# Sampling bias in multinomial logistic regression

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?