Why is taking exp(model parameters) synonymous with odds ratio?

[dougfir]

I'm working through a course on datacamp on mixed effects modeling. This particualr section is a refresher on glm and lme4::glmer wrt the binomial distribution.

We have a mixed effects model:

summary(model_out) 

Generalized linear mixed model fit by maximum likelihood (Laplace
  Approximation) [glmerMod]
 Family: binomial  ( logit )
Formula: cbind(Purchases, Pass) ~ friend + ranking + (1 | city)
   Data: all_data

     AIC      BIC   logLik deviance df.resid 
   977.4    989.9   -484.7    969.4      164 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-4.2003 -0.7846  0.0941  0.8244  4.1520 

Random effects:
 Groups Name        Variance Std.Dev.
 city   (Intercept) 0.04958  0.2227  
Number of obs: 168, groups:  city, 4

Fixed effects:
             Estimate Std. Error z value Pr(>|z|)    
(Intercept) -1.345085   0.130655 -10.295   <2e-16 ***
friendyes    0.495616   0.059344   8.352   <2e-16 ***
ranking      0.088401   0.005036  17.554   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
          (Intr) frndys
friendyes -0.256       
ranking   -0.411  0.060

We are then asked:

Extract the coefficients from model_out with fixef() and then convert to an odds-ratio by taking exponential. Repeat with confint() to get the confidence intervals.

exp(fixef(model_out))
(Intercept)   friendyes     ranking 
  0.2605175   1.6415091   1.0924257 

and:

exp(confint(model_out))
Computing profile confidence intervals ...
                2.5 %    97.5 %
.sig01      1.1240517 1.7542617
(Intercept) 0.1910566 0.3542918
friendyes   1.4614997 1.8443209
ranking     1.0817427 1.1033118

Apparently the two blocks above are converting the model output into odds ratios.

Why/how is this?

[technocrat]

@dougfir

Family: binomial ( logit )

Here's a good explainer

[dougfir]

From the linked post:

The coefficient returned by a logistic regression in r is a logit, or the log of the odds. To convert logits to odds ratio, you can exponentiate

Got it, this makes sense now, thanks for the link!