7  Understanding bounded data and Beta regression ✎ Draft

Many of the bounds methods in Chapter 13 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 chapter builds an intuition for why, by contrasting the normal distribution, which has no bounds, with the beta distribution, which does.

7.1 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.

7.2 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 9.1: 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 9.1: 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 bounds methods in Chapter 13 keep asking what the bounds of a measure imply about the statistics reported for it.

7.3 Standardized Mean Difference (Cohen’s d) effect size when data are bounded

The plot above shows one sample. A study that compares an intervention condition with a control condition has two, and the plot below extends the one above to two beta distributions. You set the mean of the control condition, the precision φ, and the mean difference between the intervention and control conditions, in scale points. The button above the sliders switches the last slider to setting the standardized mean difference instead, and the mean difference that produces it is then calculated for you; if no mean difference on the scale can produce it, the plot says so. Both conditions are given the same precision, which is what a beta regression with a predictor for condition on the mean but not on the precision assumes. With the precision held constant, the SD of each condition follows from its mean, exactly as in the plot above. The left plot shows the two distributions, with the control condition in orange and the intervention condition in blue, and their means and SDs drawn at different heights so that both can be seen. The bracket between the two means is labelled with the SMD, Cohen’s d: the mean difference divided by the pooled SD, √((SDcontrol2 + SDintervention2) / 2), assuming equal group sizes. The right plot shows both conditions as points on the same curve of constant precision. Clicking or dragging in the right plot sets the control condition’s mean and the precision, as above, and leaves the mean difference as it is.

On a 1–7 scale with a precision of 5, a control mean of 4 and a mean difference of 2, the control condition’s SD is 1.22 and the intervention condition’s is 0.91. Cohen’s d is 1.85, compared with the 1.63 that the same two-point difference would give if both conditions had an SD of 1.22. Nothing about the spread of the data differs between the conditions: they have the same precision, so they use the same share of the largest possible variance at their means. The intervention condition’s SD is smaller only because its mean is closer to the top of the scale, and the smaller SD makes the same mean difference look larger in SD units. Increase the mean difference and the gap between the two values of d widens, because the intervention condition’s SD keeps shrinking as its mean approaches the top of the scale. Now set the control mean to 2.5 and the mean difference to 3, so that the two conditions sit symmetrically either side of the middle of the scale: their SDs are equal and there is no inflation. Whether a bounded scale inflates or deflates d depends on where the two means sit relative to the middle and the ends of the scale, not only on how far apart they are.

7.4 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.

7.4.1 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.

7.4.2 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.

7.5 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.

7.6 Test yourself

Answer all the questions, then click the button below the quiz to see why each answer is right.

Means and SDs on bounded scales

  1. For normally distributed data, knowing the mean tells you nothing about what the SD can be.
  2. On a 1–7 scale, a sample of 30 people with a mean of 6.80 could have an SD of 1.50.
  3. On a bounded scale, where is the largest possible SD largest?
  4. What does the Bhatia–Davis bound describe?
  5. An article reports M = 5.5 and SD = 1.0 on a 1–7 scale. These values suggest a roughly symmetric, bell-shaped distribution.
  6. A mean and SD above the uniform-spread bound are impossible.
  7. Data whose SD lies above the uniform-spread bound must have scores piled up against at least one end of the scale.

Bounds and effect sizes

  1. Two groups whose scores have the same precision, φ, but different means, will have the same SD.
  2. An intervention that moves scores towards the top of a bounded scale can increase Cohen’s d by shrinking the intervention group’s SD, even if the data are no less spread out relative to what is possible.
  3. On a 1–7 scale with the same precision in both groups, a control mean of 4 and an intervention mean of 6 give a Cohen’s d that is:
  4. When two groups’ means lie symmetrically either side of the middle of the scale, and they have the same precision, their SDs are equal.

Real data and models

  1. Why are the means and SDs of multi-item scales usually less spread out than those of single items?
  2. A mean and SD above the uniform-spread bound are as surprising for a single item about a polarising topic as for an established multi-item scale of a personality trait.
  3. For bounded data, the mean and SD are a complete description of the distribution, as they are for normal data.
  4. Ordinary beta regression can model scores that lie exactly at the minimum or maximum of a scale without any adjustment.
  1. True. A normal distribution has no minimum or maximum, so any combination of mean and SD describes a valid normal distribution. The two parameters are not contingent on one another (Section 7.1).
  2. False. With a mean of 6.80 on a 1–7 scale, the largest possible SD is about √((6.8 − 1)(7 − 6.8)) ≈ 1.08, or 1.10 with the small-sample correction for 30 people. A mean so close to the top of the scale means that nearly every score must be close to 7 (Section 7.2).
  3. When the mean is in the middle of the scale. The largest possible SD comes from putting scores at both ends of the scale. That is easiest when the mean is in the middle, and impossible when the mean is at either end, where every score must equal that end (Section 7.2).
  4. The largest SD that any data with a given mean can have on a bounded scale. It applies to data of any kind, not only beta-distributed data, and is only reached when every score is at one end of the scale or the other (Section 7.2).
  5. False. A mean of 5.5 is only 1.5 points from the top of the scale, so with an SD of 1.0 the data must be skewed, with a longer tail towards the bottom. The mean and SD say nothing about shape on their own; it is the bounds that imply the skew (Section 7.2).
  6. False. The uniform-spread bound is a bound on plausibility, not possibility. Points above it describe data that are more spread out than scores spread evenly across the scale, which is possible, but unusual for most measures (Section 7.2).
  7. True. Above the uniform-spread bound, the beta distribution’s concentration, α + β, is below 2, which forces at least one of α and β below 1. That means scores pile up against at least one end of the scale (Section 7.2).
  8. False. With the same precision, the SD depends on the mean: the group whose mean is closer to an end of the scale has the smaller SD. It is the precision, not the SD, that is free to vary independently of the mean (Section 7.2).
  9. True. Moving the mean towards the end of the scale shrinks the SD, even when the precision is unchanged, and a smaller SD makes the same mean difference larger in SD units. This is one way that bounded scales inflate Cohen’s d (Section 7.3).
  10. Larger than if both groups had the control group’s SD. With a precision of 5, the intervention group’s SD is 0.91 rather than 1.22, so d is 1.85 rather than 1.63 (Section 7.3).
  11. True. The SD depends on the mean through μ(1 − μ), which is the same for means equally far from the middle. With symmetric means, the two SDs are equal, and the bounds do not inflate d (Section 7.3).
  12. Summing items averages out what is specific to each item, pulling scores towards the middle. Most individual differences also vary smoothly across people, whereas some attitude objects genuinely divide people into camps (Section 7.4).
  13. False. What counts as a plausible mean and SD depends on what was measured. For a single item about a polarising topic, scores that pile up at both ends are unremarkable. For an established multi-item scale of a trait, they need an explanation (Section 7.4).
  14. False. For bounded data, the mean and SD are descriptive statistics, not a complete description of the data. On their own they say nothing about skew or about the bounds (Section 7.2).
  15. False. The beta distribution is undefined at exactly 0 and 1, so scores at the minimum or maximum have to be nudged inwards. Zero–one inflated and ordered beta regression were developed to handle them (Section 7.5).