Last updated: 2025-12-05
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| Rmd | 222d2c4 | Dave Tang | 2025-12-05 | Odds ratio |
Odds ratios are widely used to measure the strength of association between two variables. Odds and probability are related but they are different:
sick <- 2
healthy <- 8
total <- sick + healthy
probability <- sick / total
probability[1] 0.2odds <- sick / healthy
odds[1] 0.25An odds ratio compares the odds of an event between two groups.
A sample of 200 people broken down into four categories.
data <- matrix(
c(20, 80, 2, 98),
nrow = 2,
byrow = TRUE,
dimnames = list(
c("Smokers", "Non-smokers"),
c("Cancer", "No Cancer")
)
)
data Cancer No Cancer
Smokers 20 80
Non-smokers 2 98Calculate odds of cancer for the two groups.
odds_smokers <- data[1, 1] / data[1, 2]
odds_nonsmokers <- data[2, 1] / data[2, 2]
odds_smokers[1] 0.25odds_nonsmokers[1] 0.02040816Calculate odds ratio.
odds_ratio <- odds_smokers / odds_nonsmokers
odds_ratio[1] 12.25Smokers have 12.25 times the odds of lung cancer compared to non-smokers.
Use fisher.test() for calculating odds ratios with
confidence intervals.
fisher.test(data)
Fisher's Exact Test for Count Data
data: data
p-value = 5.091e-05
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
2.809877 110.218221
sample estimates:
odds ratio
12.12786 Odds ratios from logistic regression models.
set.seed(1984)
n <- 200
age <- rnorm(n, mean = 50, sd = 15)
smoker <- rbinom(n, 1, 0.3)
disease <- rbinom(n, 1, plogis(-5 + 0.05*age + 2*smoker))
model <- glm(disease ~ age + smoker, family = binomial)
summary(model)
Call:
glm(formula = disease ~ age + smoker, family = binomial)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -4.53700 0.87994 -5.156 2.52e-07 ***
age 0.04302 0.01493 2.881 0.00396 **
smoker 1.83998 0.40800 4.510 6.49e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 188.56 on 199 degrees of freedom
Residual deviance: 157.34 on 197 degrees of freedom
AIC: 163.34
Number of Fisher Scoring iterations: 5Odds ratios.
or_coef <- exp(coef(model))
or_coef(Intercept) age smoker
0.01070543 1.04395510 6.29641114
sessionInfo()R version 4.5.0 (2025-04-11)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 24.04.3 LTS
Matrix products: default
BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so; LAPACK version 3.12.0
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[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
time zone: Etc/UTC
tzcode source: system (glibc)
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] lubridate_1.9.4 forcats_1.0.0 stringr_1.5.1 dplyr_1.1.4
[5] purrr_1.0.4 readr_2.1.5 tidyr_1.3.1 tibble_3.3.0
[9] ggplot2_3.5.2 tidyverse_2.0.0 workflowr_1.7.1
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