Advice on modeling rating scale data with many "round" numbers and a few in between?

Question

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1. What the student is asking (in plain language)

A tasting panel scored many fruit samples on a 0–15 scale, with the possibility of recording scores to the nearest 0.1.
In reality the panelists almost always used “round” numbers:

72 % of the scores end in .0
13 % end in .5
the remaining ~15 % are spread over the other nine decimal places.

The student wants to analyse these scores with a mixed‑effects model (panelists and/or fruits are random effects).

Two ideas have been suggested:

Scale to (0, 1), add a tiny constant to avoid exact zeros, and fit a Beta GLMM (perhaps with a hurdle for the zeros).
Round the data to the nearest whole number or 0.5 and fit a cumulative‑logit (proportional‑odds) ordinal GLMM.

The question is: Which approach (if any) is appropriate, and what other modelling options should be considered?

Below is a step‑by‑step worked solution that evaluates the options, explains why a third “latent‑continuous‑with‑rounding” model is often the most sensible, and shows how to fit it in R (or in a Bayesian framework).

2. Step‑by‑step reasoning and modelling guide

Step 1 – Understand the measurement process

What the panelists see	What they record	What we observe
A continuous sensory intensity that could in principle take any value between 0 and 15	A decision rule: they choose a value on a 0‑15 scale, but they tend to pick “round” numbers (0, 0.5, 1, 1.5, …) because that is easier for them.	A discrete set of numbers, heavily clustered at the round values.

Statistically we can think of an unobserved latent continuous variable (Y^\ast) that the panelist actually perceives, and a rounding rule that maps (Y^\ast) to the recorded score (Y).
If we ignore the rounding rule and treat the recorded scores as truly continuous, we will mis‑represent the measurement error and inflate apparent “precision”.

Key point: the data are ordinal (they have a natural order) and they are rounded versions of an underlying continuous sensation.

Step 2 – Examine the data (diagnostics)

library(ggplot2)
ggplot(df, aes(x = score)) + 
  geom_histogram(binwidth = 0.1, colour = "black", fill = "lightblue") +
  labs(title = "Histogram of raw scores (0–15, step = 0.1)",
       x = "Score", y = "Count")

Look at the frequency of each decimal part (the table the student already produced).
Check for excess zeros – here the zero‑frequency is 7 % (0.0 is a round value, not a true zero‑inflation problem).
Plot panelist‑by‑panelist histograms to see whether the tendency to use round numbers is consistent across people.

If the pattern of rounding is similar for all panelists, a single rounding mechanism can be assumed; if it varies, you may need panelist‑specific thresholds (more complex, see later).

Step 3 – Why the two proposed options are not ideal

Option	How it treats the data	Problems
Beta GLMM on (score/15) (maybe with hurdle)	Assumes the response is a continuous proportion on ((0,1)) with a smooth density.	• The Beta distribution cannot generate the massive spikes at .0, .5, .1, … • Adding a constant (e.g., 0.1) to avoid zeros is arbitrary and changes the scale. • A hurdle for zeros is unnecessary because 0 is just another round value, not a “structural zero”.
Round to nearest integer / 0.5 and fit a proportional‑odds ordinal GLMM	Treats the outcome as ordinal with a modest number of categories (16 or 31).	• Rounding discards information about the exact decimal part for the 15 % of observations that are not round. • The proportional‑odds model assumes equal spacing of the underlying latent thresholds, which may be violated when the data are heavily rounded (the latent thresholds are not equally spaced). • The model does not explicitly acknowledge the rounding mechanism, so standard errors can be biased.

Both approaches ignore the measurement process (i.e., the rounding) and therefore risk biased inference, especially for fixed‑effect estimates that are of primary interest (e.g., differences among fruit varieties).

Step 4 – A more principled model: Latent‑continuous with rounding (a “rounded” or “censored” model)

4.1 Conceptual model

Latent variable (Y_{ij}^\ast) (continuous, e.g., Normal) represents the true sensory intensity for observation (i) of panelist (j).

[ Y_{ij}^\ast = \mathbf{x}{ij}^\top\beta + \mathbf{z}{ij}^\top b_j + \varepsilon_{ij}, \qquad \varepsilon_{ij}\sim N(0,\sigma^2) ]

(\mathbf{x}_{ij}) = fixed‑effect covariates (variety, replicate, etc.)
(\mathbf{z}_{ij}) = design matrix for random effects (panelist, fruit, etc.)
Rounding rule: the observed score (Y_{ij}) is the result of rounding (Y_{ij}^\ast) to the nearest 0.1, but with a bias toward multiples of 0.5. A simple way to encode this is to define cut‑points (c_k) that separate the observed categories.

For a standard “round‑to‑nearest‑0.1” rule the cut‑points are at …, (-0.05, 0.05, 0.15, 0.25, \dots, 14.95, 15.05).

To capture the extra preference for .0 and .5 we can inflate the widths of the intervals around those values (e.g., make the interval ([0.45,0.55]) a little wider). This is equivalent to estimating unequal thresholds.
Likelihood:

[ \Pr(Y_{ij}=k) = \Phi!\bigl(\frac{c_{k+1}-\eta_{ij}}{\sigma}\bigr) - \Phi!\bigl(\frac{c_{k} -\eta_{ij}}{\sigma}\bigr), ] where (\eta_{ij}= \mathbf{x}{ij}^\top\beta + \mathbf{z}{ij}^\top b_j) and (\Phi) is the standard normal CDF.

This is precisely a cumulative probit model (or logit if you use the logistic CDF) with unequal thresholds, i.e. an ordinal mixed model that explicitly models the spacing of the categories.

When the thresholds are estimated rather than forced to be equally spaced, the model automatically accounts for the “spikes” at .0, .5, etc.

4.2 Implementation options

Platform	Typical function	How to specify unequal thresholds
R – `ordinal` package	`clmm` (cumulative link mixed model)	Use `threshold = "flexible"` (default) → thresholds are estimated freely.
R – `brms` (Bayesian)	`brm` with `family = cumulative("logit")` or `"probit"`	Set `threshold = "flexible"`; the model will estimate a separate cut‑point for each observed score.
R – `glmmTMB`	`glmmTMB(..., family = truncated_poisson(...))` not needed here.	Not directly suited for ordinal with flexible thresholds.
Stata / SAS / SPSS	`gsem` or `proc mixed` with `link=logit`	Provide the full list of cut‑points.

Why this works:

The latent variable is continuous, so all the usual GLMM machinery (random effects, covariates, heteroskedasticity) is available.
The unequal thresholds absorb the irregular spacing created by the panelists’ rounding bias.
No arbitrary constant is added, and zeros are treated just like any other category.

4.3 Example in R (frequentist)

library(ordinal)   # for clmm()
# Assume df has columns:
#   score   : observed rating (0, 0.1, …, 15)
#   variety : factor for fruit variety
#   panel   : factor for panelist (random effect)
#   replicate : numeric or factor for replicate

# Convert to an ordered factor (necessary for clmm)
df$score_ord <- ordered(df$score, levels = sort(unique(df$score)))

# Fit cumulative probit mixed model with flexible thresholds
fit <- clmm(score_ord ~ variety + (1|panel) + (1|replicate),
            data = df,
            link = "probit",          # or "logit"
            threshold = "flexible")   # default, but explicit

summary(fit)

Output interpretation

Fixed‑effects (variety): differences on the latent scale (in probit units).
Random‑effects variance (panel, replicate): how much panelists / replicates deviate from the overall latent mean.
Threshold estimates (theta1, theta2, …): the cut‑points that map the latent variable onto the observed rating categories. Large gaps between successive thresholds will correspond to the “spikes” at .0, .5, etc.

4.4 Example in R (Bayesian with `brms`)

library(brms)

# Define the formula
form <- bf(score_ord ~ variety + (1|panel) + (1|replicate))

# Fit cumulative probit model with flexible thresholds
fit_bayes <- brm(formula = form,
                 data    = df,
                 family  = cumulative("probit"),
                 prior   = c(set_prior("normal(0,5)", class = "b"),
                             set_prior("student_t(3,0,2)", class = "sd")),
                 iter    = 4000, warmup = 1000, chains = 4,
                 control = list(adapt_delta = 0.95))

summary(fit_bayes)

The Bayesian approach gives full posterior distributions for thresholds, making it easy to check whether the intervals around .0 and .5 are indeed larger than the others.

4.5 Model checking

Posterior predictive checks / simulated residuals
- In brms: pp_check(fit_bayes, ndraws = 100).
- In ordinal: residuals(fit, type = "pearson") and plot vs. fitted.
Compare to naive models
- Fit a simple Gaussian LMM (lmer(score ~ ... )).
- Fit the Beta GLMM the student suggested (glmmTMB(score/15 ~ ... , family = beta)).
- Use AIC, BIC, or LOO‑CV (for Bayesian) to see which model predicts the observed distribution best.
You will typically find that the latent‑continuous‑with‑rounded model has dramatically lower deviance and reproduces the spikes.
Check proportional‑odds assumption (if you used a logit link).
- nominal_test(fit) in ordinal or brms::hypothesis() can test whether thresholds are parallel.
- If violated, you can relax it by allowing varying slopes across thresholds (clmm2 in ordinal or brms with gr() interaction).

Step 5 – When (and how) to use the other two ideas

Situation

Recommended model

Rationale

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