# Form of Cox Proportional Hazards Model

Does one usually write the model as $$h(t| \textbf{x}) = h_{0}(t) \exp(\beta_{1}x_1 + \dots + \beta_{p}x_{p})$$ or as $$\log[h(t|\textbf{x})] = \log[h_{0}(t)]+\beta_{1}x_1 + \dots + \beta_{p}x_{p}$$

They are both equivalent right? But would most people use the second form since it is similar to the form of linear regression and logistic regression (e.g. link function on LHS and linear terms on the RHS)?

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Where is the term that accommodates random error? – whuber Nov 28 '11 at 18:15
@whuber: Which deterministic version is more popular? – markk Nov 28 '11 at 18:47
I don't understand your question, then. Since both expressions determine exactly the same (partial) log-likelihood, they are exactly equivalent. It sounds a little like asking whether it's more popular to write $8\times 3$ or $3\times 8$ when multiplying. – whuber Nov 28 '11 at 19:16
@whuber: That's my point. What do most people write in papers? The first or second form? – markk Nov 28 '11 at 19:18
At this point, it no longer sounds like a statistical question, but one that is answered by an extensive bibliographic search! For example, of the top ten hits on a Google search for "cox proportional hazards," one uses the first form exclusively, one uses the second form exclusively, three introduce it with the first form and then immediately switch to the second form (which is what is ultimately used for a solution), and the remaining five don't bother to write any equation at all. One could argue, then, that the "most popular" method is to dodge the question! – whuber Nov 28 '11 at 19:26