Ordinal Versus Nominal Multicategory Responses

Why the Meaning of Categories Matters

A response with three or more categories can look similar in a data table whether its categories are labels or levels. For example, a response might be recorded as “red,” “blue,” or “green,” or as “poor,” “fair,” or “good.” Both are multicategory outcomes, but only the second has a natural ordering. That distinction matters because an analysis can use an ordering only if the categories actually have one.

In a generalized linear mixed model, the response distribution describes how an outcome falls among its possible categories. When observations are clustered, the category probabilities may be conditional on cluster-level random effects. The response’s measurement scale remains a separate question: clustering does not turn labels into ordered levels, or ordered levels into labels.

Definition: A nominal response has categories that are distinct but have no intrinsic ranking. An ordinal response has categories with a meaningful ranking, but the gaps between successive categories are not thereby known to be equal. In either case, a single response must be assigned to one of the defined categories; the categories should be mutually exclusive and collectively exhaustive for the response being analyzed.

For notation, let \(Y\) denote one response and let its possible categories be \(c_1,\ldots,c_K\). For a nominal response, the category names can be permuted without changing their meaning. For an ordinal response, the order \(c_1<c_2<\cdots<c_K\) is part of the response definition: it says which outcomes are lower or higher. The symbols “<” here indicate category order, not numerical distances.

The earlier material on ordinal outcomes established that ordinal information concerns ordered category probabilities. The key distinction here is that a nominal representation assigns a probability to each category but does not encode a ranking among them. A set of category probabilities alone cannot establish that one category is higher than another.

What a Nominal Multinomial Representation Says

For \(n\) independent responses classified into \(K\) nominal categories, write \(n_k\) for the number in category \(c_k\), and \(p_k\) for the probability of that category. The category counts satisfy \(\sum_{k=1}^K n_k=n\), and the probabilities satisfy \(p_k\geq 0\) and \(\sum_{k=1}^K p_k=1\). The multinomial probability of the observed count vector is

P(n_1,\ldots,n_K)=\frac{n!}{n_1!\cdots n_K!}\prod_{k=1}^K p_k^{n_k}.

This distribution says how many observations fall in each named category. It does not say that category 2 is between categories 1 and 3, or that moving from category 1 to category 2 is comparable to moving from category 2 to category 3. In a GLMM, the probabilities might vary with predictors and random effects; the distinction between nominal and ordinal concerns whether the response structure uses a meaningful category order.

Why Relabeling Does Not Create an Order

The defining feature of nominal categories can be shown directly from the multinomial likelihood. Let \(\pi\) be any permutation of the category labels. If the categories are relabeled, the count and probability paired with each new label are also permuted. Since multiplication is unchanged by rearranging its factors, the probability of the data is unchanged.

\frac{n!}{\prod_{k=1}^K n_k!}\prod_{k=1}^K p_k^{n_k} = \frac{n!}{\prod_{k=1}^K n_{\pi(k)}!}\prod_{k=1}^K p_{\pi(k)}^{n_{\pi(k)}}.

This is a short justification for why the labels themselves do not supply an order: swapping “red” and “green” changes no statistical content if those names are simply labels. By contrast, swapping “mild” and “severe” in a pain scale changes the stated ordering and therefore changes the meaning of the response. This does not mean that nominal categories must have equal probabilities. It means that no rank relation among those categories is supplied by their definition.

A nominal multinomial analysis can still be applied to data whose categories happen to be ordered, if the analysis deliberately treats them as unordered. The resulting category probabilities can describe the distribution, but the model does not use the order to constrain or summarize those probabilities. That is the information trade-off: flexible category-specific description, with no built-in use of rank.

Worked Example: Classifying a Hypothetical Response Scale

Suppose, hypothetically, a clinic records the primary type of appointment requested by 40 patients: routine, urgent, or specialist. The counts are 18, 12, and 10, respectively. Is this response nominal or ordinal, and what does a multinomial summary report?

The response is nominal. “Routine,” “urgent,” and “specialist” identify different appointment types, but these labels do not define one scale on which every specialist request is higher than every urgent request. The fact that one category sounds more intensive does not, by itself, give all three categories a single unambiguous ranking.

The observed proportions are \(18/40=0.450\), \(12/40=0.300\), and \(10/40=0.250\). They sum to \(0.450+0.300+0.250=1.000\). These are descriptive proportions for the three categories; their sizes do not establish an order among the categories.

If the labels were listed in another order, the same three proportions would simply move with their corresponding labels. It would be incorrect to infer that “specialist” is a higher response than “urgent” merely because it was listed third.

Worked Examples: Using or Ignoring an Order

Worked Example: The Same Counts With an Ordinal Scale

Suppose, hypothetically, 30 participants rate a service as poor, fair, or good. The observed counts are 6 poor, 9 fair, and 15 good. Assume the wording and questionnaire define this as increasing satisfaction, so poor \(<\) fair \(<\) good. Calculate the category proportions and the cumulative proportions at the first two cut points.

The proportions are \(6/30=0.200\), \(9/30=0.300\), and \(15/30=0.500\). They sum to 1.000. The cumulative proportion at or below poor is \(6/30=0.200\); at or below fair it is \((6+9)/30=15/30=0.500\).

Those cumulative summaries have an interpretation because the categories are ordered: 20.0% reported poor, and 50.0% reported poor or fair. They do not establish equal gaps between poor and fair and between fair and good. A nominal representation could report the same three category proportions, but it would not distinguish “poor or fair” as the lower part of an ordered scale.

Worked Example: Arbitrary Codes Can Reverse a Comparison

Consider two hypothetical groups, each with three responses on an ordered scale with categories low, medium, and high. Group A has low, low, high; Group B has medium, medium, medium. Show why assigning numerical codes and averaging them does not give a coding-free comparison.

First assign codes 1, 2, and 3 to low, medium, and high. Group A’s coded mean is \((1+1+3)/3=5/3=1.667\). Group B’s is \((2+2+2)/3=6/3=2.000\), so this coding makes Group B’s mean larger.

Now use codes 0, 1, and 100 for the same ordered categories. Group A’s mean is \((0+0+100)/3=33.33\), while Group B’s is \((1+1+1)/3=1.000\). The apparent comparison reverses, even though the category order and the observed responses have not changed.

The arithmetic is correct under each assigned coding, but the codes impose different distances between levels. Since order alone does not specify those distances, neither coded mean is determined by the ordinal information alone. This example is not a reason to discard the ordering; it shows why treating ordinal labels as equally spaced quantitative measurements requires additional justification.

Choosing the Response Representation

The decision should be based on what the categories mean in the measurement process, not on which representation produces a convenient fit or a tidy numerical code. Ask whether the response categories can be ranked in a way that is meaningful for the scientific question. If not, use a nominal representation. If they can, retain the order in an ordinal representation when that order is relevant to the analysis.

Feature	Nominal response	Ordinal response
Category meaning	Distinct labels without an intrinsic rank	Levels with a meaningful rank
Effect of permuting labels	Does not alter the response meaning	May alter the meaning if it changes the order
What category probabilities describe	Probability of each separate label	Probability of each level, with an order that can also be used
What is not supplied automatically	A ranking among labels	Equal spacing between successive levels

A nominal model is often a sensible alternative when the order is genuinely ambiguous, when categories are qualitatively different, or when the research question concerns distinct types rather than “more” or “less.” An ordinal representation is preferable when the ranking is clear and relevant, because treating those categories as nominal leaves that information unused. Neither representation is universally superior: the response definition and scientific aim determine which information is legitimate to use.

In clustered data, make this decision at the level of the response for each observational unit. For example, a patient’s ordered symptom rating remains ordinal whether patients are nested in clinics or not. Random effects may account for dependence or variation across clinics, but do not resolve an ambiguous category ranking. Likewise, changing the model for clustering cannot make unordered labels ordinal.

Diagnostics and Common Failure Modes

Before fitting a model, inspect the response definition and coding. Check that every response belongs to exactly one category, that the listed categories cover the outcomes that can occur, and that the category labels have not been assigned an artificial order simply because software requires an ordering. For a proposed ordinal scale, verify that the rank is defensible in context and consistent across respondents and occasions.

Ordering by code rather than meaning. Codes 1, 2, and 3 do not establish an order unless the category definitions do. Confirm the substantive interpretation before using the codes.
Assuming a frequency ranking is an ordinal ranking. The most common category is not necessarily “highest.” Counts describe how often labels occur, not how those labels rank.
Ignoring an established order. Treating an ordered rating as nominal may be defensible for a specific question, but it does not use the rank information and can require separate description of each category.
Forcing an order onto unlike labels. Categories such as treatment type, blood group, or appointment type do not become ordered because a table lists them from left to right.
Confusing order with equal intervals. A valid ranking says which category is higher, not how far apart the categories are.

If experts or respondents disagree about the ranking, if a category has different meanings across groups, or if the response combines multiple dimensions, a single ordinal scale may not be justified. Conversely, very sparse category counts do not change a response’s measurement level; they may create estimation difficulties, but category type is determined by meaning rather than frequency.

Key takeaway: Nominal categories are interchangeable labels, while ordinal categories have an interpretable ranking. A nominal multinomial representation can describe the probability of every category, but its symmetry under relabeling shows that it does not encode that ranking.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A hypothetical survey records a respondent’s preferred mode of transport: bus, bicycle, or walking. Is the response nominal or ordinal? Explain.
A hypothetical questionnaire records pain as none, mild, moderate, or severe. What makes this ordinal, and what does the order not tell you?
In a hypothetical sample of 50, 20 responses are category A and 30 are category B. Does the larger count make B a higher category? Why or why not?
Explain why permuting category labels in a nominal multinomial description does not create a new ordering.
Why can the mean of ordinal category codes depend on the particular numerical coding?

Solutions. (1) Nominal: the three transport modes are different labels, with no general rank implied by the response definition. (2) It is ordinal because the levels are meaningfully ranked from none to severe. The ranking does not specify equal distances between adjacent levels. (3) No. The counts show that B occurred more often; frequency does not establish a rank. (4) A permutation merely changes the names attached to the same category counts and probabilities. The multinomial product is unchanged when its corresponding factors are rearranged, so no rank is supplied. (5) Numerical codes impose distances as well as order. Different increasing codes can assign different distances and therefore produce different means, even though the observed category ordering is unchanged.