Can two separate regression coefficients be added to estimate their mutual effect?

Question

Let's say I perform a Cox regression including 3 predictors that relate to the survival:

Hazard ratios (HR) for predictors

• Sex: Hazard ratio for males = HR 1.5
• Treatment: Hazard ratio for being treated, as compared with controls = HR 1.5
• Race: Being black, as compared with being white = HR 1.5.

If I wonder what the hazard will be for a black male with treatment, can I simply add upp the HRs; 1.5 + 1.5 + 1.5 = 4.5?

I understand how interactions work and how they should apply. An interaction term between these predictors can definitely be recommended, but I was wondering whether the simple addition of coefficients will convey their additive risk?

Textbooks of Cox regression often mention the terms additivity and multiplicatively, but they seldom elaborate on them.

I guess any answer to this would also extend to linear and logistic regression.

Answer 1

Let's start by writing up your hazard, $\lambda(t)$,

$$ \lambda(t) = \lambda_0(t)\exp(\beta_sX_s + \beta_tX_t + \beta_rX_r), $$

where $\beta_s, \beta_t, \beta_r$ are the coefficients to be estimated and $X_s, X_t, X_r$ are the three binary variables (takes the values $0$ or $1$). We now assumes that all coefficients are equal to $\log(1.5)$, thus, the hazard ratio for each variable is $1.5$, simply because when holding all other covariates fixed and changing only, e.g., $X_s$, we get

$$ \lambda_{1,x_t,x_r}(t)/\lambda_{0,x_t,x_r}(t) = \exp(\beta_s\cdot 1)/\exp(\beta_s\cdot 0) = \exp(\beta_s), $$

using that $\exp(a + b) = \exp(a)\exp(b)$. Now the hazard ratio only makes sense as a comparison of two sets of covariates values. If we want to compare a black male with treatment to a non-black female without treatment then we get

$$ \lambda_{1,1,1}(t)/\lambda_{0,0,0}(t) = \exp(\beta_s\cdot 1 + \beta_t\cdot 1 + \beta_r \cdot 1) = \exp(\beta_s) \exp(\beta_t ) \exp( \beta_r ). $$

This is the product of the hazard ratios when only changing one covariate value at a time (from now on, simple hazard ratios, not an official term). Thus, when we have a additive model, we can find hazard ratios between sets of covariates by multiplying the simple hazard ratios corresponding to the covariates with differing values. E.g. the hazard ratio between black males without treatment and black females with treatment is also easy to find now. The differ in sex and treatment, thus

$$ \lambda_{1,0,1}(t)/\lambda_{0,1,1}(t) = \exp(\beta_s) /\exp(\beta_t ) . $$