Writing the statistical model for extended Cox model for time dependent variables

Question

I'm writing a paper where I'm using an extended Cox model for time dependent variables. I want to see whether two labour reforms (independent variable) had any effect on the duration of job contracts (dependent variable). The subjects of the database are not people, but rather the contracts themselves. As some contracts extend both before and after the labour reforms (some do not, as they finished before the reforms were passed, and some started after), "labour reform" is a time dependent variable (because some contracts have some time "not affected" by the labour reform, then affected by the first and eventually affected by the second reform. I have also included economic crisis as a time dependent covariate, as some contracts extended during the time period of the economic crisis of 2008-2014. And then I have included other covariates for the sex of the person who had the contract, type of contract (permanent, temporary), industrial sector, etc.

Now, I want to explain my methodology very clearly, and was hoping o include the statistical model in algebraical notation. I've seen that some papers include just the general statistical model for the Cox proportional hazards model, such as this: https://imgur.com/a/u7AtY7P

But some other papers go a different way and from what I understand "go the extra mile", writing the particular version of statistical model that they are going to use, such as this: https://imgur.com/a/4jEJFGu

My question is then: is there a name for the practice of stating the particular statistical model you are using on a peper (as opposed to just the general formula)? Where you explain what variable each symbol means etc. How could I do as this latter example and write the particular version of the extended Cox model for time dependent variables that I'm going to use? Would I need to write both my version of the extended Cox model and of the hazard ratio equation? Do you know of any example of a paper that does this? (The first example I posted only states the general equations for the Cox model and the hazard ratio, plus it deals with the Cox proportional hazards model and thus I'm afraid if I follow it too closely I might mess it up, since I'm working with the extended Cox model, for time dependent variables).

Thank you so much for your help!

Answer 1

I'd suggest that you start with the form of Equation 3.1 of the classic Therneau and Grambsch text:

$$ \lambda_i(t)=\lambda_0(t) e^{X_i(t)\beta},$$

where $ \lambda_i(t)$ is the hazard for contract $i$ at time (t), $\lambda_0(t)$ is a baseline hazard over time shared among all contracts, $X_i(t)$ is the set of covariate values in place for contract $i$ at time $t$, and $\beta$ is a vector of Cox regression coefficients for the covariates. That explains the general model, with explicitly time-varying covariate values (not shown in your first example).

Then describe what each of the covariates is, and how it might be expected to change over time.

Your second example is for a different type of model, a time-series model in which it's important to specify how correlations of observations in time are handled by the model. That leads to complicated equations that should be shown.

Cox models have no such problem with correlations in time (if at most one event is possible for each case). Cox models only involve the covariate values that are in place instantaneously at event times. The covariate history doesn't matter.

That said, you then have to think about how to incorporate your covariates into the model so that their instantaneous values are associated with events. For example, if reforms were expected in the near future but hadn't yet happened, a simple binary reform-in-place covariate might not capture events that happen in anticipation of a reform being instituted.