by subgroup and look at whether or not the hazard seems to create traces with a pretty fixed proportional distance.
Nevertheless, that is solely a first step. One other method is to plot the log(-log(survival)) in opposition to log(time) graph. If the predictor is proportional, the traces must be parallel.
A extra formal statistical check for proportional hazards depends on utilizing the Schoenfeld residuals. Reasonably than estimating a single residual for every particular person statement, Schoenfeld (1982) proposed calculating a separate residual for every particular person for every covariate. Notice that the Schoenfeld residuals should not outlined for censored people. They’re calculated as covariate-value for a given particular person who truly died at time t minus the anticipated worth of the covariate for the in danger set at t. The system is beneath the place xik is the covariate worth for covariate okay for particular person i who died at time ti much less the likelihood of that covariate for the individuals within the in danger group the place pj is the likelihood of dying at time ti. A plot of the residuals can be beneath. Extra data could be discovered here.
Testing for additivity
Testing for additivity is pretty easy and is comparable to points current in any regression model. The usual method assumes that each one variables enter in with out interplay. To check for additivity, easy add in interplay phrases. As an illustration, in case you have age and gender in your Cox model, embody an age-gender interplay time period.
Testing for linearity
Covariates may additionally enter in a non-linear manner. A method to do do that is to embody nonlinear phrases (e.g., polynomial phrases like squares or cubes). One may also embody a spline time period. One can create a spline time period in R utilizing the pspline command. For instance:
Sensitivity to outliers
As in all fashions, excessive outliers (whether or not for much longer or a lot shorter survival than common) can have a giant affect on the outcomes. Wanting into the information and understanding these high-leverage factors will likely be essential as they’ll lead to violation of the three assumptions described above.
Proportional hazard fashions are a easy, clear technique for conducting survival evaluation. To be assured within the outcomes of your model, nevertheless, testing for proportional hazards, linearity, additivity and identification of high-leverage outlier factors are all wanted to assist validate the findings out of your model.