# Interpreting odds ratio of an ordinal regression when independent variables are negative percentages

I'm trying to express the results of an ordinal regression with a certain "perspective", and I'm confused.

My dependent variable is an ordinal representing the progression in a scale of negative outcome (e.g. 0 = ok, 1 = bad outcome, 2 = very bad outcome).

My independent variables are negative percentages that represent the percentual change in a measure between two time points (i.e. +X% if there was an increase, -Y% if the measure decreased).

I have the following ORs:

Percentual change in var A: 0.895 (95% CI: 0.801; 0.988)
Percentual change in var B: 0.870 (95% CI: 0.559; 1.337)
Other variable:             1.007 (95% CI: 0.995; 1.019)
Age:                        0.970 (95% CI: 0.895; 1.045)


I am interpreting these ORs as follows:

Percentual change in var B, Other variable, and age have their CIs touching 1; this means that I cannot refute the hypothesis that they have no effect on $Bad_Outcome. Percentual change in var A has a significant effect on$Bad_outcome (statistically speaking). The 0.895 OR means that an increase in var a (e.g. a change from, say, -7% to -6%) is associated with an odds 0.895 times the odds of passing from an ok outcome (0) to a bad outcome (1), or from a bad outcome (1) to a very bad outcome (2) (I am not sure if this is correct).

Given that my previous interpretation is correct, how can I "translate" this results taking as reference the decrease in the percentual change in var A?

In plain English: I'd like to express the results as the odds of having a worse outcome (from ok to bad, and from bad to very bad) when var A decreases (say, from -6% to -7%), but I'm confused by this "reversal of perspective", and I'm not sure how to "convert" the OR. I was thinking of 1/OR, but I'm really uncertain.

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You make a lot of assumptions without verification. It is seldom the case that % change is the correct summary measure for two time points. It assumes that effects are through a ratio but % change is an asymmetric measure (as opposed to log ratio). You didn't check the Tukey mean-difference (Bland-Altman) plot to see if the log ratio is the proper summary of the two as opposed to the difference or other measure. You didn't check that the coefficients of the two measures (or their logs) have a ratio of -1.0 if the two were used as separate variables in the regression model.

Once you get the proper model formulated you can plot the two variables (or their log ratio or difference or some nonlinear function of them using regression splines) against the outcome to get a better interpretation than odds ratios as two arbitrary settings.

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Thank you for your observations, they are extremely interesting. I have a bit of googling ahead of me before I can fully appreciate your suggestions :) Now I have a few questions... Can you expand a bit about the asymmetricity of % change? Also, by using log ratio are you suggesting that I should transform my data in log and then compute the ratio? (eg. log(measure_at_time2)/log(measure_at_time1) – greymatter0 Aug 29 '14 at 14:19
Googling? You need to get some good books and spend a great deal of time studying them. Your desire to take the ratio of two logs points out that you need to start with algebra. Regarding references about change measures see biostat.mc.vanderbilt.edu/ManuscriptChecklist – Frank Harrell Aug 29 '14 at 14:32
I'm a medical student, and at the moment I have only basic textbooks (Martin Bland's An introduction to medical statistics, which was a required textbook when I took my biostatistics lessons a few years ago, and Motulsky's Intuitive biostatistics, which I purchased on my own initiative seeing a lot of positive reviews on amazon). I recognize that they are inadequate for anything beyond the basics, so I was planning to purchase intermediate-to-advanced statistical textbooks - if you have any suggestions, I'm listening. I am aware of my deficiencies in basic math; I'm working on it. – greymatter0 Aug 31 '14 at 9:03
It would be great if you could find a local biostatistician to collaborate with. Getting back to your question, the link I provided above provides a lot of information. The basic problem with % change related to asymmetry is shown in an example. Suppose that one subject began with X=1 and changed to X=2. That is a 100% increase. To get back to where she started (X=1) requires a 50% decrease. Thus in the data a 100% increase is balanced by a 50% decrease, but the average over two subjects (one with X=1 going to X=2 and the other X=2 going to X=1) is the wrong answer of 25%. – Frank Harrell Aug 31 '14 at 12:57