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I an trying to estimate an ordered logit model where the DV is a likert-scale response (1-5) and I have 6 independent dummy variables representing whether an observation belongs to one of six mutually-exclusive clusters. For example, dummy c1 = 1 if it belongs to cluster 1, and 0 otherwise, and the same for all other clusters. There is an additional dummy (h) that I would like to interact with the effect of each of these clusters, hence the model:

ologit y = beta0 + beta1*c1 + beta2*c1*h + 
 beta3*c2 + beta5*c2*h + ... + beta*c6 + beta*c6*h

Since the goal of this model is to assess the effect of belonging to each cluster, the issue that arises is that one dummy must be dropped. I am considering ways to overcome this, although I am a bit stuck. While I can do separate models for each cluster, this would lead to zero-inflation due to unbalanced classifications associated with each cluster. Moreover, an approach like seemingly unrelated regression seems infeasible due to this approach being specific to linear (not ologit) models.

Any advice on how to tackle this?

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    $\begingroup$ First, you have one variable with six levels. Second, there are many ways to parameterize categorical variables. Look up how to do different contrasts for whatever software you are using. $\endgroup$
    – Peter Flom
    Commented Dec 23, 2023 at 23:04

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