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

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
``````

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?

Here's a good explainer