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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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  • $\begingroup$ Is this the same thing as each study group having a hazard function that is a positive multiple of the baseline hazard, $r \times h_{0}(t)$? $\endgroup$ – robbertp Mar 13 '12 at 1:03
  • $\begingroup$ A somewhat problematic question given the context of a Cox model which doesn't estimate a baseline hazard. $\endgroup$ – Fomite Mar 13 '12 at 1:12
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    $\begingroup$ You can estimate the baseline hazard nonparametrically in the context of the Cox model, although noisily. Another way to translate the PH assumption is that survival curves for any two individuals are connected by a power transformation, i.e., to get from $S_{1}(t)$ to $S_{2}(t)$ you exponentiate the first by the hazard ratio for the two. $\endgroup$ – Frank Harrell Mar 13 '12 at 11:50

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