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I start my question with an example. Assume we are calculating HR associated with biological sex in a Cox proportional hazard model. let's say the ratio is 2. Since it has only two levels, we can say the chance of dying an individual in males in two times as females.

  1. If we have a muli-level categorical variable, what is the reference that we should compare? In the previous example, we had 2 levels, and we compared males or females against each other.

  2. My second question is about continues variables. What does hazard ratio mean for a continues variable? I mean what is the reference for a continues variable? For example, in the previous example, the reference was males or females.

I would appreciate if you guys explain these 2 questions.

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  1. If you have a categorical variable with $K$ levels, then new $K-1$ variables are created and the output presents results for each level of a categorical variable with respect to the variable's baseline level. You will have one less coefficient than you have levels of the variable. The choice of reference level is arbitrary.

  2. For continuous variables, hazard ratio tells you what is the unique (assuming other variables are constant) proportional effect of a unit increase in the variable on the hazard. The reference value for a continuous variable in regression models is usually 0, but that can cause numerical problems in fitting Cox models if the actual values are far from 0. Depending on the particular software, a different reference value might be chosen internally, to match a corresponding baseline hazard estimate that holds for that reference value.

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