Why Table Design Matters
A posterior-results table is more than a compact way to display numbers. It is an argument about what the trial estimated and how the evidence answers a clinical question. A reader should be able to identify the target quantity, the summaries for each arm, the treatment contrast, the uncertainty around that contrast, and any probability tied to a clinical decision. If these are mixed together or left implicit, even correctly computed results can mislead.
The first task is to make the estimand explicit: the population-level quantity targeted by the analysis, including the outcome, time point, treatment comparison, and relevant handling of events that affect outcome interpretation. The table then reports summaries of that quantity. The earlier material on posterior summaries, credible intervals, and clinically important effects supplies the underlying interpretations; here the focus is how to put them together without obscuring their meaning.
A basic layout often uses columns for the arm-level result, the treatment contrast, a posterior credible interval, and one or more posterior probabilities. The table note should specify the endpoint and time point, the analysis population, which direction favors treatment, the interval level, and whether reported quantities are posterior means, medians, or other summaries. Do not use a generic label such as “effect” when the scale could be a risk difference, risk ratio, mean difference, or hazard ratio.
Keep Arm Summaries and Contrasts Distinct
An arm-level summary describes an outcome within one treatment group. A treatment contrast compares groups. These quantities may have different scales: for example, each arm’s event risk is a proportion, while the difference in risks is measured in percentage points. A table should name each scale rather than placing apparently comparable numbers side by side without explanation.
The contrast must also have a stated order. If \(\Delta=p_T-p_C\), positive values indicate a higher event risk in the treatment arm; whether that is favorable depends on whether the event is beneficial or harmful. For a continuous endpoint, \(\Delta=\mu_T-\mu_C\) has units of the outcome, and the direction of benefit must be stated. A relative effect has a different scale and usually a different interpretation from an absolute effect.
Worked Example: Binary Outcome on the Absolute Scale
Hypothetical data. In a randomized trial, 45 of 100 participants assigned to treatment and 35 of 100 assigned to control have the favorable binary outcome. Suppose the arm-specific risks have independent \(\mathrm{Beta}(1,1)\) priors and binomial likelihoods. The resulting posterior distributions are \(p_T\mid data\sim\mathrm{Beta}(46,56)\) and \(p_C\mid data\sim\mathrm{Beta}(36,66)\). Define the treatment contrast as the risk difference \(\Delta=p_T-p_C\), so positive values favor treatment for this favorable outcome.
For a \(\mathrm{Beta}(a,b)\) distribution, the posterior mean is \(a/(a+b)\). Thus the posterior arm means are \(46/102=0.45098\), or 45.098%, for treatment, and \(36/102=0.35294\), or 35.294%, for control. The posterior mean risk difference is
At the displayed precision, the arm summaries are 45.10% and 35.29%; subtracting those displayed values gives 9.81 percentage points. The unrounded posterior means differ by about 9.804 percentage points. These values are consistent up to rounding, but the table should not claim that the displayed arm values subtract to 9.804.
For an approximate 95% credible interval, use a moment-matched normal approximation to the posterior difference. The variance of a \(\mathrm{Beta}(a,b)\) variable is \(ab/[(a+b)^2(a+b+1)]\). Since the two arm posteriors are independent,
The approximate interval is \(0.09804\pm1.96(0.06798)=(-0.03520,0.23128)\), or about −3.52 to 23.13 percentage points. Under the same approximation, \(P(\Delta>0\mid data)\approx\Phi(0.09804/0.06798)=\Phi(1.442)\approx0.925\), and \(P(\Delta>0.05\mid data)\approx\Phi((0.09804-0.05)/0.06798)=\Phi(0.707)\approx0.760\). The latter probability concerns an improvement of more than five percentage points, not merely any improvement.
| Estimand and scale | Treatment arm posterior mean | Control arm posterior mean | Contrast, treatment − control | Approximate 95% credible interval for contrast | Posterior probability |
|---|---|---|---|---|---|
| Favorable-outcome risk; absolute scale | 45.10% | 35.29% | 9.81 percentage points | −3.52 to 23.13 percentage points | \(P(\Delta>5\text{ percentage points}\mid data)\approx0.760\) |
The interval and probabilities are explicitly marked as approximate because the difference of two beta-distributed variables is not generally normal. For a final analysis, posterior simulation or an appropriate numerical calculation can provide the interval and probabilities directly. The table note should preserve that distinction rather than presenting an approximation as an exact result.
Report Each Estimand on Its Own Scale
A table can contain more than one estimand, but it should not imply that unlike quantities are directly interchangeable. A risk difference is measured in percentage points; a risk ratio is unitless; a mean difference is measured in the outcome’s units; and a survival probability at a specified time is a probability. Give separate labels and, where needed, separate rows or panels. If both absolute and relative effects are relevant, identify each clearly and avoid placing one interval under a column labelled only “effect interval.”
For continuous outcomes, include the outcome’s units and the treatment-minus-control convention. For binary outcomes, distinguish proportions from percentage points. For time-to-event outcomes, name the time point when reporting survival probabilities, and do not confuse those probabilities with a hazard ratio. In all cases, a posterior interval belongs to the estimand named in its row; it should not be read as an interval for an arm summary or a different effect scale.
Worked Example: Continuous Outcome and a Clinical Threshold
Hypothetical analysis output. In a trial of a symptom score measured in points, lower scores are better. At the specified follow-up visit, the posterior mean score is 28.6 points for treatment and 32.8 points for control. Define \(\Delta=\mu_T-\mu_C\), so negative values favor treatment. The posterior contrast is approximated by a normal distribution with mean −4.20 points and standard deviation 1.48 points.
The displayed arm means give \(28.6-32.8=-4.2\) points, matching the posterior mean contrast. An approximate 95% credible interval is
Suppose a reduction of at least 3 points is the prespecified clinically important threshold. Under the stated normal posterior approximation,
A table can therefore report arm means in points, the difference in points, the interval in points, and the probability of a reduction exceeding 3 points. It should not label the contrast as a percentage, nor should it state that 79.1% of participants improved by at least 3 points: this probability is about the posterior treatment contrast, not an individual response rate.
| Estimand and direction | Treatment posterior mean | Control posterior mean | Contrast | Approximate 95% credible interval | Posterior probability |
|---|---|---|---|---|---|
| Follow-up symptom score; lower is better; \(\Delta=\mu_T-\mu_C\) | 28.6 points | 32.8 points | −4.20 points | −7.10 to −1.30 points | \(P(\Delta<-3\text{ points}\mid data)\approx0.791\) |
Include Time Points and Clinical Events Explicitly
For repeated or time-to-event outcomes, a table should make the time horizon part of the estimand label. A survival probability at 12 months is not the same quantity as a hazard ratio, and a probability difference at 12 months is not a difference in hazards. Even when multiple summaries arise from one analysis, label each by its own definition and scale.
Worked Example: Twelve-Month Survival Probability Difference
Hypothetical analysis output. Suppose the posterior mean probability of being alive at 12 months is 0.72 in the treatment arm and 0.64 in the control arm. Define \(\Delta=S_T(12)-S_C(12)\), in which positive values favor treatment. The posterior mean difference is \(0.72-0.64=0.08\), or 8.0 percentage points. Suppose posterior computation gives an approximately normal distribution for \(\Delta\) with mean 0.080 and standard deviation 0.045.
The approximate 95% credible interval is \(0.080\pm1.96(0.045)=(-0.0082,0.1682)\), or −0.82 to 16.82 percentage points. If a clinically important increase is defined as more than 5 percentage points, then
The table should call this a 12-month survival probability difference and give the contrast in percentage points. It should not describe the 0.748 probability as the probability that a particular participant survives, or treat it as a hazard-ratio probability.
| Estimand and time point | Treatment posterior mean | Control posterior mean | Contrast | Approximate 95% credible interval | Posterior probability |
|---|---|---|---|---|---|
| Survival probability at 12 months; \(\Delta=S_T(12)-S_C(12)\) | 72% | 64% | 8.0 percentage points | −0.82 to 16.82 percentage points | \(P(\Delta>5\text{ percentage points}\mid data)\approx0.748\) |
Checks, Alternatives, and Common Pitfalls
Before finalizing a table, check that the estimand, arm summaries, contrast, interval, and probability refer to the same population, endpoint, time point, and analysis. Verify the contrast direction by subtracting the arm summaries where that arithmetic applies, and check that the interval and probability use the stated contrast scale. Rounding can make displayed values differ slightly from calculations using unrounded results; state this when it matters rather than forcing numbers to agree artificially.
A table is not the only reporting format. A prose summary can explain a complicated estimand or important caveat, while a figure can reveal posterior skewness, multimodality, or other distributional features that a few summaries conceal. These approaches complement rather than replace a well-labelled table. When the posterior is skewed or multimodal, a mean and symmetric interval may not describe its shape well; use summaries appropriate to the posterior and make the choice clear.
- Mixing scales. Wrong: report a risk difference in a column headed “risk ratio.” Correction: label the contrast and its units explicitly; use a separate entry for each scale.
- Leaving direction implicit. Wrong: report “treatment effect = −4.2” without defining the subtraction order or whether lower values are better. Correction: state both the contrast definition and favorable direction.
- Attaching a probability to an unstated event. Wrong: report “posterior probability = 0.76.” Correction: specify the event, such as \(P(\Delta>5\text{ percentage points}\mid data)\), and identify the clinical threshold.
- Equating a contrast probability with an individual outcome probability. Wrong: interpret \(P(\Delta>0\mid data)\) as the probability that a patient benefits. Correction: explain that it is a posterior probability about the population-level contrast.
- Overstating rounded arithmetic. Wrong: claim displayed arm values subtract exactly to a contrast computed from unrounded values. Correction: show the arithmetic at the displayed precision and note any difference caused by rounding.
Check Your Understanding
Use the definitions and table checks above. Each answer is given below the questions.
- A favorable event occurs in 30% of the treatment arm and 24% of control. What is the displayed absolute risk difference if it is defined as treatment minus control?
- A table reports a mean difference of −2.5 points but does not give the contrast order. What information is needed before the direction can be interpreted?
- In a posterior analysis, \(P(\Delta>0\mid data)=0.90\). Does this mean that 90% of treated patients benefit?
- A 95% credible interval for a risk difference is −2 to 12 percentage points, and the table also reports \(P(\Delta>5\text{ percentage points}\mid data)=0.70\). Are these statements necessarily inconsistent?
- Why should a 12-month survival probability difference not be labelled a hazard ratio?
Solutions. (1) \(30\%-24\%=6\) percentage points, with positive values favoring treatment for this favorable event. (2) The table must state whether the contrast is treatment minus control or control minus treatment, and which score direction is favorable. (3) No. It is a posterior probability about the defined population-level contrast, not the proportion of individuals who benefit. (4) No. The interval allows values both below and above zero, while the probability above 5 percentage points summarizes posterior mass beyond that separate threshold; both can hold. (5) A survival probability difference compares probabilities of being alive at a specified time and is measured in percentage points. A hazard ratio compares hazards and is a distinct, unitless estimand.