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I am following a tutorial in which we are looking at a risk regression model (the Cox’s proportional hazards model. In particular, the hazard rate is modelled as $$ \lambda(t) = \lambda_0(t) \exp(\mathbf{x}\beta)$$ where $\mathbf x$ is a vector of covariates and $\beta$ are the coefficients.

The tutorial goes on to say that $\mathbf x$ should not include a constant term corresponding to an intercept. If $\mathbf x$ includes a constant term corresponding to an intercept, the model becomes unidentifiable. They illustrate this:

Suppose the model does include an intercept term, i.e. $$ \lambda(t) = \lambda_0(t) \exp(\beta_0 + \mathbf{x} \beta) = \lambda_0(t) \exp(\beta_0) \exp(\mathbf{x} \beta) $$ If $\tilde{\beta}_0 = \beta_0 + \delta$ and $\tilde{\lambda}_0(t) = \lambda_0(t) \exp(-\delta)$, then $$ \lambda(t) = \tilde{\lambda}_0(t) \exp(\tilde{\beta}_0 + \mathbf{x} \beta) $$ as well, making the model unidentifiable.

But I do not get how this makes it unidentifiable? I suppose I don't really understand what it means to be unidentifiable in the first place.

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    $\begingroup$ Your model already has an intercept term. You can see it when you write $\log\lambda(t)=\log \lambda_0(t) + \mathbf x \beta:$ it corresponds to $\log\lambda_0(t).$ $\endgroup$
    – whuber
    May 11 at 18:37

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