# Equivalence between interaction and second difference measures of heterogeneity

I am trying to understand a paper used by Frank Harrell in his biostats course to demonstrate how NOT to present results from an analysis of heterogeneity of treatment effects (HTE). Frank's point is that the main measure of HTE (and its variance) are missing, the ratio of hazard ratios (RHR).

They report the relevant results as:

"(hazard ratio with clopidogrel among carriers, 0.69; 95% CI, 0.49 to 0.98; hazard ratio among noncarriers, 0.72; 95% CI, 0.59 to 0.87; P=0.84 for the interaction)"

The NEJM paper is clear that they tested for an interaction term as far as 'significance' goes, but unclear what model(s) the HR point estimates refer to. More generally, what hazard ratios should be included in a second difference RHR? I see one of two options:

1. The RHR is the quotient of two separate Cox models subset by trial status, where an interaction term is not explicitly modeled.
2. The quotient is from one Cox model, with the HR associated with a covariate that is interacted with treatment, divided by the non-interacted HR associated with the same variable.

I understand there is an equivalence between second differences and interaction terms. But is scenario one above equivalent to scenario two above? Or to what extent can one approximate the other? Is scenario one above even appropriate?

Scenario 1 and 2 are not equivalent (as posed in the question).

The Ratio of Hazard Ratios (using the quotient of two HR point estimates for the same subgroup, but from two models subset by treatment status) is equivalent to the multiplicative interaction, in a single model, between the subgroup in question and treatment status. (Where the single model includes both treatment and control). There is no quotient of point estimates involved at all in the conventional multiplicative interaction model.

I'm still not sure what scenario 2 in my question is equivalent to (or even whether it has any utility or other information).