Odds Ratio (OR)

The odds ratio represents the ratio of the odds of an outcome occurring in an exposed group to the odds of it occurring in an unexposed group. In PRS contexts, it typically compares individuals in a high-risk percentile (e.g., top 10%) to a reference population (e.g., the remaining 90%).

• OR = 1: No association between PRS stratum and disease
• OR > 1: Increased odds of disease in the high PRS group
• OR < 1: Decreased odds (protective effect)
• Example: An OR of 2.5 for the top 5% of a PRS distribution means those individuals have 2.5 times the odds of developing the disease compared to the general population

The odds ratio is not the same as a risk ratio (relative risk), though they are often misinterpreted interchangeably. The distinction is critical when communicating results to clinicians.

• Odds: The probability of an event divided by the probability of its absence (p / (1-p))
• Risk: The probability of an event occurring (p)
• When disease is rare (prevalence < 10%), the OR approximates the risk ratio
• When disease is common, the OR overestimates the risk ratio
• Example: If baseline disease risk is 20% and OR = 2.0, the actual risk ratio is approximately 1.67, not 2.0

The odds ratio emerges naturally from logistic regression, the most common framework for PRS validation. The model estimates the log-odds of disease as a linear function of predictors.

• The coefficient β from logistic regression equals ln(OR)
• OR = exp(β): Exponentiating the coefficient yields the odds ratio
• This mathematical relationship ensures the OR is always positive
• Logistic regression assumes a linear relationship between the PRS and the log-odds of disease
• Example: A β coefficient of 0.693 corresponds to an OR of exp(0.693) ≈ 2.0

Every odds ratio estimate must be accompanied by a 95% confidence interval (CI) to convey the precision of the estimate. Wide intervals indicate uncertainty, often due to small sample sizes in extreme PRS percentiles.

• CI crossing 1.0: The association is not statistically significant at α = 0.05
• Narrow CI: High precision, typically from large sample sizes
• CI calculation: Based on the standard error of the log(OR): CI = exp(ln(OR) ± 1.96 × SE)
• Example: OR = 3.0 (95% CI: 2.5–3.6) indicates a robust, precisely estimated threefold increase in odds

Odds ratios in PRS studies are typically reported as adjusted ORs, accounting for potential confounders that could bias the association. Unadjusted ORs may reflect confounding rather than true genetic risk.

• Common covariates: Age, sex, and principal components (PCs) of ancestry
• PC adjustment: Corrects for population stratification, ensuring the OR reflects genetic risk, not ancestry differences
• Clinical covariates: Models may further adjust for known clinical risk factors to assess the incremental value of the PRS
• Example: An unadjusted OR of 2.8 may attenuate to 2.3 after adjusting for the first 10 ancestry PCs

The odds ratio is a relative measure and does not convey absolute risk. Two PRS models can have identical ORs but vastly different clinical utility depending on the baseline disease prevalence.

• Comparative ORs: Used to benchmark different PRS methods on the same cohort
• Per-standard-deviation OR: Reports the increase in odds per 1 SD increase in the PRS, enabling comparison across studies with different PRS distributions
• Percentile-based OR: Reports odds for extreme quantiles (e.g., top 1% vs. remainder), directly relevant for clinical risk stratification
• Example: A PRS with a per-SD OR of 1.5 may yield a top-1% OR of 5.0, identifying a small but extremely high-risk group

While often used interchangeably in casual discussion, risk ratio (relative risk) and odds ratio are distinct measures. Risk ratio directly compares the probability of disease in exposed vs. unexposed groups. Odds ratio compares the odds. In case-control studies fundamental to GWAS, disease prevalence is fixed by design, making risk incalculable; the odds ratio is the mathematically valid measure of association. When the disease is rare (prevalence <10%), the OR approximates the risk ratio.

The statistical workhorse underlying most GWAS and PRS analyses. Logistic regression models the log-odds of a binary outcome (case/control) as a linear combination of predictors. The exponentiated coefficient for a genetic variant directly yields the odds ratio for that allele. Key properties:

• Handles binary dependent variables naturally
• Coefficients are estimated via maximum likelihood
• The model is additive on the log-odds scale, meaning each risk allele contributes a fixed increment to the log-odds of disease

A range around the point estimate of an odds ratio that likely contains the true population value. Conventionally reported as the 95% confidence interval. Interpretation:

• If the interval crosses 1.0, the association is not statistically significant at α=0.05
• A narrow interval indicates high precision, typically driven by large sample size
• In PRS reporting, ORs should always be accompanied by their 95% CI to convey estimate uncertainty

In GWAS summary statistics, the beta coefficient represents the change in log-odds of the outcome per copy of the effect allele. The odds ratio is derived by exponentiating beta: OR = e^β. Key relationships:

• β = 0 → OR = 1 (no effect)
• β > 0 → OR > 1 (risk-increasing allele)
• β < 0 → OR < 1 (protective allele)
• The standard error of beta determines the confidence interval width and p-value

The foundational epidemiological design that necessitates the use of odds ratios in genetic association studies. Researchers ascertain a group of cases (individuals with the disease) and a group of controls (without the disease), then retrospectively compare allele frequencies. Because the case-to-control ratio is artificially set by the investigator, absolute risk cannot be computed. The odds ratio remains valid because it is symmetric—the OR for disease given genotype equals the OR for genotype given disease.

A public health metric extending the odds ratio to quantify the proportional reduction in disease incidence if a risk factor were eliminated. PAF incorporates both the strength of association (OR) and the prevalence of the exposure (risk allele frequency). A variant with a modest OR but high population frequency can have a greater public health impact than a rare variant with a large OR. PAF is critical for translating PRS findings into population-level health strategies.