# Comparison of two populations by analyzing the dependence of a single dependent variable on several independent ones

I am looking at two populations on which I have measured 4 independent variables ($X_1$, $X_2$, $X_3$, $X_4$) and one dependent variable ($Y$) along with each measurement or observation. I suspect that the two populations are different in the sense that the second population's $Y$ shows a dependence on an additional non-considered factor (I expect $Y$ to be systematically lower, in fact). Considering that:

• the dependent variables are highly correlated;
• the dependence of $Y$ on the $X$'s is unknown and arguably non-parametric;
• the number of observables/measurements is small (~100 for the first group, ~30 for the second).

I feel that multi linear regression would assume some kind of underlying structure and would bias my analysis, so I would like to avoid it. I tried to use PCA on both and compare the PC's: nothing significant comes out in the sense of dimensionality. Could you suggest a test that would check if the two populations are really different in respect to $Y$ while keeping at a minimum the number of assumptions on its functional dependence?

Thank you very much,

P.

Despite your feelings, you might want to try multiple regression nevertheless. Put your two populations into one regression and add an indicator for group membership (i.e. to which population does the observation belong). This allows you to test if there are any significant mean differences across the populations after controlling for your explanatory variables. If you think that the two populations' dependency on the explanatory variables are different as well (i.e. different $\beta$s), you can also easily interact the variables with your group membership dummy, which allows you to formally test if the parameters are equal across the two populations.

• @coffeinejunky: thank you for your answer, but let me note that I have no idea on what the dependence of $Y$ on the $X$'s looks like. Do you have a suggestion on how should I proceed in this situation? – pedrofigueira May 1 '14 at 0:43
• Personally, as a first (not necessarily final) step, I always start with a linear regression (assuming your variables are continuous). Linear regression is basically a fancy way to compare means, and comparing means often makes sense. ;) – coffeinjunky May 1 '14 at 0:50
• I understand your comment, but I would use it as my last option, as I would have to neglect several properties of my data, such as the correlation between the $X_i$ and the fact that $Y$ is probably a non-parametric function of $X_i$. I would not know what to make of the results. – pedrofigueira May 1 '14 at 0:59
• Just a small remark: multiple regression does not assume that the $X_i$ are uncorrelated. See e.g. here – coffeinjunky May 1 '14 at 1:33

PCA is generally used to reduce the number of dimensions. Considering that you have only 4, I don't think it would help much.

Can you tell me more about your Y variable? Is it classification or a regression that you are trying to perform?

If its classification, I would suggest you try using trees.

• The variable $Y$ is a continuous and always positive variable that is associated to the $X$'s in a very complicated way. I do not think this is a classification situation, but I am merely interested in checking if the populations are different in $Y$. – pedrofigueira May 1 '14 at 0:47