How to specify a risk model?

Question

I'm reading this article. The authors indicated that for a random sample of 12 characteristics and 3600 patients affected in two arms (control and treatment arm), they fitted a risk model consisting of all 12 characteristics in the patients in the control arm only, as well as in the whole sample, blinded to treatment. But, they don't give a mathematical formula.

1. Does the risk model fitted in the patients in the control arm only have the following form?

$\pi = \beta_1*x_1+... +\beta_{12}*x_{12} $, $(1)$

2. Does the risk model fitted in the whole sample blinded to treatment have the following form?

$\pi = \beta_1*x_1+... +\beta_{12}*x_{12} + \gamma*Trt $, $(2)$

or this form without treatment effect ($\gamma*Trt $) like $(1)$:

$\pi = \beta_1*x_1+... +\beta_{12}*x_{12} $, $(1)$

Answer 1

A risk model is a model that models risk.

The models you show as $\pi = \beta_1 x_1 + \ldots $ are additive risk models. Which is interesting because they perform so badly with even modest numbers of covariates.

In both cases, the models are presumably the same in terms of the number of covariates and the expression of the written form. The difference between the two statements is the analysis population involved. Model 1 drops all treated individuals. Model 2 includes all treated individuals. (Or randomized, depending on ITT vs PP effect).