flexsurvreg log logistic intercept in R

Question

I'm trying to fit a log-logistic AFT model with flexsurvreg(). With AFT model,

$$\log T = \gamma_0 + \gamma_1 z_1 + \gamma_2 z_2 + \sigma W,$$ where W ~ standard logistic distribution.

Below is an example output from flexsurvreg

Estimates: 
       data mean     est      L95%     U95%     se       exp(est)  L95%     U95%   
shape       NA     3.0638   1.6929   5.5447   0.9273       NA        NA       NA
scale       NA    18.9428  10.7900  33.2557   5.4394       NA        NA       NA
z1     -0.0378    -0.3767  -0.6690  -0.0843   0.1492   0.6861    0.5122   0.9191
z2      0.4444    -0.9479  -2.0764   0.1806   0.5758   0.3875    0.1254   1.1979

How to get $\gamma_0$ and its 95% CI?

What does shape in the output refer to?

Scale is $\sigma$, right?

Answer 1

The flexsurvreg() documentation specifies that the "location" parameter for a log-logistic model is the "scale" parameter returned by the model. The "location" represents the value of $\log T$ around which the $\sigma W$ term adds variability.

So $\gamma_0$ is related to the reported scale value for a situation with your covariates at reference values. That "location" is then modified by the $\gamma_1 z_1$ and $\gamma_2 z_2$terms. There might be a log/exponential relationship between the reported "scale" parameter and your $\gamma_0$; examine the documentation or do some plots to check that out. This answer illustrates the situation for a log-logistic model fit by the survreg() function; I'm not sure if flexsurvreg() reports coefficients in the same way.

The "shape" value is related to $\sigma$ in your AFT formula of $\log T$. These course notes provide a succinct summary of these relationships for many types of parametric models.