Solved – How to implement ordinal logistic regression for a factorial design

interactionkruskal-wallis test”logisticordered-logitstata

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 2×2 factorial design since K-W is the non-parametric equivalent of one-way ANOVA.

My situation: I have a 2×2 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:

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

  2. 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?

  3. 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?

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

  5. 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.

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

Best Answer

  1. 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.

  2. 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.

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

  4. 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.

  5. 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.

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

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