Why Integer Responses Need Interpretation
A response recorded as 0, 1, 2, or 3 can represent several different things. It might be the number of events observed, or it might be a rating selected from four ordered categories. Both responses are integer-valued and ordered, but their numbers mean different things. The distinction determines what a one-unit change represents and which response likelihood is appropriate in a generalized linear mixed model (GLMM).
The decisive question is not “Are the values ordered?” Both counts and ordinal ratings are ordered. Ask instead: Does each value state how many events occurred, or does it identify an ordered category? A count of three is three occurrences of a defined event. A rating coded 3 is a category label; it need not mean three units of anything.
Meaning, Support, and Response Likelihood
For an event count, the data-generating definition identifies what is counted and the unit over which it is counted: for example, the number of falls per resident during a month. The possible values are constrained by that definition. A count over a fixed time window can often take 0, 1, 2, and higher values; a count of successes among five trials is bounded by 0 and 5, but it remains a count because each success is an event.
For an ordinal rating, the response options define a finite set of categories, such as “none,” “mild,” “moderate,” and “severe.” Coding these categories 0, 1, 2, and 3 preserves their order, but does not establish that the distance from “none” to “mild” equals the distance from “moderate” to “severe.” Even if a rating uses numbers, its values are not event totals unless the response process genuinely records a number of events.
This difference appears in the likelihood. A count model assigns probability to possible event totals. For example, under a conditional Poisson model, \(Y_{ij}\mid b_i\sim\operatorname{Poisson}(\lambda_{ij})\), so the response is supported on nonnegative integers. An ordinal model assigns probabilities to the ordered categories. In a cumulative link model, if \(F\) is the chosen cumulative distribution function and \(\theta_k\) is the threshold for category \(k\), category probabilities can be written as differences between cumulative probabilities:
Here \(Y_{ij}\) is the response for unit \(i\) at occasion \(j\), \(b_i\) is a unit-level random effect, and \(\eta_{ij}\) is the linear predictor. The thresholds are ordered, and the cumulative probabilities must be nondecreasing so that each category probability is nonnegative. The first and last categories use the corresponding boundary cumulative probabilities 0 and 1. This expression is not a count likelihood: it models the probability of selecting each category. Cumulative link likelihoods are a standard ordinal approach; the response-generating meaning must still justify using an ordinal model.
The random effect accounts for dependence among responses from the same unit; it does not change what the response measures. A repeated response that counts events remains a count conditional on \(b_i\). A repeated rating remains ordinal conditional on \(b_i\). Clustering is not a reason to turn one response type into the other.
Worked Examples
Worked Example: Episodes Recorded During a Fixed Window
Suppose, in a hypothetical repeated-measures study, six participants each report the number of migraine episodes they experienced during the same seven-day window. Their recorded totals are 0, 1, 1, 3, 2, and 0. Classify the response and summarize what the numbers mean.
Each value is the number of occurrences of a defined event—an episode—within one participant’s seven-day window. Thus the response is an event count. Zero means no episode occurred in the window; three means three episodes occurred. The difference between 3 and 2 corresponds to one additional episode, not merely a move to a higher label.
The total number of reported episodes is \(0+1+1+3+2+0=7\). The mean count is \(7/6=1.167\) episodes per participant per seven-day window. This average has a direct event-count interpretation. If responses from the same participants were collected over multiple weeks, a GLMM could represent participant-level dependence with a random effect while retaining a count likelihood for the conditional response.
The fixed window makes the unit of interpretation explicit. If windows differed in length, the raw totals would not describe the same exposure duration; the observation window would need to be accounted for in the analysis. That issue does not make the outcome ordinal: the recorded values still count episodes.
Worked Example: An Ordered Symptom Rating
In a hypothetical questionnaire, 20 patients select one category for symptom severity: 0 = none, 1 = mild, 2 = moderate, 3 = marked, or 4 = severe. The observed category frequencies are 2, 3, 6, 7, and 2, respectively. Is this an event count?
No. Each response identifies a selected severity category; it does not report how many symptom events occurred. The response is a finite ordinal rating because the categories have a meaningful order. The empirical category probabilities are the counts divided by 20: \(2/20=0.10\), \(3/20=0.15\), \(6/20=0.30\), \(7/20=0.35\), and \(2/20=0.10\). For example, the cumulative proportion at or below category 2 is \((2+3+6)/20=11/20=0.55\). The category-2 proportion can also be recovered by subtraction: \(0.55-0.25=0.30\), where \(0.25=(2+3)/20\) is the proportion at or below category 1.
The numeric mean of these codes is \((0\times2+1\times3+2\times6+3\times7+4\times2)/20=44/20=2.20\). But this calculation treats adjacent codes as equally spaced. To see why that assumption is not supplied by the ordering, recode the same categories as 0, 1, 2, 4, and 7. The order is unchanged, while the code average becomes \((0\times2+1\times3+2\times6+4\times7+7\times2)/20=57/20=2.85\). The category frequencies and their order have not changed; the numerical average has. A cumulative ordinal likelihood uses the ordered categories without requiring those codes to measure equal distances.
Worked Example: A Bounded Total That Is Still a Count
Suppose a hypothetical assessment gives each of four participants exactly five opportunities to complete a task. The recorded numbers of successful completions are 0, 1, 3, and 5. Because the possible values are limited to 0 through 5, should the response be treated as a finite ordinal rating?
No. Each value counts successful events among five defined opportunities. The response is a bounded count, not a rating category. The observed total is \(0+1+3+5=9\), so the average is \(9/4=2.25\) successful events per participant across five opportunities. A value of 5 means five successes, not simply the highest category.
If, as an additional model assumption, the five opportunities are independent and have a common success probability \(p\) for a participant, the conditional count can have a binomial likelihood: \(Y_i\mid b_i\sim\operatorname{Binomial}(5,p_i)\). Its support is exactly \(0,1,2,3,4,5\), and its conditional mean is \(E(Y_i\mid b_i)=5p_i\). The count classification comes from what was recorded; the binomial likelihood additionally requires the stated trial assumptions. If the same labels 0 through 5 instead stood for six ordered satisfaction categories, the response would be ordinal, not binomial.
A Short Justification: Why Ordering Alone Is Not Enough
An ordinal response supplies an ordering of categories. If its category codes are replaced by any strictly increasing set of labels, the order remains unchanged. For example, 0, 1, 2, 3, 4 and 0, 1, 2, 4, 7 encode the same order. Therefore, a model intended to use only ordinal information should derive its category probabilities from that order, rather than treating the code differences as measured distances.
A count has a different structure. Its value \(k\) is defined by the number of events, so the statement \(Y=k\) has a specific meaning: exactly \(k\) events occurred. Replacing count values by arbitrary increasing labels would destroy that meaning and would no longer describe event totals. In a cumulative ordinal model, the probability of category \(k\) is obtained by subtracting adjacent cumulative probabilities:
This follows because the event \(\{Y\leq k\}\) consists of the disjoint events \(\{Y\leq k-1\}\) and \(\{Y=k\}\). Subtracting the probability of the first event from the combined probability leaves \(P(Y=k)\). The identity is useful for ordered categories, but it does not make their codes event counts.
How to Decide and What Can Go Wrong
When a data dictionary shows integers without explaining their meaning, inspect the response protocol before selecting a likelihood. Ask what an observer or respondent did to produce each value, what zero means, whether one unit represents one additional occurrence, and whether the possible values arise from counting events or choosing among response options. The study form and operational definition often settle the question more reliably than a frequency table.
Some response scales are genuinely ambiguous from the stored values alone. A scale labeled “number of symptoms” may be a count if each reported symptom is tallied, but it may instead be a rating if respondents choose a level whose codes happen to be 0, 1, 2, and 3. A maximum value also does not settle the matter: five successes among five opportunities is a bounded count, whereas the highest of six severity labels is a rating.
- Wrong reasoning: “The values are integers from 0 to 4, so they are counts.” Correction: Integers are just a storage format. Establish whether each value counts occurrences or names a category.
- Wrong reasoning: “The categories are ordered, so a one-unit increase has the same meaning everywhere.” Correction: Order does not establish equal spacing. A move between adjacent rating categories need not represent the same change in severity.
- Wrong reasoning: “The ratings have a finite number of values, so they are bounded counts.” Correction: A count may be bounded by a fixed number of opportunities, but only when the values count events within those opportunities.
- Wrong reasoning: “Repeated observations make a rating a count because there are several responses per person.” Correction: Repeated measurement affects dependence; it does not redefine the response. Choose the conditional response likelihood according to what each response measures.
The main alternative is not simply “integer model versus non-integer model.” For event counts, use a likelihood for event totals, with its support and assumptions matched to the counting process. For finite ordered ratings, use an ordinal likelihood, such as a cumulative link likelihood, that represents category probabilities and respects order without assuming equal spacing. A nominal multicategory likelihood is an option when categories have no meaningful order, but it discards the ordering information. Treating ordinal codes as a continuous measurement makes additional spacing assumptions and should not be justified merely by numeric coding.
The choice has limitations in either direction. A count model is inappropriate if a value is a subjective category label, even when the labels are integers. An ordinal model is inappropriate if it discards the event-total meaning of genuine counts. And a correct response classification does not by itself guarantee a suitable fitted model: the likelihood’s other assumptions and the dependence structure must also be considered. Classification is the first decision, not a complete model diagnostic.
Practical Checklist
State the event, rating question, or task that produced each value, including the unit or response options.
Ask whether it means one additional event or movement to the next category. Do not infer equal spacing from ordered codes.
Use a count likelihood for event totals and an ordinal likelihood for ordered category selection, checking the assumptions of the chosen likelihood.
In a GLMM, represent within-unit dependence separately from the response type; a random effect does not convert a rating into a count or vice versa.
Check Your Understanding
For each item, classify the response and explain the decisive feature. Solutions follow the questions.
- A laboratory records the number of bacterial colonies on each plate. Is this a count or an ordinal rating?
- A survey records satisfaction as 1 = very dissatisfied through 5 = very satisfied. What is the response type, and what does the code 4 mean?
- Workers each receive exactly four inspection opportunities; the recorded response is the number of defects found. Does the upper bound make this an ordinal rating?
- A nurse records “pain level” using codes 0, 1, 2, and 3, but the data file contains no definitions. What information is needed before choosing a likelihood?
- Participants select one of four ordered categories at each of three visits. Does repeated measurement make the response a count?
Solutions. (1) A count: each value is the number of colonies, which are occurrences of the defined event. (2) An ordinal rating: code 4 identifies the fourth ordered response category, “satisfied,” not four satisfaction events or a distance of four units. (3) It is a bounded count: values tally defects among the four opportunities; the bound does not change their meaning. (4) Obtain the questionnaire wording and category definitions, including whether the values are event totals or ordered severity categories. Integer storage alone is insufficient. (5) No. Each response remains an ordinal category; repeated visits create within-participant dependence that a GLMM may represent separately.