# Is there a multivariate version of logistic regression?

Based on readings with logistic regression, it appears that you could use this analysis to make predictions about categorical variables. Does logistic regression allow you to predict multiple dependent variables with one independent variable, or just one? For example, if I were looking at predicting the gender of someone based on their emotional intelligence scores, could I also predict gender and race at the same time? How does this work?

### Reference

Stevens, J. P. (2009) Applied Multivariate Statistics for the Social Sciences (5th Edition) New York: Routledge Academic. Chapter3.

You can do this with a multilevel model or a regression which takes clustering into account, or a structural equation model. You convert your data from wide to long, so each person has two rows in the dataset, and you have a variable that identifies a person.

You could also do it with multinomial logistic regression - predict member of (say) 4 groups - minority male, minority female, majority male, majority female.

I don't know much about SEMs, but I've heard they're pretty complicated. An easy to understand alternative would be a multinomial logistic regression. In R, you can formulate this with the vglm() function from the VGAM package. Here's a quick gist:

In binomial logistic regression, your model looks like this:

$$\log\left(\frac{p}{1-p}\right) = \mathbf{X\beta}$$

In a multinomial logistic regression, you need to have a baseline factor level (I think R defaults to the last one alphabetically). So, if you have $k$ prediction factors, you predict $p_1, ..., p_k$, where $p_k = 1 - p_1 - ... - p_{k-1}$. Your model then looks like this:

$$\log\left(\frac{p_i}{p_k}\right) = \mathbf{X\beta}_i$$

Notice here that $\mathbf{\beta}_i$ is itself a vector. So if you have $p$ predictor variables, you now have $p\times(k-1)$ slope estimates.