# 11.2.4 Odds ratios

To start, let’s just look at how the odds of dying may depend on sex
(ignoring concentration for the time being). Note that now we have to
use the `data`

argument to tell **R** that the variables `dead`

,
`alive`

, and `sex`

are within the dataframe titled `d`

.

+ data = d)

> summary(m2)

Call:

glm(formula = cbind(dead, alive) ~ sex, family = binomial("logit"),

data = d)

Deviance Residuals:

Min 1Q Median 3Q Max

-4.7887 -2.9371 0.1015 2.3400 4.9522

Coefficients:

Estimate Std. Error z value Pr( > |z|)

(Intercept) -0.4754 0.1878 -2.532 0.0113 *

sexm 0.6425 0.2623 2.449 0.0143 *

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 124.88 on 11 degrees of freedom

Residual deviance: 118.80 on 10 degrees of freedom

AIC: 152.91

Number of Fisher Scoring iterations: 4

Like in the ANOVA models we have seen before, **R** is using treatment
contrasts to develop a dummy variable that codes for sex. Thus, the
intercept is the $ln(\frac{p}{1-p})$, or the log-odds of death, for
females (i.e., $ln(\text{Odds}_{\text{females}})$). The estimate for `sexm`

represents the change in the log-odds going from female to male. In
fact, it is:

We can use `exp()`

to get rid of the log (remember that ’log’ in **R**
refers to the natural log).

sexm

1.901186

Thus, the odds (not the probability!) of death for males is 1.9 times
the odds of death for females. Another way of interpreting this is that
the odds of death for males is 90% greater than the odds of death for
females. This number is referred to as the *odds ratio* (OR). It is

(See if you can’t calculate this by hand using the raw data.) The interpretation of coefficients from logistic regression as odds ratios is an important concept! An OR of 1 would indicate that the odds of death are equal for the two groups being compared.