Contingency Table

A standard 2x2 contingency table organizes data by exposure status (rows) and outcome status (columns).

• Cell a: Exposed individuals with the outcome
• Cell b: Exposed individuals without the outcome
• Cell c: Unexposed individuals with the outcome
• Cell d: Unexposed individuals without the outcome

Marginal totals represent row and column sums, providing the denominators for risk calculations. This structure is the universal input for federated cohort discovery queries.

The odds ratio (OR) is the primary measure of association derived from a contingency table, calculated as (a/b) / (c/d) or equivalently ad / bc.

• OR = 1: No association between exposure and outcome
• OR > 1: Exposure associated with higher odds of outcome
• OR < 1: Exposure associated with lower odds of outcome

In federated analytics, each site computes its local contingency table, and odds ratios are pooled using inverse variance weighting in a meta-analysis engine without sharing raw patient counts.

The chi-squared test evaluates whether two categorical variables are statistically independent by comparing observed frequencies to expected frequencies under the null hypothesis.

• Expected frequency for each cell: (row total × column total) / grand total
• The test statistic sums (observed - expected)² / expected across all cells
• A significant p-value rejects the null hypothesis of independence

In distributed settings, federated chi-squared tests aggregate only the test statistic and degrees of freedom, preserving patient privacy.

Stratification involves constructing separate contingency tables for each level of a potential confounding variable to isolate the true exposure-outcome relationship.

• Mantel-Haenszel estimator computes a weighted average of stratum-specific odds ratios
• Stratification addresses Simpson's Paradox, where aggregated results reverse direction within subgroups
• Federated systems require each site to compute stratum-specific tables locally before secure aggregation

This is essential for observational studies where randomization is absent.

In a cross-silo federated learning network, contingency tables are computed locally at each institution and only aggregated counts are shared with the central distributed query engine.

• Each site executes the same computable phenotype definition to identify cohorts
• Local 2x2 tables are transmitted through a secure aggregation protocol
• The central node sums cell counts to construct a global contingency table

This approach complies with HIPAA and GDPR by ensuring patient-level data never leaves the originating institution.

Beyond the odds ratio, contingency tables support multiple effect size metrics for clinical interpretation.

• Risk Ratio (RR): (a/(a+b)) / (c/(c+d)) — used in cohort studies
• Risk Difference (RD): (a/(a+b)) - (c/(c+d)) — absolute measure of effect
• Number Needed to Treat (NNT): 1 / RD — clinical relevance metric

In federated meta-analysis, these measures are computed from each site's table and combined using random-effects or fixed-effects models depending on heterogeneity assessment results.

A measure of association derived directly from a 2x2 contingency table. It quantifies the odds that an outcome will occur given a particular exposure, compared to the odds of the outcome occurring without that exposure. In a case-control study, the odds ratio estimates the relative risk when the disease is rare. Calculation: (a/b) / (c/d), where 'a' and 'c' are exposed cases and controls, and 'b' and 'd' are unexposed cases and controls.

A statistical hypothesis test that evaluates whether there is a significant association between the two categorical variables displayed in a contingency table. It compares the observed frequencies in each cell to the expected frequencies under the null hypothesis of independence. A significant p-value indicates the variables are likely dependent, though it does not measure the strength of the association.

An extraneous variable that correlates with both the exposure and the outcome, potentially creating a spurious association within a contingency table. If not controlled for through stratified analysis or regression, a confounder can distort the true relationship. The Mantel-Haenszel procedure is a classic method for calculating an adjusted odds ratio across stratified contingency tables to remove this bias.

An alternative to the chi-squared test used when sample sizes are small. It calculates the exact probability of observing a specific distribution of frequencies in a 2x2 contingency table, given fixed marginal totals. This test is essential when any expected cell frequency is less than 5, a scenario where the chi-squared approximation becomes unreliable.

A quasi-experimental technique used to balance observed covariates between treatment and control groups before constructing a contingency table. By matching subjects with similar propensity scores—the probability of receiving treatment given observed covariates—researchers aim to mimic randomization and reduce selection bias, allowing for a less confounded calculation of the treatment effect.

Key performance metrics for a binary classification test, calculated from a 2x2 contingency table that cross-tabulates predicted outcomes against actual outcomes. Sensitivity (True Positive Rate) measures the ability to correctly identify positives. Specificity (True Negative Rate) measures the ability to correctly identify negatives. These metrics are foundational for evaluating diagnostic models in federated clinical analytics.