# How can I assess the effect of several categorical predictors on an outcome variable?

I have a function of a dozen discrete variables, $$y = f(x_1, ..., x_n)$$ and $k$ samples of $y$ (a lot), $y$ being continuous.

I can use Matlab to analyse the data, with the statistics toolbox. The goal is to analyse the relationship between $y$ and my variables. For example, which variables $x_m$ explain best variations of $y$? (I was thinking of using an Principal Components Analysis, but I can't think of a way to adapt the method to it) To what extent do they have an influence on $y$? Can I find clusters?

My question is: do you have methods I should look into and hints about how to adapt them to my analysis?

Thank you

• Since you want to find the effect of the predictors - $x_1, ..., x_n$ on a univariate response $y$, this seems like a classical multiple regression modeling problem (not multivariate, which would imply you have multiple dependent variables). Is there any reason multiple regression doesn't do what you need? Jul 9, 2012 at 13:58
• Well the main problem is that I have no idea of the form of f, and it's quite difficult to observe so many variables. Also, I am mostly trying to see which variables are relevant to explain changes in y instead of having an explicit formula that is very likely to be wrong. Lastly, my variables are discrete, which doesn't seem to be very suited for this kind of analysis. Jul 9, 2012 at 14:01
• OK but since your predictor variables are categorical, the most general function form (the "saturated" model) is just to have a different function value for each combination of the predictor values, which can be accomplished with a regression model with interactions between all of the variables. This is probably overkill though - pairing down the model would have to involve some substantive theory which is impossible to comment on without more information about what the data actually is - can you provide some background? Jul 9, 2012 at 14:07
• Well for example you can take y: sales for x_1: date, x_2: product type, x_3: product size, x_4: store size Jul 9, 2012 at 14:11
• Have you looked into regression trees? They seem like a good fit to your question. Jul 9, 2012 at 14:47