# Can I specify a random intercept in a conditional logit model?

In conditional logit models, global intercepts cannot be estimated as they do not influence the conditional probability of a positive outcome within groups. I understand the intercept term gets canceled out in the derivation of the equation, just as any coefficients for which predictor values are constant within groups; however, would it make sense to estimate the variance among group-intercepts when group is specified as a random term?

Example: I'm looking at consumer choice between two alternative products, one is always chosen. The products differ in their attributes and I'm modeling the conditional probability that one is chosen based on the combination of those attributes. The outcome (being chosen) is grouped by test (strata). Some consumers appear more than once in the data, and some products appear in more than one test. I expect certain products to be more likely to be chosen due to inherent, unobserved attributes. I also expect certain consumers to have inherent preferences (e.g. value attribute1 more than other consumers).

| Test | Consumer | Product | Chosen | Attr1 | Attr2|
| ---- | -------- | ------- | ------ | ----- | ---- |
| 1    | A        | a       | 0      |  0    | 0.5  |
| 1    | A        | b       | 1      |  1    | 0.8  |
| 2    | B        | c       | 1      |  1    | 0.0  |
| 2    | B        | a       | 0      |  1    | 0.1  |
| 3    | A        | d       | 1      |  0    | 0.0  |
| 3    | A        | e       | 0      |  1    | 0.8  |

1. Can I include a random intercept for Product?
2. Does it make sense to include random terms for Consumer given there are no between-test estimations?

I'm working in R, rstanarm (but my question is about model specification so feel free to chime in with examples from frequentist approach). stan_clogit() does give me estimates for both randomgroup level intercepts (i.e. Product and Consumer), but I'm not sure if these are meaningful.

Estimates:                              mean   sd   10%   50%   90%
b[(Intercept) Product:a]                0.0    1.1 -1.3   0.0   1.4
b[(Intercept) Product:b]               -0.4    1.3 -2.1  -0.4   1.1
...
b[(Intercept) Consumer:P]               -0.1    1.4 -1.3   0.0   1.1
b[(Intercept) Consumer:Y]               0.0    1.4 -1.2   0.0   1.3
Sigma[Product:(Intercept),(Intercept)]  2.0    1.6  0.6   1.6   4.0
Sigma[Consumer:(Intercept),(Intercept)] 2.0    4.9  0.0   0.5   5.1

MCMC diagnostics                           mcse Rhat n_eff
b[(Intercept) Product:a]                   0.0  1.0  5589
...
b[(Intercept) Consumer:Y]                  0.0  1.0  3509
Sigma[Product:(Intercept),(Intercept)]     0.0  1.0  1510
Sigma[Consumer:(Intercept),(Intercept)]    0.1  1.0  4280
log-posterior                              0.3  1.0  1126