# Multinomial logistic regression with unequal categories

I'm trying to solve a statistical problem in my research. The research was conducted to test the relationship between the experience (happy, unhappy) of a traveler and his awareness of digital communication channels in the airport. My dependent variable consists of 3 responses from 200 people; happy, unhappy and neutral. The predictor is the number of digital channels that they are aware of (total of 9). There are several other variables that have also been taken into account, such as age, sex, Residence (dummy variable), whether traveling first time (dummy variable), travel purpose (dummy variable), place where data was collected such as ticketing, baggage claim, parking lot etc. (categorical variable) and whether traveling with a companion (dummy variable). My hypothesis is that awareness of digital communication channels improves the quality of experience (happy unhappy or neutral). I want to perform a regression analysis to see test my hypothesis and also to see which of the predictors are the strongest. I tried Multinomial logistic regression I get a statistically significant model, however, none of the parameter estimates are significant. I feel that the problem might be because my three groups (happy, unhappy and neutral) are unequal in size (happy = 125, unhappy=33 and neutral = 42). How do I carry on a meaningful regression analysis?

• Wouldn't ordinal regression be more appropriate in your setting? There seems to be a natural ordering of your categories: unhappy --> neutral --> happy. You could start by looking at your (ordered) outcome variable to see how it relates, in turns, to each individual predictor. Then consider the ordered logistic regression model which includes all predictors at once. – Isabella Ghement Oct 19 '18 at 0:10
• Thank you! Even in this case, wouldnt the unequal categories pose a problem? – user182250 Oct 19 '18 at 0:30
• That is why I suggested you look at one predictor at a time first - you can build an intuition for what may be happening. Is a predictor significant by itself but not in the presence of other predictors? – Isabella Ghement Oct 19 '18 at 0:38
• number of digital channels: You treat it as categorical or continue covariate? – user158565 Oct 19 '18 at 3:23
• @a_statistician I treat it as a covariate – user182250 Oct 19 '18 at 3:41

## 1 Answer

I don't think unequal class is the problem. Multinomial logistics seeks coefficients that maximizes the probability of a class in relative to the baseline class, i.e.

$$\hat{\beta} = \max_\beta \ln \sum_{i=1}^{K-1} \dfrac{P(C=i)}{P(C=K)}$$

And $$\dfrac{P(C=i)}{C=K}$$ is the size of class $$i$$ relative to class $$K$$, so the difference in class sizes convey information that is (and ought to be) taken into account.

Imagine if one of your class has only 1 data point (say only one customer responded "unhappy"). You would expect your algorithm to be really careful to classify a new observation as "unhappy" -- and any features that qualify an observation as "unhappy" ought to be highly salient. If two customers are identical in every observed dimension, and one is "happy" and the other is the only case of "unhappy", then the feature dimensions would have no explanatory power for classifying "happy" vs "unhappy". The algorithm will simply classify observations with these features to "happy" if that's the majority case.

In short, you want your model to be trained on whatever class sizes that come from the field. You may want to look into your X variables, and consider increasing your sample size overall. I do not think it is a good idea to hand pick samples from each class to artificially create equal classes.