4.5 The odds ratio
The odds ratio is a widely used measure for studying the association between two categorical variables, especially when the interest is in comparing the probability of an event occurring across different population groups. Its interpretation is based on comparing the relative odds of an event occurring between two analysis categories.
In this sense, this parameter makes it possible to evaluate how those odds change across different groups of interest and is a fundamental tool in social, economic, and health studies. In addition, this measure can also be used to quantify the relationship between the levels of a variable and a categorical factor by comparing the relative odds of the event occurring in each group (Heeringa et al., 2017).
For example, suppose the aim is to study the association between people’s sex and poverty status. In particular, the interest is in evaluating whether the relative odds of belonging to the non-poor group versus the poor group differ between women and men. To do this, the following odds ratio can be defined:
\[ \frac{ P(\text{Sex}=\text{Female} \mid \text{pobreza}=0) \big/ P(\text{Sex}=\text{Female} \mid \text{pobreza}=1) }{ P(\text{Sex}=\text{Male} \mid \text{pobreza}=1) \big/ P(\text{Sex}=\text{Male} \mid \text{pobreza}=0) } \]
The numerator expression represents the relative odds of observing women not living in poverty relative to observing women living in poverty. Analogously, the denominator represents the relative odds of observing men living in poverty relative to men not living in poverty. Therefore, the odds ratio compares both relative odds and makes it possible to quantify whether the association between sex and poverty favors one group more than the other.
A value equal to one would indicate the absence of association between the variables, that is, that the relationship between sex and poverty status is similar for men and women. Values greater than one would indicate greater relative odds for women compared with men, while values below one would suggest greater relative odds for men. In the context of household surveys, this type of measure is especially useful for studying sociodemographic inequalities and gaps in well-being conditions across different population groups.
The procedure for carrying this out in R is first to estimate the proportions of the cross-tabulation between the sex and poverty variables, presented in Table 4.22:
sex_poverty_interaction_proportion <- survey_design %>%
group_by(Sex, poor) %>%
summarise(proportion = survey_prop(vartype = c("se", "ci")))
sex_poverty_odds_contrast <- svymean(x = ~interaction(Sex, poor),
design = survey_design,
se = TRUE, na.rm = TRUE, ci = TRUE,
keep.vars = TRUE)| Sex | poor | proportion | proportion_se | proportion_low | proportion_upp |
|---|---|---|---|---|---|
| Female | 0 | 0.611 | 0.032 | 0.547 | 0.671 |
| Female | 1 | 0.389 | 0.032 | 0.329 | 0.453 |
| Male | 0 | 0.605 | 0.037 | 0.531 | 0.675 |
| Male | 1 | 0.395 | 0.037 | 0.325 | 0.469 |
Then, the contrast is performed by dividing each of the elements of the expression shown above:
odds_ratio <- quote(
(`interaction(Sex, poor)Female.0` /
`interaction(Sex, poor)Female.1`) /
(`interaction(Sex, poor)Male.0` /
`interaction(Sex, poor)Male.1`)
)
svycontrast(
stat = sex_poverty_odds_contrast,
contrasts = odds_ratio
)## nlcon SE
## contrast 1.02 0.1The result is that the estimated odds that a woman is not living in poverty, compared with a man, is equal to 1.02. This means that, without considering other survey variables, women have odds of not being in poverty that are approximately 2% higher than those observed among men. In other words, the relative probability of not living in poverty is slightly higher for women than for men.