Could anyone kindly provide an explanation (mathematically or non-mathematically) about the non-existence of the intercept term in conditional logistic regression? Is the interpretation of the coefficients similar to that of (unconditional) logistic regression?
Thank you for your help.
Conditional logistic regression compares cases to controls. The coefficients multiply differences in factor values between cases and controls. The intercept terms cancel in the likelihood and therefore play no identifiable role in the model. See the section on "Conditional Logistic Regression" in Ying So's tutorial for SAS.
Conditional logit models for unit $i$ in group $j$ of size $G$, $P(Y_{ij}=1|\sum_{k=1}^GY_{gj}=M)$ for some $0<M<G$. So, suppose you have a group of size 2 in which there is one success. Then, group member number 1's contribution to the likelihood is given by $A^{Y_{1j}}B^{1-Y_{1j}}$, where,
$$
\begin{matrix}
A & = P(Y_{1j}=1|Y_{1j}+Y_{2j}=1) = \frac{P(Y_{ij}=1 \cap Y_{1j}+Y_{2j}=1)}{P(Y_{1j}+Y_{2j}=1)} \\
& = \frac{P(Y_{1j}=1 \cap Y_{2j}=0)}{P(Y_{1j}=1 \cap Y_{2j}=0)+P(Y_{1j}=0 \cap Y_{2j}=1)}.
\end{matrix}
$$
The term $B$ is derived in a similar manner. Specifying the response probabilities in terms of a logit model,
$$
\begin{matrix}
A & = \frac{\frac{\exp(\alpha + \beta'x_{1j})}{1+\exp(\alpha + \beta'x_{1j})}\frac{1}{1+\exp(\alpha + \beta'x_{2j})}}{\frac{\exp(\alpha + \beta'x_{1j})}{1+\exp(\alpha + \beta'x_{1j})}\frac{1}{1+\exp(\alpha + \beta'x_{2j})} + \frac{1}{1+\exp(\alpha + \beta'x_{1j})}\frac{\exp(\alpha + \beta'x_{2j})}{1+\exp(\alpha + \beta'x_{2j})}}\\
& = \frac{\exp(\alpha)\exp(\beta'x_{1j})}{\exp(\alpha)\exp(\beta'x_{1j})+\exp(\alpha)\exp(\beta'x_{2j})}\\
& = \frac{\exp(\beta'x_{1j})}{\exp(\beta'x_{1j})+\exp(\beta'x_{2j})},
\end{matrix}
$$
that is, the intercept cancels out due to the conditioning. In fact, all additively separable group specific effects cancel in this way, which is why conditional logit is sometimes called "fixed effects logit" model (with "fixed effects" defined in the econometrics sense, not the mixed models sense). This property is also why conditional logit is so useful for matched case control data, effectively partialling out potential sources of confounding due to heterogeneity across matching strata.
The coefficients are indeed on the "usual" logit scale.
