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Suppose we have the following Cox Model: $$g(t,d,x ,\beta) = \ln[h_{0}(t)]+d_{1}\beta_{1}+d_{2}\beta_{2} + d_{1}d_{2} \beta_{3}$$ where $d_1$ and $d_2$ are binary variables that take $1$ or $0$. What is the difference between the following hazard ratios: $$\exp(\beta_{1}+\beta_3)$$ $$ \exp(\beta_1+\beta_2 +\beta_3)$$

If we are interested in comparing $d_1 = 1$ vs. $d_1=0$ given that $d_2=1$, why is $\exp(\beta_1+\beta_3)$ the correct hazard ratio? What happens to the $d_{2}\beta_{2}$ term?

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If $d_2 = 1$ then you are comparing the following cases:

If $d_1 = 1$, then:

$1 \times \beta_1 + 1 \times \beta_2 + 1 \times 1 \times \beta_3 = \beta_1 + \beta_2 + \beta_3$

else if $d_1 = 0$, then:

$0 \times \beta_1 + 1 \times \beta_2 + 0 \times 1 \times \beta_3 = \beta_2$

Note that:

$\beta_1 + \beta_2 + \beta_3 - \beta_2 = \beta_1 + \beta_3$

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