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Let's start by fitting a logistic model:

fit <- lme4::glmer(vs ~ wt + (1|gear), data=mtcars, family="binomial")
summary(fit)

The output is the following:

Call:
glm(formula = vs ~ wt, family = "binomial", data = mtcars)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.9003  -0.7641  -0.1559   0.7223   1.5736  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)   
(Intercept)   5.7147     2.3014   2.483  0.01302 * 
wt           -1.9105     0.7279  -2.625  0.00867 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 43.860  on 31  degrees of freedom
Residual deviance: 31.367  on 30  degrees of freedom
AIC: 35.367

Number of Fisher Scoring iterations: 5

I understood that the values in the "Estimate" column are log odds ratios. I would like to interpret their size using rules of thumb, such as (in)famous Cohen's (1988) for standardized differences and such.

I found three sources of information:

  • This suggests using 1.5, 3.5 and 9 as cutoffs.
  • Here it is mentioned 1.5, 2.5 and 4.30.
  • Chen (2010) suggests 1.68, 3.47 and 6.71.

As you can see, those seem to refer to "odds ratio" rather than to the "log odds ratio" returned by the logistic regression. No problem, I thought that I could transform these log odds into odds by running exp(estimates).

And this is where I got stuck. For example, in the regression above, the "effect" of wt is negative, meaning that each increase of wt lowers the probability of the outcome. However, "simply" transforming this coefficient by exp() returns a very small, yet positive odds ratio (which is expected). It would mean that a negative effect could never have a large effect size, which is not right.

My question is twofold:

1) How to "transform" these log odds to odds ratios so we can use the mentioned rules of thumb?

2) If not applicable, are there any rules of thumb for log odds ratios?

DISCLAIMER: I am aware of all the "we shouldn't use such rules of thumb", "they are dependent on the context", and mostly agree with it (yet need it anyway).

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To transform log odds ratios to odds ratios, just (as you thought) use exp().

If you want to use those rules of thumb for an odds ratio <1, just take the reciprocal to compare with the threshold (or use exp() on the absolute value of the log odds ratio).

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