Quantifying marginal effect of dose in survival analysis Suppose I have overall survival data for patients who are A) untreated; B) treated with a dosage of x for some treatment; and C) treated with dosage of 2x. I want to determine if there is a statistically significant marginal effect of increasing dosage from 1x to 2x, and also to quantify that effect. What is the appropriate way to do this?
My naive idea (not being an expert in survival analysis) was that I would just have to use pairwise Cox proportional hazard models comparing (1) untreated vs. 1x-treated and (2) 2x-treated vs 1x-treated, and take the ratio of the coefficient from (2)/the coefficient from (1) as a measure of the marginal effect of doubling the dosage. I suspect this may not be the 'correct' way to do this analysis though, so any suggestions are appreciated.
There are basically three things I want to know: 1) is survival with treatment with 1x significantly better than untreated? 2) is treatment with 2x significantly better than treatment with 1x? and 3) a measure that quantifies the effect on survival of treatment with 2x relative to treatment with 1x. I'm assuming, for the sake of simplicity, that the effect of treatment with 2x doesn't go in the opposite direction of 1x; so doubling dosage either improves outcome, or does nothing at all.

Provided that the proportional hazards assumption holds, this could all be done with a single Cox model in which treatment is a 3-level categorical predictor, with levels untreated, 1X, 2X. That's a more efficient use of data than separate pairwise models.
Set untreated to be the reference level of that predictor. You get 2 coefficients from that predictor, for the log-hazard of each of 1X and 2X versus untreated. The coefficient for 1X answers your first question, effect of 1X versus untreated. A subsequent statistical comparison of the 1X and 2X coefficients addresses your second question. The difference between the 1X and 2X coefficients give the log-hazard difference between those treatment levels; exponentiating that log-hazard difference gives the corresponding hazard ratio.
