# Intercepts in stan_clogit

Dear all,

I’m relatively new to “rstanarm” and I have a question about “stan_clogit” function.

Usually, “stan_clogit” does not estimate intercepts, which is similar to “clogit” in “survival” package.
But, when we add group-level terms, intercepts for each group are estimated.
In this case, what is the meaning of intercepts?
Can we interpret them the same way as we do in other multilevel models?

Thanks.

Best,
Daisuke

Hey there!

My guess is that it’s the group specific Intercepts you are seeing there,

…so I’d say you are right here.

Usually you can write a (simple) hierarchical model like

\begin{align} \mu &= \alpha + \alpha_\texttt{group} + X\beta \\ \alpha_\texttt{group} &\sim \text{Normal}(0, \sigma_\texttt{group}) \\ &... \end{align}

where \alpha is your overall intercept and \alpha_\texttt{group} is the random intercept for each group. In a model “without” intercept \alpha = 0, so

\begin{align} \mu &= 0 + \alpha_\texttt{group} + X\beta \\ \alpha_\texttt{group} &\sim \text{Normal}(0, \sigma_\texttt{group}) \\ &... \end{align}

in such a model the group effects are not deviations from a main/overall intercept, but rather deviations from “zero”. They then act as intercepts for their specific group.

Hope this helps!
Cheers,
Max

Dear Max,

If “stan_clogit” presents the group specific Intercepts, I wonder why we can estimate them.
In conditional logistic regression, the intercept cancels out due to the conditioning.

In standard logistic regression,

\mathrm{Pr(positive \; outcome)}= \frac{\mathrm{exp}(x_{i}\beta + \alpha)}{1 + \mathrm{exp}(x_{i}\beta + \alpha)}

In conditional logistic regression, if there are two observations in each group and there must be one positive outcome in each group,

\mathrm{P(obs. \; 1 \; positive \; and \; obs. \; 2 \; negative\: | \: one \; positive \; outcome)} \\ =\frac{ \mathrm{P(obs.\; 1\; positive)}\mathrm{ P(obs.\; 2\; negative)}}{\mathrm{P(obs\; . 1\; positive)} \mathrm{P(obs.\; 2 \; negative)}+\mathrm{P(obs.\;1\; negative)} \mathrm{P(obs.\;2\; positive)}} \\ = \frac{\mathrm{exp}(x_{1}\beta + \alpha)}{\mathrm{exp}(x_{1}\beta + \alpha)+\mathrm{exp}(x_{2}\beta + \alpha)} \\ = \frac{\mathrm{exp}(x_{1}\beta)}{\mathrm{exp}(x_{1}\beta)+\mathrm{exp}(x_{2}\beta)}

So, in conditional logistic regression, the intercept cannot be estimated.
I wonder if it holds true for the group specific intercepts.

I would appreciate it if you could help me about it.

Daisuke

Hey Daisuke! I did not see your reply since a few days ago, I’m sorry!

I’ll have to admit, that I’m not expert in clogit models. However, I think group specific effects can still be estimated because of the implicit sum-to-zero constraint of the (common) group variance.

For example in multilevel linear regression you can estimate all group specific intercepts, while in classical linear regression you’d have to apply an explicit sum-to-zero constraint or drop out a reference level for identification.

Does this make sense?

Cheers,
Max

Dear Max,

I understand that we can estimate the group specific effects in this case.
Thank you!

Best,
Daisuke

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