12 Bounds methods ✎ Outline
12.1 Bounded scales and the beta distribution
Most of the methods in this chapter rest on one fact: on a scale with a minimum and a maximum, the mean and the SD are not free to take any values independently of one another. This section builds an intuition for why, by contrasting the normal distribution, which has no bounds, with the beta distribution, which does.
12.1.1 Mean and SD when data are unbounded
The left plot below is the one from Section 5.1: a single sample drawn as a normal distribution, with sliders for its mean and its SD. The right plot shows the same sample as a single point, with its mean on the horizontal axis and its SD on the vertical axis.
Move the two sliders and watch the point, or click anywhere in the right plot to move the point there, and drag to move it continuously. Changing the mean moves the point left and right and changing the SD moves it up and down, and neither slider ever restricts the other. Every point on the right plot, wherever it is, describes a valid normal distribution. Knowing the mean of a normal distribution tells you nothing about what its SD can be, and vice versa: the two parameters are not contingent on one another.
The mean and SD are also a complete description of a normal distribution. If you know that data are normally distributed and you know their mean and SD, you know everything about the distribution: it is symmetric, it has no minimum or maximum, and the proportion of scores in any interval follows from the mean and SD alone. This is why reporting a mean and an SD feels like a full summary of a sample. It is, but only when the data are something like normal.
12.1.2 Mean and SD when data are bounded
Scales in psychology have a minimum and a maximum. A natural distribution for data that lie between two bounds is the beta distribution. It is defined on the interval from 0 to 1, and can be stretched to any other range: a score on a 1–7 scale is a beta-distributed value multiplied by 6 and shifted up by 1. The left plot below is a single-sample version of the plot in Section 8.1: one beta distribution on a 1–7 scale, whose parameters are calculated from the mean and SD you choose with the sliders. The grey areas are outside the scale. The scale starts as 1–7, and you can change its logical minimum and maximum in the two boxes above the sliders. The right plot is the same space of means and SDs as above, and you can again click or drag anywhere in it to set the mean and SD.
Start with the mean in the middle of the scale and increase the SD: the point can rise until the SD is half the range of the scale, which is 3 on a 1–7 scale. Now move the mean towards either end of the scale. The point is no longer free to go anywhere. Above the solid line, no combination of mean and SD is possible, and if you push the SD slider into that region the left plot reports that the combination cannot occur. On a bounded scale, the largest possible SD is a function of the mean.
The solid line is the Bhatia–Davis bound (Bhatia & Davis, 2000). For any data whatsoever on a scale from min to max, not only beta-distributed data, the variance cannot exceed (M − min) × (max − M), so the SD cannot exceed the square root of this. The bound is only reached when every score is at one end of the scale or the other. It is largest when the mean is in the middle of the scale, and shrinks to zero as the mean approaches either end: a sample whose mean is 6.9 on a 1–7 scale must have nearly every score at 7, and so must have a very small SD. The region of the right plot in which a beta distribution exists is exactly the region under the bound. The dashed line marks a second, softer boundary, which we will call the uniform-spread bound. Along this line the data are exactly as spread out, relative to the largest SD their mean allows, as scores spread evenly across the whole scale would be. A uniform distribution uses a third of the largest possible variance, so the dashed line is the Bhatia–Davis bound scaled down by 1/√3, to about 58% of its height at every mean. In terms of the beta distribution, this is where its concentration, α + β, equals 2; the concentration is the same quantity as the precision φ described below. Above the line the concentration is below 2, which forces at least one of α and β below 1, so the scores pile up against at least one end of the scale: the data are less concentrated than even a flat distribution. Below the line the data are at least as concentrated as a uniform distribution, although away from the middle of the scale some of these distributions still have scores piled up against the nearer end, as with mild floor and ceiling effects; the message under the plot says when this is the case. The uniform-spread bound is therefore a fairly liberal threshold for plausibility. Points above it describe data that are possible, but unusual for most measures, and whether they are plausible for a given measure is a question worth asking.
The beta distribution is not described by a mean and an SD but by two other parameters, which can be written in two equivalent ways, both shown under the left plot:
- The shape parameters α and β. Roughly, α pulls scores towards the top of the scale and β pulls them towards the bottom. When they are equal the distribution is symmetric around the middle of the scale, and the larger they both are, the more tightly the scores cluster. When either of them is 1 or less, scores pile up against that end of the scale.
- The mean–precision parameters μ and φ, where μ = α / (α + β) is the mean expressed as a proportion of the way along the scale, and φ = α + β is the precision. This is the form used in beta regression (Ferrari & Cribari-Neto, 2004; Smithson & Verkuilen, 2006). Just as logistic regression models a probability on the logit scale, beta regression models μ on the logit scale and, usually, φ on the log scale, so the intercepts of an intercept-only beta regression are logit(μ) and log(φ). A coefficient for a predictor is then a change in the log-odds of where the mean sits between the two ends of the scale, not a change in scale points.
The contrast with the normal distribution lies in the precision. The variance of a beta distribution, on the 0 to 1 scale, is μ(1 − μ) / (1 + φ). The numerator, μ(1 − μ), is the Bhatia–Davis bound at that mean, so 1 / (1 + φ) is the share of the largest possible variance that the data actually use. In a beta distribution it is φ, not the SD, that is free to vary independently of the mean. The SD is not free, because it depends on both. Two samples with the same precision but different means will therefore have different SDs, and the one whose mean is closer to an end of the scale will have the smaller SD. This is the mechanism behind Section 8.1: an intervention that moves scores towards the end of a scale also shrinks their SD, which inflates Cohen’s d.
The plot below makes this concrete. Instead of setting the mean and the SD, you set the mean and the precision φ, and the SD follows from them. The right plot shows the same space of means and SDs as above, with the Bhatia–Davis bound as a solid black line and the uniform-spread bound as a dashed line. The blue curve shows every combination of mean and SD that has the precision you have chosen. Clicking or dragging anywhere in the right plot sets the mean and the precision whose curve passes through that point. Hold φ constant and move the mean: the point slides along the blue curve, and the SD shrinks as the mean approaches either end of the scale, even though the precision of the data, and so the share of the largest possible variance that they use, has not changed at all.
On a 1–7 scale, try a precision of 5 and move the mean from 4 to 6.5. The SD falls from 1.22 to 0.68, which is almost halved, although the precision is the same throughout. If these were the control and intervention conditions of a study, the intervention condition would look much less variable than the control condition only because its mean is closer to the top of the scale. Setting φ to 2 puts the blue curve exactly on the uniform-spread bound, and any smaller precision puts it above.
A mean and an SD can still be calculated for any beta distribution: M = min + (max − min) × μ, and SD = (max − min) × √(μ(1 − μ) / (1 + φ)). These are the numbers that articles report, and the method of moments used above simply runs these formulas in reverse. But for bounded data the mean and SD are descriptive statistics rather than a complete description of the data generating process. On their own they say nothing about skew or about the bounds. A reader who sees M = 5.5 and SD = 1.0 tends to picture a symmetric bell curve, but on a 1–7 scale these data are necessarily skewed (see the skewness reported under the first beta plot), and the bar marking one SD above the mean can extend beyond the top of the scale: try a mean of 6 and an SD of 1.5. The information about the shape of the data does not come from the mean and SD themselves. It comes from combining them with the bounds of the scale and with an assumption about the form of the distribution. This is why the rest of this chapter keeps asking what the bounds of a measure imply about the statistics reported for it.
12.1.3 Where real scales fall
The plots above show which combinations of mean and SD are possible and which are plausible. The plots below show where the means and SDs of real data actually fall, for two quite different kinds of measure. The data come from openly available datasets (Bainbridge et al., 2022; Brysbaert, 2024) and the Attitudes, Identities, and Individual Differences (AIID) study, as collated in the Granary repository of psychological datasets.
In each plot, every point on the right is one sample’s scores on one measure. Because the measures have different lengths and response formats, both axes of the right plot are expressed as proportions of each measure’s range: the mean as the proportion of the way from the minimum to the maximum, and the SD as a proportion of the range. On these axes the Bhatia–Davis bound and the uniform-spread bound are the same for every measure. Hover over a point to see which measure and sample it is, and click it to keep the label in place. The left plot shows, in the measure’s own units, the beta distribution that has that sample’s mean and SD. This is the shape that the mean and SD imply if the data are beta-distributed, not a histogram of the observed responses.
12.1.3.1 Multi-item scales for individual differences
The first plot shows established multi-item scales that measure individual differences, such as personality traits, emotional and clinical symptoms, and beliefs. Each participant’s score is the sum of their responses to all of the scale’s items. There are 217 samples of 135 scales, with sample sizes from 99 to 19,509.
Almost every sample falls well inside the uniform-spread bound. Typical SDs are around 15% of the scale’s range, or about a third of the largest SD that each sample’s mean would allow, and most means fall in the middle 60% of the scale. One sample of a PTSD symptom checklist (the PCL-5), whose mean is close to the bottom of the scale, sits on the uniform-spread bound, less than 1% above it.
The one clear exception is the Edinburgh Handedness Inventory (EHI, Oldfield, 1971), which measures whether people prefer to use their left or right hand. Its SD is about 28% above the uniform-spread bound, and the beta distribution with its mean and SD, shown on the left, has scores piled up at both ends of the scale. This is not an error, but a consequence of what the scale measures. Handedness is bimodal: most people are strongly right-handed, a minority are strongly left-handed, and few fall in between. It is the only scale in these data for which a bimodal distribution is a plausible description of the construct being measured.
12.1.3.2 Single-item attitude ratings
The second plot shows a different kind of measure: single-item ratings, on a 1–10 scale, of attitudes towards or preferences for 187 objects, such as social groups, public figures, political and religious positions, brands, and foods, from the AIID study. Each point is one attitude object, rated by between 835 and 2,683 participants.
These data are much more spread out than the multi-item scales. The typical SD is about 23% of the range, or about half of the largest SD that each mean allows, and 17 of the 187 attitude objects (9%) fall above the uniform-spread bound. Most of these are topics on which people are polarised: creationism, evolution, capital punishment, abortion, gun control, religion, and a former US president. People tend to feel strongly for or against them, so ratings pile up at both ends of the scale, as the beta distribution on the left shows for creationism. A few others are matters of taste on which people are divided, such as coffee, cats, and meat.
Two things make single-item attitude ratings more spread out than multi-item scales. First, summing several items averages out much of what is specific to each item, which pulls scores towards the middle of the possible range; a single item has no such averaging. Second, some attitude objects genuinely divide people into camps, whereas most individual differences, such as traits, vary smoothly across people, with most of them in the middle.
The lesson for forensic work is that what counts as a plausible mean and SD depends on what was measured. For an established multi-item scale measuring an individual difference, a mean and SD above the uniform-spread bound are possible, but they need an explanation of the kind that handedness provides: a reason to expect most scores to sit at the ends of the scale. For most such scales, no such reason exists. For a single item about a polarising topic, the same mean and SD are unremarkable.
12.1.4 A family of models for bounded data
The beta distribution is not the only way to model bounded data, and beta regression is better thought of as one member of a family of models. Each makes different assumptions, and each can or cannot represent particular features of real data: scores exactly at the minimum or maximum of the scale, data whose SD is very small, and data that can only take whole-number values, such as a single Likert item or a sum score. The table below summarises some of the most common members.
| Model | Designed for | Scores exactly at the minimum or maximum | Very small SDs | Notes |
|---|---|---|---|---|
| Beta regression (Ferrari & Cribari-Neto, 2004; Smithson & Verkuilen, 2006) | Continuous data strictly between the bounds | No: the likelihood is undefined at exactly 0 or 1, so such scores are usually nudged inwards by an ad hoc transformation | Yes | The simplest member. Results can be sensitive to how scores at the bounds were nudged. |
| Zero–one inflated beta (ZOIB) (Ospina & Ferrari, 2012) | Continuous data, including scores at the bounds | Yes, modelled as separate processes from the scores between the bounds | Yes | Has separate parts, each with its own coefficients, for whether a score is at a bound and for where it falls otherwise, which makes results harder to interpret. |
| Ordered beta regression (Kubinec, 2023) | Continuous data, including scores at the bounds, such as slider scales | Yes, through cutpoints that join the bounds and the interior in a single process | Yes | Fewer parameters than ZOIB, at the cost of assuming that one process drives both the bounds and the interior. |
| Beta-binomial regression | Whole numbers out of a known maximum, such as sum scores or the number of correct answers | Yes, as ordinary values | No: its SD cannot be smaller than that of a binomial distribution with the same mean | Captures data that are more spread out than a binomial distribution, but not data that are less spread out. |
| Ordinal (cumulative) regression (Bürkner & Vuorre, 2019; Liddell & Kruschke, 2018) | Ordered categories, such as a single Likert item | Yes, as ordinary categories | Yes: all responses can fall in one category | Does not assume that response options are equally spaced. Its parameters describe a latent variable rather than the response scale. |
It is not necessary to understand the differences between these models in detail to follow the rest of this book. The point is that a whole universe of models exists for bounded data, and that the familiar combination of a mean and an SD, together with the normal distribution it implicitly invokes, is only an approximation to all of them. It is often a good enough approximation, but it is worst exactly where many forensic questions arise: when means sit close to the ends of a scale, when SDs are very large or very small, and when effects are large.
12.2 Means bounds and SD bounds
- TIDES + power + false positives
- Plausibility as well as possibility
12.3 CI bounds vs. estimates
- Estimate in CIs
- Harder test: estimate in middle of CIs. Bounds of the mean given the CIs, i.e., is it in the bounds that define the middle given rounding
12.4 Mean inside range
- Is the reported mean inside the possible range of the measure?
12.5 Metric bounds
- E.g., correlations must be in [-1, 1]
12.6 Correlation matrix positive definiteness
- Checking whether reported correlation tables are positive definite