6.4 Estimands

The first estimand of interest is the regression coefficient \(\beta\) associated with the simulated binary treatment, to see if and how misspecification of the baseline hazard or frailty distribution affects relative risk estimates. Second, I am interested in comparing estimates of the frailty variance - and therefore of the unobserved heterogeneity - obtained from each model. Finally, I am going to compare two measures of absolute risk:

  1. marginal survival difference at time \(t\), defined as \(S(t)_{\text{diff}} = S(t | x = 1) - S(t | x = 0)\);

  2. integrated marginal survival difference, defined as \(iS_{\text{diff}} = iS(x = 1) - iS(x = 0)\).

\(S(t)_{\text{diff}}\) is obtained by fixing the time \(t\) (I am using 1, 2, 3, and 4 years), and then integrating out the frailty term from the conditional survival function as explained in Chapter 2. Conversely, \(iS_{\text{diff}}\) requires integrating the marginal survival function for both treatment groups and then computing their difference. I estimate it as follows:

  1. I estimate marginal survival for a treatment arm (e.g. for \(x = 0\)) at 1000 equally spaced values over the follow-up time \(t\);

  2. I fit an interpolating natural spline over the 1000 estimates from step (1) using the splinefun R function;

  3. I integrate the resulting spline function using Gauss-Kronrod quadrature as implemented in the integrate function;

  4. I compute the difference of the integrated marginal survival for the two treatment groups.

The integral of a survival function can be interpreted as life expectancy; hence, the quantity I am computing can be interpreted as the difference in 5-years life expectancy between treated and untreated individuals.