Posterior Inference for Paired Continuous Outcomes

Why Analyze the Within-Patient Difference?

In a paired trial, each patient contributes two related continuous measurements: for example, an outcome before and after treatment, or measurements under two conditions. The central quantity is the change within each patient, not the difference between two unrelated group averages. Pairing preserves the information about how the two measurements move together and can make the treatment contrast more precise.

Choose a direction and keep it fixed. For a two-condition trial, define \(d_i=Y_{i,T}-Y_{i,C}\), where \(Y_{i,T}\) and \(Y_{i,C}\) are the measurements for patient \(i\) under treatment and control. For a before-and-after study, one might instead define follow-up minus baseline. The sign then has a context-specific interpretation: if lower values are better, a negative treatment-minus-control difference favors treatment; if higher values are better, the interpretation reverses.

The analysis below assumes that the observed differences are independent across patients and arise from a normal distribution with a common mean and variance: \(d_i\mid\mu_d,\sigma_d^2\sim N(\mu_d,\sigma_d^2)\). This model concerns the distribution of the differences; it does not require the two measurements within a patient to be independent. Indeed, their association is one reason to form differences. The model is appropriate when the paired contrast is the target, the patient pairs are sampled or assigned in a way that supports the intended inference, and the difference distribution is reasonably represented by a normal model. It also assumes the recorded pairs are analyzed consistently; missing or selectively observed measurements require additional justification.

Normal-Inverse-Gamma Model and Posterior

Use a normal prior for the mean difference conditional on its variance and an inverse-gamma prior for that variance:

Here \(\kappa_0>0\), \(a_0>0\), and \(b_0>0\). The inverse-gamma convention used here has density proportional to \((\sigma_d^2)^{-(a_0+1)}\exp(-b_0/\sigma_d^2)\). The prior mean for \(\mu_d\), conditional on the variance, is \(m_0\); \(\kappa_0\) controls its concentration. The parameters \(a_0,b_0\) control prior information about the variance. These choices are assumptions, not consequences of the trial data, and should be considered in relation to the outcome scale.

Let \(\bar d=\frac{1}{n}\sum_{i=1}^n d_i\), and let \(S_d=\sum_{i=1}^n(d_i-\bar d)^2\). The posterior parameters are

Thus \(\sigma_d^2\mid d_1,\ldots,d_n\sim\operatorname{Inv\text{-}Gamma}(a_n,b_n)\), and \(\mu_d\mid\sigma_d^2,d_1,\ldots,d_n\sim N(m_n,\sigma_d^2/\kappa_n)\). Integrating out the unknown variance gives a Student t posterior for the mean difference:

The second argument is the Student t scale, not its standard deviation. The degrees of freedom are \(2a_n\). A central \(100(1-\alpha)\%\) credible interval is

where \(t_{\nu,q}\) is the \(q\)-quantile of a Student t distribution with \(\nu\) degrees of freedom. Given the model and prior, the posterior probability that \(\mu_d\) lies in this interval is \(1-\alpha\). This is a probability statement about the parameter under the posterior, not a claim that repeated intervals contain a fixed parameter with a particular long-run frequency.

Why the Posterior Has This Form

The likelihood for the differences is proportional to \((\sigma_d^2)^{-n/2}\exp[-\sum_i(d_i-\mu_d)^2/(2\sigma_d^2)]\). The identity

separates variation around the observed mean from variation between the observed mean and the parameter. Multiplying likelihood and prior adds the quadratic terms in \(\mu_d\). Completing the square gives

The first term determines the conditional normal posterior for \(\mu_d\). The remaining term adds to the inverse-gamma scale, along with \(S_d\), yielding \(b_n\); the likelihood contributes \(n/2\) to the shape, yielding \(a_n\). Finally, integrating the normal conditional distribution over the inverse-gamma variance produces the stated Student t distribution. This derivation also explains why the interval accounts for uncertainty in both the mean and the unknown variance.

Worked Analyses and Interpretation

Assumptions, Diagnostics, and Alternatives

The paired model is useful because it analyzes the relevant within-patient contrast directly. In a random-effects or repeated-measures description, the variance of a difference reflects the variability of both measurements and their covariance: \(\operatorname{Var}(Y_T-Y_C)=\operatorname{Var}(Y_T)+\operatorname{Var}(Y_C)-2\operatorname{Cov}(Y_T,Y_C)\). Forming patient-level differences automatically retains this covariance information. Treating the two measurements as independent groups discards it and answers a different variance calculation.

Before relying on the posterior, examine the differences themselves. A plot of the differences can reveal marked skewness, outliers, clusters, or a spread that changes across clinically relevant subgroups. With a small sample, such checks have limited power: a roughly symmetric plot does not prove normality. Confirm that each difference uses the intended measurements, that units and direction are consistent, and that one patient has not contributed multiple dependent pairs as if they were independent. A few extreme differences can strongly affect both \(\bar d\) and \(S_d\), and thus the posterior location and scale. Investigate data errors and clinical explanations; do not remove observations solely because they weaken a desired conclusion.

Pairwise analysis also has a data limitation: a patient without both measurements does not have an observed difference. Excluding such patients is not automatically harmless. If pair availability depends on outcomes or prognosis, the complete-pair analysis may be biased for the intended trial population. The simple model does not solve informative missingness; the missing-data mechanism and the estimand must be considered separately.

A common mistake is to analyze the treatment and control measurements as independent observations even though they come from the same patients. That reasoning ignores the pairing and can misstate uncertainty. The correction is to compute one consistently directed difference per patient and model those differences. Another mistake is to interpret a 95% credible interval as giving a 95% posterior probability that each patient's effect lies inside it. The interval concerns the population mean difference \(\mu_d\), not individual patient effects.

The posterior is conditional on the chosen prior and normal model. In particular, a prior with larger \(\kappa_0\) gives more weight to \(m_0\) in the posterior mean, and the scale prior can matter when \(n\) is small. Report the outcome units, contrast direction, prior, and interval. If conclusions are sensitive to plausible prior choices or to a small number of observations, say so rather than presenting one interval as assumption-free. This analysis estimates a mean paired difference; it does not by itself establish causal effects unless the trial design and analysis support a causal interpretation.