Information in Paired Comparisons

Why Pairing Changes Information

In a paired comparison, each unit contributes two related measurements, such as outcomes under two conditions or measurements before and after an intervention. The analysis does not treat these measurements as independent observations. Instead, it forms one difference for each pair and estimates the mean of those differences. The precision of that estimate depends on the within-pair variance, which in turn depends on the correlation between the two measurements.

This matters for information-based planning: two designs with the same number of pairs can have different information if their within-pair correlations differ. Positive correlation often reduces the variability of the differences and increases information. But pairing does not automatically improve precision; the correlation and the comparability of pairs matter.

From Within-Pair Correlation to Information

Assume the pairs are independent of one another and identically distributed. Within a pair, the outcomes may be correlated. Let \(X_i\) and \(Y_i\) have variances \(\sigma_X^2\) and \(\sigma_Y^2\), and within-pair correlation \(\rho\). The variance of one difference follows from the variance of a subtraction:

Because the \(n\) differences are independent, \(\operatorname{Var}(\bar{D})=\sigma_D^2/n\). Its standard error is therefore \(\sigma_D/\sqrt{n}\), and the reciprocal of its variance is \(n/\sigma_D^2\). This gives the information expression above. If both marginal standard deviations are \(\sigma\), then \(\sigma_D^2=2\sigma^2(1-\rho)\). Holding the marginal variability fixed, a larger positive \(\rho\) makes differences less variable, so each pair contributes more information.

The normal-model justification is direct. If \(D_i\sim N(\delta,\sigma_D^2)\) with known variance, the log likelihood for \(\delta\), apart from terms not involving \(\delta\), is \(-\sum_i(D_i-\delta)^2/(2\sigma_D^2)\). Its score is \(\sum_i(D_i-\delta)/\sigma_D^2\), whose variance is \(n/\sigma_D^2\). Thus the Fisher information is \(I=n/\sigma_D^2\). Without normality, the same formula describes the reciprocal variance of the sample mean under the stated finite-variance, independent-pairs model; it is the usual large-sample information measure for planning.

In a group sequential design with a fixed planned difference variance, \(n_k\) complete pairs at look \(k\) provide \(I_k=n_k/\sigma_D^2\). If the final analysis has \(N\) pairs, the planned information fraction is \(I_k/I_{\mathrm{final}}=n_k/N\). A pair supplies one difference, not two independent outcome contributions. The number of enrolled individuals alone is not the relevant count if some do not provide both measurements.

Using Information for Sample-Size Planning

For a fixed-sample, two-sided test of \(H_0:\delta=0\) at significance level \(\alpha\), a normal-approximation benchmark for the number of pairs needed to detect a target difference \(\delta_1\) with power \(1-\beta\) is

where \(z_q\) is the \(q\)-quantile of the standard normal distribution. For a prespecified one-sided test, replace \(z_{1-\alpha/2}\) with \(z_{1-\alpha}\). This is a fixed-sample, known-variance normal approximation. A group sequential design’s final target information must also reflect its sequential testing plan; the formula here is a benchmark, not a substitute for that calibration.

When the marginal standard deviations are equal, substituting \(\sigma_D^2=2\sigma^2(1-\rho)\) makes the role of pairing explicit. Relative to an independent two-arm design with the same marginal standard deviation, a positive within-pair correlation can substantially reduce the required number of pairs. The benefit comes from comparing measurements that share pair-specific variation, not simply from collecting two outcomes per pair.

Information Fractions, Diagnostics, and Alternatives

With a fixed \(\sigma_D^2\), information accumulates linearly with the number of complete pairs. For example, if the planned final analysis has \(N\) pairs, the information fraction after \(n_k\) pairs is \(n_k/N\). This simple relation depends on the difference variance remaining as planned. If the variance of differences is higher than anticipated, the same number of pairs yields less information than planned.

Useful checks and failure modes include the following:

• Check the pairing. Each \(X_i\) must be matched with the corresponding \(Y_i\) for the intended comparison. Incorrect matching changes the differences and their variance.
• Check the variance input. The relevant quantity is \(\sigma_D^2\), not either marginal variance alone. Ignoring the covariance term can greatly misstate precision.
• Check independence across pairs. Shared sites, households, or other dependencies between pairs can make \(n/\sigma_D^2\) too large as an information estimate.
• Check that correlation is plausible for the target setting. A planning value borrowed from a different population or measurement process may not predict the variance of future differences. A negative correlation increases \(\sigma_D^2\) when the marginal variances are fixed.
• Check the estimand and study conditions. Pairing supports inference about the mean difference for the defined pairs; it does not, by itself, justify a causal interpretation. In an intervention study, assignment and design assumptions still matter.

The main alternative is an unpaired comparison, which estimates a difference between two group means without using within-pair covariance. It can be appropriate when observations cannot be meaningfully matched or when pairing is infeasible. A paired analysis can be more precise when the pairs are well formed and the correlation is positive, but may be less efficient if correlation is negative or pairing adds substantial variability. The choice should follow the design and target comparison, not a presumption that paired data are always superior.