Why the Choice of Plot Matters
A posterior summary can answer more than one clinical question. A density plot shows the shape of uncertainty, an interval plot emphasizes a selected range of plausible effects, and a cumulative distribution plot makes threshold probabilities directly visible. A figure is useful only if its scale, labels, and annotations make the intended question clear without implying more than the analysis supports.
Before plotting, define the treatment contrast and its direction. In the examples below, \(\Delta\) is defined so that positive values favor the experimental treatment. For a continuous outcome where larger values are better, one possible definition is the experimental-arm mean minus the control-arm mean. For an outcome where smaller values are better, such as symptom duration, a benefit-oriented contrast might instead be the control-arm mean minus the experimental-arm mean. The definition and units belong on the figure; “treatment effect” alone does not specify either.
Matching the Plot to the Question
A density plot is appropriate when shape matters: for example, when a posterior is skewed, has several modes, or puts appreciable density near a clinically important boundary. For a continuous parameter, probability is represented by area under the density over a range, not by the height at one point. The density’s vertical units are the inverse of the horizontal units. Thus a density for a risk difference measured in percentage points has vertical units of one per percentage point.
An interval plot is compact when the main message is a point summary and a selected credible interval. It should identify the point summary (such as posterior median) and the interval’s probability level and construction (such as equal-tailed). It does not show the full posterior shape: two posteriors can share the same median and interval while differing in skewness or modality.
A cumulative distribution plot is especially useful for threshold questions. Its vertical coordinate at \(c\) is \(F(c)=P(\Delta\leq c\mid\text{data})\). For a continuous posterior, the probability of being strictly below \(c\) equals \(F(c)\), because the probability at a single point is zero. For a posterior that can have point mass at \(c\), however, strict and non-strict inequalities differ:
If the clinical question is whether benefit is at least a threshold \(c\), the relevant probability is \(P(\Delta\geq c\mid\text{data})=1-F(c)+P(\Delta=c\mid\text{data})\). State whether the threshold comparison is strict or inclusive when point mass is possible. A continuous density curve alone cannot show a point mass; a discrete spike or another explicit indication is needed.
Constructing a Clear Treatment-Contrast Figure
A comparison between arms should generally show the posterior distribution of the contrast, not just separate arm summaries. If posterior draws are available, calculate the contrast for each joint draw: for draw \(m\), set \(\Delta^{(m)}=\theta_T^{(m)}-\theta_C^{(m)}\), or use the reverse subtraction if that is the defined benefit direction. The joint draws retain the posterior dependence between arms. Subtracting endpoints of two marginal intervals does not generally produce a credible interval for the contrast.
This distinction can also be seen from the variance identity. For a contrast \(\Delta=\theta_C-\theta_T\),
The covariance term matters. In a randomized trial, shared model components or other structure can make arm-specific posterior estimates dependent. A plot of separate arm intervals does not show that dependence and cannot, on its own, quantify uncertainty in their difference.
Write the contrast, identify which sign favors treatment, and give the measurement units.
Use a density for shape, an interval plot for selected summaries, or a cumulative distribution for threshold probabilities.
Label the no-effect value and any clinically important threshold, with a concise explanation of each.
State the interval definition, posterior probability, or other numerical summary needed to interpret the display precisely.
Worked Example: Annotating a Density and Interval Plot
Suppose a hypothetical trial estimates \(\Delta\), the treatment-related increase in symptom-free days over a fixed follow-up period, in days. Positive values favor the experimental treatment. Assume the fitted posterior is \(N(4,2.5^2)\), with posterior mean 4 days and posterior standard deviation 2.5 days. This normal posterior assumption is made here to demonstrate the plot calculations; an actual analysis should use its fitted posterior rather than assume normality for convenience.
A 95% equal-tailed interval under this normal posterior is approximately the mean plus or minus \(1.96\) standard deviations:
Suppose 3 additional symptom-free days is the prespecified threshold for a clinically important benefit. Then
A density plot should label the horizontal axis in days, show a vertical line at zero for no effect and another at 3 days for the clinical threshold, and identify the area to the right of 3 days as about 0.655. An accompanying interval plot can show the posterior mean or median and the 95% interval, with both endpoints labeled. The interval includes zero, while the calculated posterior probability of exceeding 3 days is about 0.655; neither display should be described as proving that the treatment has an important effect.
Worked Example: Comparing Arms Through Joint Posterior Draws
In a hypothetical symptom-severity analysis, lower scores are better. The posterior means are 10.0 points for the experimental arm and 11.5 points for control, so define \(\Delta=\theta_C-\theta_T\), in score points; positive values favor the experimental treatment. Suppose the joint posterior is bivariate normal, with a standard deviation of 1.0 point for each arm and correlation 0.800.
The marginal 95% intervals are approximately \(10.0\pm1.96(1.0)=[8.04,11.96]\) points for the experimental arm and \(11.5\pm1.96(1.0)=[9.54,13.46]\) points for control. These intervals overlap. But the contrast has mean \(11.5-10.0=1.5\) points, and the covariance is \(0.800(1.0)(1.0)=0.800\) square points. Therefore,
Under the stated normal joint posterior, the contrast’s 95% interval is \(1.5\pm1.96(0.6325)=[0.260,2.74]\) points. Its posterior probability of a positive contrast is
The contrast interval excludes zero even though the arm-specific intervals overlap. This is possible because the arm estimates are positively correlated. A figure should display the contrast directly, label positive values as favoring experimental treatment, and explain the arm-level summaries only as supplementary information. In a real analysis, the contrast interval and probability should be calculated from the specified joint posterior.
Worked Example: Reading a Cumulative Distribution With a Point Mass
Consider a hypothetical posterior for a treatment contrast \(\Delta\), in percentage points of absolute risk reduction, with probability 0.100 at \(-2\), 0.200 at \(0\), 0.500 at \(4\), and 0.200 at \(8\). This discrete distribution is useful for illustrating threshold conventions. Its cumulative distribution jumps at each listed value.
At the no-effect value \(c=0\), \(F(0)=P(\Delta\leq0\mid\text{data})=0.100+0.200=0.300\). The probability of a strictly negative effect is only \(P(\Delta<0\mid\text{data})=0.100\), while \(P(\Delta>0\mid\text{data})=1-F(0)=0.700\). The inclusive probability \(P(\Delta\geq0\mid\text{data})\) is \(0.200+0.500+0.200=0.900\). Thus \(F(0)\) is not the probability of being strictly below zero when there is mass at zero.
At a clinically important threshold \(c=4\), \(F(4)=0.100+0.200+0.500=0.800\). Consequently, \(P(\Delta>4\mid\text{data})=0.200\), whereas \(P(\Delta\geq4\mid\text{data})=0.500+0.200=0.700\). A cumulative distribution plot should be right-continuous, show jumps at the mass points, and label whether the target event is “greater than” or “at least” the threshold.
Diagnostics, Alternatives, and Limitations
Check that the horizontal scale spans the posterior support relevant to the clinical question and that labels include units. A truncated axis can hide tails; a density curve with an arbitrary vertical scale can make two posteriors look more or less concentrated than they are. If curves are compared, use a common horizontal scale and make clear whether their areas each equal one. For interval plots, identify the point estimate and interval endpoints rather than relying on color or position alone. For cumulative plots, verify that the curve is nondecreasing, ranges from zero to one (including any limiting jumps), and uses the intended inequality at each threshold.
A density plot provides a view of shape, but threshold probabilities may be easier to read from a cumulative distribution or to report numerically beside the figure. An interval plot is space-efficient but can conceal skewness, multiple modes, and point masses. A cumulative distribution makes probabilities at thresholds transparent but can be less intuitive for readers interested mainly in the overall shape. These displays are complements, not competing estimators: all should represent the same posterior and estimand.
Do not infer that two treatments are equivalent merely because their marginal intervals overlap, or that they differ because separate intervals do not overlap. Those visual comparisons do not replace the posterior distribution of the specified contrast. Similarly, a density’s height at a point is not the probability of that point, and a curve crossing a threshold does not by itself give the desired probability unless its axes and event definition are read correctly. Visual summaries also cannot repair an unsuitable model, unrepresentative data, or inadequate posterior computation. The displayed posterior should be checked against the model and computation used for the analysis.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A density is plotted for a continuous treatment contrast in millimeters. What does the area between 1 and 3 millimeters represent?
- For a posterior with \(F(2)=0.70\) and \(P(\Delta=2\mid\text{data})=0.10\), calculate \(P(\Delta<2\mid\text{data})\), \(P(\Delta>2\mid\text{data})\), and \(P(\Delta\geq2\mid\text{data})\).
- Why can overlapping marginal credible intervals for two arms coexist with a contrast interval that excludes zero?
- Which plot is most direct for showing \(P(\Delta>c\mid\text{data})\) over a range of clinical thresholds \(c\)?
- A figure labels positive values as favoring control, but its caption says positive values favor the experimental arm. What should be corrected before interpreting the plot?
Solutions. (1) It represents the posterior probability that the contrast lies between 1 and 3 millimeters, obtained as the area under the density over that interval. (2) \(P(\Delta<2\mid\text{data})=F(2)-P(\Delta=2\mid\text{data})=0.70-0.10=0.60\); \(P(\Delta>2\mid\text{data})=1-F(2)=0.30\); and \(P(\Delta\geq2\mid\text{data})=0.30+0.10=0.40\). (3) The contrast depends on the joint posterior, including covariance between arm estimates; marginal intervals do not encode that covariance. (4) A posterior cumulative distribution plot is direct: \(P(\Delta>c\mid\text{data})=1-F(c)\). (5) Resolve the inconsistent direction by checking the contrast definition and relabeling the axis and caption consistently before drawing any clinical interpretation.