# Showing a regression model is unidentifiable? [duplicate]

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.

• 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 $\log\lambda_0(t).$
– whuber
May 11 at 18:37