I am interested in estimating a set of coefficients in a multinomial logistic model. However, I only observe a subsample of the true sample in which base category $A$ was chosen. I have no way of knowing whether my subsample is a random or a biased draw from the true sample. (It is probably biased.) What consequences does this have for my inference on the coefficients?

The more formal setup: Let $y$ be a variable with 3 nominal outcomes: categories $\{A,B,C\}$. The categories are unordered. Consider also the explanatory variables $X$.

I want to estimate a set of coefficients $\beta^A, \beta^B, \beta^C$ corresponding to each outcome. For identifiability, let's set $A$ as the base category and set the coefficient vector $\beta^A$ to 0.

I am now interested in estimating $\beta^B$ and $\beta^C$: $$\frac{P(y = B)}{P(y = A)} = \exp(X\beta^B) \text{ and} \\ \frac{P(y = C)}{P(y = A)} = \exp(X\beta^C)$$

So far, this is just the standard setup of a multinomial logistic model. Now let $S_A$, $S_B$ and $S_C$ be the true samples in which categories $A,B,C$ are chosen.

My problem now is that I only observe $S_A' \subset S_A$. I know for sure that I observe the full $S_B$ and $S_C$. The recorded $y$ available to me come from $S_A' \cup S_B \cup S_C$. I also do not know whether $S_A'$ is a random subsample of $S_A$ or biased.

My intuition is that if $S_A'$ were a random subsample, my estimators for $\beta^B$ and $\beta^C$ would simply be inflated by however large the difference between $S_A'$ and $S_A$ is. What happens if it is not a random subsample? Is there any way to do inference in such a setting?


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