How to implement ordinal logistic regression for a factorial design I have read a few answers (e.g. this, this and this) recommending to use Ordinal Logistic Regression (OLR) as a generalized method when Kruskal-Wallis is not suitable. For instance, a 2x2 factorial design since K-W is the non-parametric equivalent of one-way ANOVA.
My situation: I have a 2x2 design with two treatment variables and each has two levels (so 4 treatments in total). Each variable represents the level of exposure of subjects to some environmental factor. There are two possible values: low and high, and I encode them as 0 (low) and 1 (high) in Stata.
I also have several other variables measuring different characteristics of the subjects after being exposed to different combinations of levels of exposure. I want to explore the individual as well as the interaction effects of the two treatment variables on these characteristics.
I am convinced that I should use OLR to do this, but I have a few questions regarding its implementation:


*

*Is it correct to run one regression per characteristic individually (so each characteristic as the dependent variable)?

*These characteristic variables are continuous, so is it appropriate to use them as the dependent variable in a OLR? I guess it is fine, as OLR handles an ordinal variable and a continuous variable is also ordinal?

*How do we run the regression exactly? For example, if an interested dependent variable is a, and the treatment/independent variables are b and c, do we simply do ologit a i.b i.c in the case of Stata? (http://www.ats.ucla.edu/stat/stata/dae/ologit.htm) But then how do we find out about the interaction effect?

*When do we include other variables in the regression in addition to the (categorical) treatment variables? For example, independent variables like age, sex, etc.

*Do we need to adjust for anything when using OLR this way? I see a few options to use with ologit in Stata (http://www.stata.com/manuals13/rologit.pdf), but am inexperienced to decide which to use, if any.

*Finally, is it possible to achieve pairwise comparison across treatments using OLR? If so, how do we do it in Stata?
 A: *

*Yes. Though you can't answer questions about how those characteristics are correlated, or how their correlation might change depending on the predictor values, unless you use a multivariate model.

*Yes. You're modelling $\log\frac{\Pr(Y \geq y_i|x_1,x_2,\ldots)}{\Pr(Y < y_i|x_1,x_2,\ldots)}=\beta_{0i} + \beta_1 x_1 + \beta_2 x_2 + \ldots$ for all the observations $y_i$. So the intercepts $\beta_{0i}$ define a kind of baseline distribution for $Y$ (when all the predictor values $x_1, x_2, \ldots$ are zero) as a step function, which you can use when you're making predictions. The assumption that a change in a predictor has the same effect on the log odds  for all $i$ is a strong one, & you need to check it.

*Model interaction effects just as you would for any other regression model.

*When you're interested in them, or want to control for them, & have enough data to do so. Just as for any other regression model.

*You probably want to present it a bit differently, not wanting a big table of intercepts. And there are some computational issues owing to having so many intercepts. The orm function in the rms package for R deals with them; I don't know about STATA.

*Yes. Once you've fitted the model you can look at any contrasts you like.
