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I'm running a latent class (profile) analysis, and there is still a thing not clear to me. What happens when I don't include the constant in the membership function? Is the interpretation different? Should be the choice justified? And if so, when should I include it or not include it? Thank you

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The modelling of the class membership works as a "standard" regression model - By default you would only have the constant as predictor and then you could add some other variables.

In first case the estimate for the constant(s) will tell you something about probability of belonging to the different classes (Strictly speaking it is possible to compute the membership proba by plugging the estimates in the logit formula, but best to follow the Bayesian rule to predict membership at individual level and then computing % of indiv in the different classes).

In the second case, the constant will capture the baseline effect (i.e. when all the predictors are set to their reference level).

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I'm not clear what you are asking. Latent class or profile models estimate two things: a model for the indicator means in each latent class, and a multinomial model for the probability of membership in each latent class. I assume that "membership function" means the latter. For a two class model:

$$P(C = 1) = \frac{e^{\gamma_1}}{e^{\gamma_1} + e^{\gamma_2}}$$ $$P(C = 2) = \frac{e^{\gamma_2}}{e^{\gamma_1} + e^{\gamma_2}}$$

${\gamma_1}$ and ${\gamma_2}$ are intercepts, or constants, and by convention, ${\gamma_1}$ is set to zero (you have to set one or the other to zero, anyway). So ... there is no way not to included the intercepts in the multinomial part of the equation. It does not work. Can you clarify if you were asking about something else?

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