Suppose we have a Cox-PH model and death is the outcome of interest. Suppose we have the predictors $X_1, X_2$ and $X_3$. Then does the PH assumption mean that for any person (i.e. arbitrary values of $X_1,X_2$ and $X_3$) the hazard of death is $c \cdot h_{0}(t)$ where $c$ is a constant? So the hazard function is the same for everyone?
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PH model: $$ h_i(t) = h_0(t) \exp(x_{1i}\beta_1 + x_{2i}\beta_2 + x_{3i}\beta_3). $$ The baseline hazard, $h_0(t)$, is common to all individuals; it plays the role of the intercept in a linear model. But the second term, the exponential of the linear predictor, adjusts the hazard of the $i$th individual for observed risk factors. PH assumption According to the PH model, for any two individuals with covariate vectors $(x_{1i}, x_{2i}, x_{3i})'$ and $(x_{1j}, x_{2j}, x_{3j})'$, the hazards ratio is $$ \frac{h_i(t)}{h_j(t)} = \frac{h_0(t) \exp(x_{1i}\beta_1 + x_{2i}\beta_2 + x_{3i}\beta_3)}{h_0(t) \exp(x_{1j}\beta_1 + x_{2j}\beta_2 + x_{3j}\beta_3)} = \exp\left((x_{1i} - x_{1j})\beta_1 + (x_{2i} - x_{2j})\beta_2 + (x_{3i} - x_{3j})\beta_3 \right),$$ which does not depend on time. This is the PH assumption. Typically, all covariates are taken to be the same except one, say $x_{1}$, which differs by one unit, so that $$ \frac{h_i(t)}{h_j(t)} = \exp(\beta_1).$$ This leads to a nice interpretation for the regression coefficients as (conditional) log hazards ratios. |
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The hazard function isn't the same for everyone. The hazard functions for each covariate need to be proportional - hence the name. So, regardless of how h(t) bounces around, the ratio of the hazard function at time t between the levels of the covariate is constant. |
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