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I am trying to compare the distribution of one group versus the distribution of all other groups. To that end, I have a nonlinear model that I am estimating two coefficients on both for a combined model (the group is lumped with all other groups) and separately.

Given that I'm using a non-linear model, is it appropriate to calculate likelihood using the normal likelihood model? E.G., $r_i = y_i - \hat{y_i} = \epsilon_i \sim N(0,\sigma^2)$, and I'm calculating likelihood of the model as $\sum_{i=1}^n loglik(r_i|\sigma^2)$, estimating $\sigma^2$ from the residuals.

If this is valid, is it valid to do a likelihood ratio test where the alternative model is using the coefficient values calculated separately, and the null model is using the coefficient values calculated jointly?

If this is valid, what would be the degrees of freedom for that test?

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