Binary Outcomes in Matched Pairs

Why Binary Outcomes Need a Paired View

A binary outcome records one of two categories, often coded 1 for an event and 0 for no event. When the same individuals, or deliberately matched units, each contribute an outcome under two conditions, the observations come in pairs. The two condition-specific proportions still describe how often the event occurred, but they do not by themselves show how responses correspond within pairs.

That correspondence matters. If the outcome is 1 under both conditions, the pair contributes to both proportions but not to their difference. If it changes from 0 to 1, or from 1 to 0, it contributes to the difference in opposite directions. A useful analysis therefore keeps both the marginal totals and the within-pair pattern visible.

Organizing the Outcomes

A two-by-two table displays the outcome under condition A in rows and the outcome under condition B in columns. The orientation must be stated because it determines which discordant cell represents an event in A but not B.

Call the cell counts \(n_{11}\), \(n_{10}\), \(n_{01}\), and \(n_{00}\), where the first subscript is the outcome under A and the second is the outcome under B. Thus \(n_{10}\) counts pairs with an event under A and no event under B; \(n_{01}\) counts pairs with no event under A and an event under B. These discordant cells are the only cells that change the difference between the two event totals.

The sample proportion of events under A is \(\hat{p}_A=(n_{11}+n_{10})/n\), and under B it is \(\hat{p}_B=(n_{11}+n_{01})/n\). Subtracting gives:

This identity is the central descriptive result for matched binary data. The paired difference in sample proportions is the net number of discordant pairs favoring A, divided by the number of pairs. Pairs with both outcomes 1 cancel in the subtraction, as do pairs with both outcomes 0.

What the Pairing Adds

The paired comparison is not a comparison of two unrelated collections of outcomes. Each pair contributes a joint outcome: one of \((1,1)\), \((1,0)\), \((0,1)\), or \((0,0)\). A compact way to represent that contribution is the within-pair difference \(D_i=A_i-B_i\). It equals 1 for an A-only event, -1 for a B-only event, and 0 for a concordant pair.

This definition separates two ideas. The marginal probabilities \(p_A=P(A_i=1)\) and \(p_B=P(B_i=1)\) describe event rates under each condition. Their difference is \(\Delta=p_A-p_B\). The paired data also tell us how those outcomes co-occur. Two studies can have the same marginal proportions but different patterns of concordance and discordance.

Why an Independent-Proportion Comparison Is Different

An ordinary independent-proportion comparison treats the observations contributing to the two sample proportions as independent groups. That assumption does not describe a matched-pair design: the A and B outcomes within each pair may be related, and the design intentionally preserves their pairing. The marginal proportions can be calculated correctly from the table, but an independent-groups standard error or test does not use the observed within-pair relationship.

The distinction can be seen from the variance of the pair-level difference. If pairs are independent of one another and come from the target population, then the \(D_i\) values are the independent units for estimating the variability of the paired mean. Since \(D_i\) is 1 in the \(n_{10}\) cell, -1 in the \(n_{01}\) cell, and 0 in the concordant cells, its sample variance is calculated from those pair-level values. For example, with \(s_D\) denoting their sample standard deviation, the estimated standard error of \(\bar{D}\) is \(s_D/\sqrt{n}\), under the usual independent-pairs sampling framework.

This is a description of the relevant structure, not a complete inferential procedure. A specific confidence interval or test also requires its own conditions and method. In particular, it is not valid to call the \(2n\) condition outcomes independent merely because there are two records per pair: the matching means those two records belong to the same unit or matched set.

A Short Derivation of the Pair-Level Variance

The pair-level formulation gives a direct justification for focusing on discordance. Let \(q_{10}=P(A_i=1,B_i=0)\) and \(q_{01}=P(A_i=0,B_i=1)\). Because \(D_i\) takes values 1, -1, or 0, its expectation is:

Squaring \(D_i\) makes both kinds of discordance contribute 1, while concordance still contributes 0. Hence \(E(D_i^2)=q_{10}+q_{01}\). Using the definition of variance, \(Var(D_i)=E(D_i^2)-[E(D_i)]^2\), gives:

Thus the variability of a paired difference depends on the total probability of discordance and on the net difference between the two discordant probabilities. The concordant outcomes contribute zero to each individual \(D_i\); they do not create variation in the paired difference. This does not mean concordant pairs should be discarded from the table: they establish the total number of pairs and the marginal event rates, and their presence helps explain the data structure.

Common Mistake and Practical Checklist

A common mistake is to say, “There are 80 outcomes under A and 80 under B, so I can compare them as two independent samples.” That reasoning counts the marginal records but ignores how the design created the observations. When each A observation is linked to a specific B observation, treating all records as independent discards the information about concordance and discordance, and can give an inappropriate measure of uncertainty.

Another mistake is to focus only on the number of discordant pairs without keeping their direction. The total \(n_{10}+n_{01}\) describes how often pairs disagree, but the difference in event proportions depends on \(n_{10}-n_{01}\). If the two discordant counts are equal, the sample difference is zero even if many pairs disagree.

Before moving from description to inference, check that the design and data support a paired analysis:

• Identify what makes each A outcome belong with its corresponding B outcome.
• State which outcome value represents the event and which condition is in the table rows.
• Check that the four cell counts sum to the number of complete pairs.
• Calculate the two marginal proportions and verify that their difference equals \((n_{10}-n_{01})/n\).
• Recognize the pair, rather than each condition-specific record separately, as the observational unit for the paired difference.

Matched binary outcomes are therefore summarized both by their marginal event rates and by their joint table. The difference in proportions is determined by the balance between the two discordant cells, while the pair-level differences retain the structure needed for appropriate paired analysis.