Last updated: 2026-02-04
Checks: 7 0
Knit directory: muse/
This reproducible R Markdown analysis was created with workflowr (version 1.7.1). The Checks tab describes the reproducibility checks that were applied when the results were created. The Past versions tab lists the development history.
Great! Since the R Markdown file has been committed to the Git repository, you know the exact version of the code that produced these results.
Great job! The global environment was empty. Objects defined in the global environment can affect the analysis in your R Markdown file in unknown ways. For reproduciblity it’s best to always run the code in an empty environment.
The command set.seed(20200712) was run prior to running
the code in the R Markdown file. Setting a seed ensures that any results
that rely on randomness, e.g. subsampling or permutations, are
reproducible.
Great job! Recording the operating system, R version, and package versions is critical for reproducibility.
Nice! There were no cached chunks for this analysis, so you can be confident that you successfully produced the results during this run.
Great job! Using relative paths to the files within your workflowr project makes it easier to run your code on other machines.
Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility.
The results in this page were generated with repository version bfee866. See the Past versions tab to see a history of the changes made to the R Markdown and HTML files.
Note that you need to be careful to ensure that all relevant files for
the analysis have been committed to Git prior to generating the results
(you can use wflow_publish or
wflow_git_commit). workflowr only checks the R Markdown
file, but you know if there are other scripts or data files that it
depends on. Below is the status of the Git repository when the results
were generated:
Ignored files:
Ignored: .Rproj.user/
Ignored: data/1M_neurons_filtered_gene_bc_matrices_h5.h5
Ignored: data/293t/
Ignored: data/293t_3t3_filtered_gene_bc_matrices.tar.gz
Ignored: data/293t_filtered_gene_bc_matrices.tar.gz
Ignored: data/5k_Human_Donor1_PBMC_3p_gem-x_5k_Human_Donor1_PBMC_3p_gem-x_count_sample_filtered_feature_bc_matrix.h5
Ignored: data/5k_Human_Donor2_PBMC_3p_gem-x_5k_Human_Donor2_PBMC_3p_gem-x_count_sample_filtered_feature_bc_matrix.h5
Ignored: data/5k_Human_Donor3_PBMC_3p_gem-x_5k_Human_Donor3_PBMC_3p_gem-x_count_sample_filtered_feature_bc_matrix.h5
Ignored: data/5k_Human_Donor4_PBMC_3p_gem-x_5k_Human_Donor4_PBMC_3p_gem-x_count_sample_filtered_feature_bc_matrix.h5
Ignored: data/97516b79-8d08-46a6-b329-5d0a25b0be98.h5ad
Ignored: data/Parent_SC3v3_Human_Glioblastoma_filtered_feature_bc_matrix.tar.gz
Ignored: data/brain_counts/
Ignored: data/cl.obo
Ignored: data/cl.owl
Ignored: data/jurkat/
Ignored: data/jurkat:293t_50:50_filtered_gene_bc_matrices.tar.gz
Ignored: data/jurkat_293t/
Ignored: data/jurkat_filtered_gene_bc_matrices.tar.gz
Ignored: data/pbmc20k/
Ignored: data/pbmc20k_seurat/
Ignored: data/pbmc3k.csv
Ignored: data/pbmc3k.csv.gz
Ignored: data/pbmc3k.h5ad
Ignored: data/pbmc3k/
Ignored: data/pbmc3k_bpcells_mat/
Ignored: data/pbmc3k_export.mtx
Ignored: data/pbmc3k_matrix.mtx
Ignored: data/pbmc3k_seurat.rds
Ignored: data/pbmc4k_filtered_gene_bc_matrices.tar.gz
Ignored: data/pbmc_1k_v3_filtered_feature_bc_matrix.h5
Ignored: data/pbmc_1k_v3_raw_feature_bc_matrix.h5
Ignored: data/refdata-gex-GRCh38-2020-A.tar.gz
Ignored: data/seurat_1m_neuron.rds
Ignored: data/t_3k_filtered_gene_bc_matrices.tar.gz
Ignored: r_packages_4.4.1/
Ignored: r_packages_4.5.0/
Untracked files:
Untracked: .claude/
Untracked: CLAUDE.md
Untracked: analysis/bioc.Rmd
Untracked: analysis/bioc_scrnaseq.Rmd
Untracked: analysis/chick_weight.Rmd
Untracked: analysis/likelihood.Rmd
Untracked: bpcells_matrix/
Untracked: data/Caenorhabditis_elegans.WBcel235.113.gtf.gz
Untracked: data/GCF_043380555.1-RS_2024_12_gene_ontology.gaf.gz
Untracked: data/SeuratObj.rds
Untracked: data/arab.rds
Untracked: data/astronomicalunit.csv
Untracked: data/femaleMiceWeights.csv
Untracked: data/lung_bcell.rds
Untracked: m3/
Untracked: women.json
Unstaged changes:
Modified: analysis/isoform_switch_analyzer.Rmd
Modified: analysis/linear_models.Rmd
Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.
These are the previous versions of the repository in which changes were
made to the R Markdown (analysis/odds_ratio.Rmd) and HTML
(docs/odds_ratio.html) files. If you’ve configured a remote
Git repository (see ?wflow_git_remote), click on the
hyperlinks in the table below to view the files as they were in that
past version.
| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | bfee866 | Dave Tang | 2026-02-04 | Add more background material |
| html | 34a78ed | Dave Tang | 2025-12-05 | Build site. |
| Rmd | 222d2c4 | Dave Tang | 2025-12-05 | Odds ratio |
Odds ratios are widely used to measure the strength of association between two variables, particularly in medical research, epidemiology, and case-control studies. Before understanding odds ratios, we need to understand the difference between probability and odds.
Probability and odds are related but different:
sick <- 2
healthy <- 8
total <- sick + healthy
probability <- sick / total
probability[1] 0.2odds <- sick / healthy
odds[1] 0.25In this example:
You can convert between probability (P) and odds using these formulas:
\[\text{Odds} = \frac{P}{1 - P}\]
\[P = \frac{\text{Odds}}{1 + \text{Odds}}\]
Convert probability to odds.
prob <- 0.2
odds_from_prob <- prob / (1 - prob)
odds_from_prob[1] 0.25# Convert odds back to probability
prob_from_odds <- odds_from_prob / (1 + odds_from_prob)
prob_from_odds[1] 0.2To build intuition, let’s see how odds change across different probabilities:
probs <- c(0.1, 0.2, 0.5, 0.8, 0.9)
odds_values <- probs / (1 - probs)
data.frame(
Probability = probs,
Odds = round(odds_values, 3),
Interpretation = c("1 to 9", "1 to 4", "1 to 1 (even)", "4 to 1", "9 to 1")
) Probability Odds Interpretation
1 0.1 0.111 1 to 9
2 0.2 0.250 1 to 4
3 0.5 1.000 1 to 1 (even)
4 0.8 4.000 4 to 1
5 0.9 9.000 9 to 1Notice that when P = 0.5, the odds are 1 (even odds). When P > 0.5, odds > 1; when P < 0.5, odds < 1.
An odds ratio (OR) compares the odds of an event between two groups. It answers the question: “How many times higher (or lower) are the odds of the outcome in one group compared to another?”
Interpreting odds ratio values:
Let’s work through a classic example examining the association between smoking and lung cancer. We have a sample of 200 people broken down into a 2x2 contingency table:
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 98The general structure of a 2x2 table is:
| Disease+ | Disease- | |
|---|---|---|
| Exposed | a | b |
| Not Exposed | c | d |
In our example: a=20, b=80, c=2, d=98.
# Odds of cancer for smokers = a/b
odds_smokers <- data[1, 1] / data[1, 2]
odds_smokers[1] 0.25# Odds of cancer for non-smokers = c/d
odds_nonsmokers <- data[2, 1] / data[2, 2]
odds_nonsmokers[1] 0.02040816The odds ratio is simply the ratio of these two odds:
odds_ratio <- odds_smokers / odds_nonsmokers
odds_ratio[1] 12.25Interpretation: Smokers have 12.3 times the odds of developing lung cancer compared to non-smokers. This is a strong positive association.
For a 2x2 table, the odds ratio can also be calculated directly using the cross-product formula:
\[OR = \frac{a \times d}{b \times c}\]
Using the cross-product formula.
or_cross <- (data[1, 1] * data[2, 2]) / (data[1, 2] * data[2, 1])
or_cross[1] 12.25This gives the same result and is often more convenient for quick calculations.
A point estimate of the odds ratio alone doesn’t tell us how precise our estimate is. We need confidence intervals to understand the range of plausible values for the true odds ratio.
The fisher.test() function calculates the odds ratio
with a 95% confidence interval:
result <- fisher.test(data)
result
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 Key outputs:
Interpreting the confidence interval: Since the 95% CI does not include 1, we can conclude there is a statistically significant association between smoking and lung cancer at the 0.05 significance level.
Logistic regression is commonly used to model binary outcomes (yes/no, disease/no disease). The coefficients from logistic regression are log odds ratios, which can be converted to odds ratios by exponentiation.
Logistic regression models the log odds (also called logit) of the outcome:
\[\log\left(\frac{P}{1-P}\right) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + ...\]
The coefficients (\(\beta\)) represent the change in log odds for a one-unit increase in the predictor.
Let’s simulate data where disease risk depends on age and smoking status:
set.seed(1984)
n <- 200
age <- rnorm(n, mean = 50, sd = 15)
smoker <- rbinom(n, 1, 0.3)
# True model: log odds = -5 + 0.05*age + 2*smoker
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: 5The model coefficients are log odds ratios. To get odds ratios, we exponentiate them:
or_coef <- exp(coef(model))
or_coef(Intercept) age smoker
0.01070543 1.04395510 6.29641114 Interpretation:
For a complete picture, we should report odds ratios with confidence intervals:
# Get confidence intervals for coefficients
ci <- confint(model)Waiting for profiling to be done...# Combine into a nice table
or_table <- data.frame(
OR = exp(coef(model)),
Lower_CI = exp(ci[, 1]),
Upper_CI = exp(ci[, 2])
)
round(or_table, 3) OR Lower_CI Upper_CI
(Intercept) 0.011 0.002 0.054
age 1.044 1.015 1.076
smoker 6.296 2.878 14.392Key points about odds ratios:
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
locale:
[1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
[5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
[7] LC_PAPER=en_US.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[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
loaded via a namespace (and not attached):
[1] sass_0.4.10 generics_0.1.4 stringi_1.8.7 hms_1.1.3
[5] digest_0.6.37 magrittr_2.0.3 timechange_0.3.0 evaluate_1.0.3
[9] grid_4.5.0 RColorBrewer_1.1-3 fastmap_1.2.0 rprojroot_2.0.4
[13] jsonlite_2.0.0 processx_3.8.6 whisker_0.4.1 ps_1.9.1
[17] promises_1.3.3 httr_1.4.7 scales_1.4.0 jquerylib_0.1.4
[21] cli_3.6.5 rlang_1.1.6 withr_3.0.2 cachem_1.1.0
[25] yaml_2.3.10 tools_4.5.0 tzdb_0.5.0 httpuv_1.6.16
[29] vctrs_0.6.5 R6_2.6.1 lifecycle_1.0.4 git2r_0.36.2
[33] fs_1.6.6 pkgconfig_2.0.3 callr_3.7.6 pillar_1.10.2
[37] bslib_0.9.0 later_1.4.2 gtable_0.3.6 glue_1.8.0
[41] Rcpp_1.0.14 xfun_0.52 tidyselect_1.2.1 rstudioapi_0.17.1
[45] knitr_1.50 farver_2.1.2 htmltools_0.5.8.1 rmarkdown_2.29
[49] compiler_4.5.0 getPass_0.2-4