Why Distribution Shape Matters
A predictive distribution is more than a point estimate with an uncertainty range. Its shape can show whether outcomes are concentrated near one value, spread unevenly toward a tail, or divided into distinct groups. Those differences matter clinically: a long tail may represent a small chance of a substantially worse outcome, while separate peaks may reflect two plausible response patterns. Reporting only a mean and a symmetric interval can hide both.
The object being summarized must first be clear. Here, let \(Y\) denote a specified clinical prediction for a defined population, treatment scenario, endpoint, and time. Predictive draws \(y_1,\ldots,y_n\) approximate the distribution of \(Y\) under that setup. Their interpretation depends on the predictive construction and its weights: an unweighted sample should represent equally weighted draws, while a weighted sample must retain its assigned weights. A summary cannot repair a mismatch between the simulated target and the clinical question.
For an empirical distribution with equally weighted draws, the empirical CDF is \(\hat F(y)=n^{-1}\sum_{i=1}^{n} I(y_i\leq y)\). This tutorial uses the empirical quantile convention \(\hat q_p=\inf\{y:\hat F(y)\geq p\}\), equivalent to selecting the observation at rank \(\lceil np\rceil\) after sorting. Other software may interpolate between observations; when exact reported values matter, state the convention used. With weights \(w_i\) summing to one, replace the empirical CDF by \(\hat F_w(y)=\sum_i w_i I(y_i\leq y)\).
Summaries That Respect Shape
Quantiles describe positions in the predictive distribution without requiring symmetry. The median marks the halfway point; the 10th and 90th percentiles, for example, mark the lower and upper boundaries of the central 80% equal-tailed interval. For a continuous distribution, \(P(q_{0.10}\leq Y\leq q_{0.90})=0.80\), subject to the usual quantile convention. In a discrete or empirical distribution, ties can make the probability in that interval differ from exactly 80%; the quantile boundaries still have the stated CDF definition.
A mode describes a peak, not a typical value in every possible sense. For continuous outcomes, a density mode is where the probability density is locally or globally greatest; the probability of observing exactly that value may nevertheless be zero. For discrete outcomes, a mode is a value with greatest probability mass. A multimodal distribution has multiple peaks, which may correspond to distinct response patterns, subpopulations, or model-supported regimes. Peaks estimated from simulated draws can depend on histogram bin width or density-estimation bandwidth, so report how they were identified and avoid treating every small bump as a meaningful subgroup.
Probability mass at a clinically relevant threshold often complements quantiles. For example, \(P(Y>c)\) directly describes the predicted fraction or probability exceeding a safety limit \(c\), provided the prediction and target population are defined appropriately. It does not explain the full distribution: the same threshold probability can arise from predictions barely above \(c\) or from a severe upper tail. Pair it with quantiles or a plot when that distinction matters.
Worked Example: Summarizing a Right-Skewed Prediction
Suppose 1,000 equally weighted hypothetical predictions of a clinical biomarker change, in units, have the following frequency distribution. The values and frequencies are invented for illustration.
| Prediction (units) | Frequency | Cumulative frequency |
|---|---|---|
| 1 | 100 | 100 |
| 2 | 200 | 300 |
| 3 | 250 | 550 |
| 4 | 200 | 750 |
| 5 | 100 | 850 |
| 10 | 100 | 950 |
| 50 | 50 | 1,000 |
The mean prediction is the frequency-weighted sum divided by 1,000:
The median is the 500th ordered observation. Since the cumulative frequency is 300 at 2 units and 550 at 3 units, \(\hat q_{0.50}=3\) units. The 5th percentile is observation 50, which is 1 unit; the 95th percentile is observation 950, which is 10 units. Thus the empirical 90% equal-tailed interval is [1, 10] units under the stated rank convention. Its empirical coverage is 950 observations, or 95%, because the observation at the upper boundary is included and the quantile convention creates a tie at that point. This illustrates why empirical interval coverage need not equal its nominal level exactly.
If values of at least 10 units are clinically important, the predicted probability is \((100+50)/1000=0.15\), or 15%. The mean of 6.05 units alone would conceal that 15% of the draws are at or above 10 units and that a small tail reaches 50 units. The median and threshold probability make those features clearer.
Why a Mean Can Fall Between Distinct Outcomes
The mean is an expectation: it averages values according to their probability. It need not identify a region where predictions concentrate. For a two-component mixture, if outcome \(Y\) comes from component 1 with probability \(\pi\) and component 2 with probability \(1-\pi\), then
When the component means are separated, this weighted average can lie in a region with little or no predictive probability. This is not an error in calculating the mean: it is a limitation of using one location summary to represent a distribution with distinct groups.
Worked Example: A Symmetric Interval Spanning a Multimodal Distribution
Consider 1,000 hypothetical, equally weighted predictions of a biomarker change in units. Their frequencies are 100 at each of 2, 3, 4, 5, and 6 units, and 100 at each of 10, 11, 12, 13, and 14 units. This creates two separated clusters with a gap from 7 through 9 units.
The mean is \([100(2+3+4+5+6)+100(10+11+12+13+14)]/1000\). Each five-value group sums to 20 and occurs 100 times, so the mean is \(100(20+20)/1000=4\) units. This calculation is incorrect if the purpose is to average the two groups: each group sums to 20, but the second group actually sums to 60. Correctly, the mean is \([100(20)+100(60)]/1000=8\) units. The median is the average of observations 500 and 501: observation 500 is 6 units and observation 501 is 10 units, so the conventional even-sample median is 8 units.
The modes are 2, 3, 4, 5, 6, 10, 11, 12, 13, and 14 under the exact discrete frequency definition: each occurs 100 times. If the values instead represented binned measurements, a coarser histogram could display two peaks, one for each cluster. The key feature is the separated mass, not an artificial claim that one exact value is uniquely modal. Half of the predictions are at or below 6 units and half are at or above 10 units, while none fall from 7 through 9 units. The mean and median of 8 units therefore describe the center numerically but not a typical observed prediction. The empirical 90% equal-tailed interval is [2, 14] units, symmetric about 8 units; it includes the empty gap and conceals the two clusters if reported without distributional context.
Deriving the Coverage of an Equal-Tailed Interval
For a continuous predictive distribution with quantiles defined by \(F(q_p)=p\), the probability between the \(a\)-quantile and the \((1-a)\)-quantile is
Thus \(a=0.05\) gives a central 90% equal-tailed interval. This result does not require a symmetric distribution. Its justification does require a continuous distribution with the stated quantile equalities; for discrete distributions, jumps in the CDF mean the interval's actual probability may differ. Neither case makes the interval a statement that there is a 90% probability the fixed model parameter lies inside it. It summarizes the predictive distribution of \(Y\), conditional on the specified model and predictive setup.
A highest-density region is a different alternative: it selects values with the greatest density or mass until the desired probability is accumulated. For a multimodal distribution, it may consist of separated ranges rather than one interval. Such a region can avoid including a low-density gap, but it depends on density estimation for simulated continuous outcomes and may be less straightforward to communicate. Equal-tailed quantile intervals are easier to define and reproduce; neither summary replaces a plot or threshold probability when those features answer the clinical question.
Worked Example: Comparing the Mean With a Clinical Threshold
Suppose a separate hypothetical set of 1,000 predictions of a biomarker reduction, in percentage points, has these frequencies: 50 at 2, 100 at 3, 300 at 4, 200 at 5, and 50 at 6; also, 30 at 10, 60 at 11, 150 at 12, 45 at 13, and 15 at 14. The higher cluster represents a distinct predicted response pattern. Treat a reduction of at least 10 percentage points as the clinically relevant threshold for this example.
The weighted sum for the lower cluster is \(50(2)+100(3)+300(4)+200(5)+50(6)=2900\). For the upper cluster it is \(30(10)+60(11)+150(12)+45(13)+15(14)=3555\). Therefore the mean is \((2900+3555)/1000=6.455\) percentage points. The proportion at or above the threshold is \((30+60+150+45+15)/1000=300/1000=0.30\), or 30%.
The frequency is greatest at 4 percentage points in the lower cluster and at 12 percentage points in the upper cluster. The exact peak values depend on the discrete values used here; in continuous simulated predictions, mode estimates would require a stated density or binning method. Reporting only the 6.455-point mean would not communicate that 30% of predictions reach the specified threshold or that the predicted outcomes include a separate higher-response cluster. A useful summary therefore pairs the mean, if required, with the threshold probability and a shape-aware display or quantiles.
Diagnostics, Alternatives, and Limitations
Before interpreting a shape-aware summary, inspect the predictive draws or a plot of their distribution. Check for an unexpectedly long tail, gaps, isolated extreme values, and peaks that appear or disappear when histogram bins or density bandwidth change. For a multimodal pattern, check whether the peaks represent substantial probability mass and whether the pattern is stable across reasonable plotting choices. A few outlying simulation draws can create a misleading tail or apparent peak; conversely, too few draws can blur a real feature. Monte Carlo precision diagnostics address simulation stability, but do not establish that the model or predictive target is clinically appropriate.
The main alternatives answer different questions:
| Summary | What it conveys | Limitation |
|---|---|---|
| Mean and standard deviation | Average prediction and overall spread | Can be unrepresentative for skewed or multimodal outcomes |
| Median and quantiles | Distributional positions and asymmetric uncertainty | Do not by themselves identify clusters or clinical threshold risk |
| Mode or density peaks | Locations of high concentration | Can be sensitive to binning or smoothing and may be non-unique |
| Threshold probability | Probability or predicted fraction meeting a defined criterion | Depends on the chosen threshold and omits outcomes' distance beyond it |
| Plot or highest-density region | Shape, gaps, or separated high-density regions | May be less concise; density estimation can be sensitive to method |
A distribution's shape also depends on scale. A monotone transformation preserves quantile ordering, though transformed quantile values change. Density modes, however, are not generally preserved by a nonlinear transformation because the density changes with the transformation's Jacobian. Consequently, a mode reported on a log scale need not transform into the mode on the original clinical scale. State the scale and interpret the mode there.
These summaries describe predictions conditional on the model and scenario used to generate them. A quantile interval does not by itself diagnose model misspecification, and a threshold probability is not automatically a clinical decision rule. The predicted target, population, treatment conditions, and endpoint definition must remain attached to the interpretation. For multimodal predictions, investigate whether distinct modes correspond to meaningful subgroups or mechanisms before assigning them a clinical explanation; the distribution alone does not establish a cause.
Common Pitfall
Another mistake is to call a mode a probability. For a continuous distribution, density at a point is not the probability of that exact value. Report probability over a range or above a threshold when the question concerns probability mass. Similarly, do not compare mode estimates from different scales or smoothing choices as if they were invariant facts.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In an equally weighted sample of 100 predictions, the 10th ordered value is 3 units and the 90th ordered value is 9 units. What empirical equal-tailed interval do these ranks define under the stated nearest-rank convention, and what is its nominal central probability?
- A predictive distribution has two well-separated peaks near 4 and 12 units and almost no mass near 8 units. Why can a mean near 8 units be a poor description of a typical prediction?
- Out of 800 equally weighted predictions, 120 are at or above a clinically specified threshold. Calculate the empirical probability of meeting or exceeding that threshold.
- Why might a density mode estimated from predictions change when the outcome is transformed to a nonlinear scale?
- What diagnostic concern arises if an apparent second mode disappears when histogram bin width is changed modestly?
Solutions. (1) The interval is [3, 9] units; the 10th and 90th percentiles define a nominal central 80% equal-tailed interval. Actual empirical coverage can differ because of ties and the quantile convention. (2) The mean averages the two separated groups and may fall in a region with little predictive mass, so it need not indicate a concentrated or typical outcome. (3) \(120/800=0.15\), or 15% of the predictions. (4) A nonlinear transformation changes density through its Jacobian, so the point of greatest density need not map to the mode on the transformed scale. (5) The apparent peak may be a binning artifact rather than a stable feature; assess its robustness and whether it represents substantial predictive mass before interpreting it clinically.