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).
