Why Collapsing Categories Changes the Question
An ordered response can distinguish several degrees or levels of an outcome. Turning it into a binary response may make a specific decision easier to analyze—for example, whether a rating is high enough to trigger follow-up—but it removes distinctions among responses on the same side of the cutoff. The key issue is therefore not simply whether categories can be combined. It is whether the question after combining them is still the scientific question of interest.
Suppose \(Y_{ij}\) is an ordinal response for observation \(j\) in cluster \(i\), with ordered categories \(1,\ldots,K\). A cutoff \(c\), where \(1\leq c<K\), defines the binary response \(Z_{ij}=I(Y_{ij}>c)\). Thus \(Z_{ij}=1\) means the response is above the cutoff, and \(Z_{ij}=0\) means it is at or below it. The cutoff is part of the definition of the outcome, not a neutral data-cleaning choice.
In a GLMM, probabilities may be specified conditional on a cluster-specific random effect \(b_i\) and covariates \(x_{ij}\). Write \(p_{ijk}=P(Y_{ij}=k\mid x_{ij},b_i)\). The probability of crossing the cutoff is then
The binary response model concerns \(q_{ij}(c)\). It does not directly describe the individual probabilities \(p_{ijk}\). A binary coefficient therefore describes change in the chance of being above the chosen cutoff, conditional on the model's covariates and random effects. It does not, by itself, describe how probability is redistributed among all the ordinal categories.
What the Collapsed Likelihood Retains
Assume that, conditional on the random effects and covariates, the ordinal observations have the specified category probabilities and are independent across observations. For one observation, the ordinal likelihood contribution is \(p_{ijk}\) if its observed category is \(k\). Once only \(Z_{ij}\) is retained, the likelihood contribution is instead \(q_{ij}(c)\) for \(Z_{ij}=1\), or \(1-q_{ij}(c)\) for \(Z_{ij}=0\). The binary likelihood records which side of the cutoff occurred, not the category that generated it.
For a group of observations with the same modeled probabilities, let \(n_k\) be the number in category \(k\), \(n=\sum_{k=1}^K n_k\), and \(m=\sum_{k=c+1}^K n_k\). The ordinal likelihood contribution is proportional to \(\prod_{k=1}^K p_k^{n_k}\). After collapsing, the corresponding binary contribution is proportional to \(q^m(1-q)^{n-m}\), where \(q=\sum_{k=c+1}^K p_k\). The binary data retain \(m\) and \(n-m\), but not the separate counts \(n_k\) within either side. For a GLMM, these expressions are conditional on the random effects; the likelihood for the observed data must account for the random-effects distribution rather than treating the effects as known.
This is a many-to-one mapping: different ordinal data sets can produce exactly the same binary data. Consequently, the collapsed response cannot be used to reconstruct the lost category distinctions. The loss concerns the information available in the response and the estimand, not an absolute claim that a binary analysis must have lower power for every possible question. If the scientific target is specifically crossing one threshold, a binary analysis may be directly aligned with that target.
Worked Example: Two Different Distributions, One Binary Summary
Consider hypothetical counts from two groups, each with 100 responses in five ordered categories. Use cutoff \(c=2\), so categories 3, 4, and 5 count as above the cutoff.
| Group | Category 1 | Category 2 | Category 3 | Category 4 | Category 5 | At or below 2 | Above 2 |
|---|---|---|---|---|---|---|---|
| A | 10 | 20 | 30 | 25 | 15 | 30 | 70 |
| B | 25 | 5 | 30 | 25 | 15 | 30 | 70 |
For group A, the above-cutoff proportion is \((30+25+15)/100=0.70\). For group B, it is also \((30+25+15)/100=0.70\). Thus both groups have the same binary summary: 30 responses at or below 2 and 70 above 2.
The ordinal distributions are not the same: group A has 10 responses in category 1 and 20 in category 2, while group B has 25 and 5. The binary summaries cannot show that difference. In likelihood terms, the binary contribution for either group at a modeled crossing probability \(q\) is \(q^{70}(1-q)^{30}\), whereas the ordinal contributions retain separate factors for each category, such as \(p_1^{10}p_2^{20}p_3^{30}p_4^{25}p_5^{15}\) for group A. The table is descriptive; it does not account for clustering or support an inferential comparison by itself.
When Is a Binary Threshold Meaningful?
A cutoff is most defensible when it corresponds to a clearly stated decision or scientific distinction: for example, a rating that triggers additional assessment, a level regarded as adequate, or a symptom severity threshold specified by a protocol. In those cases, “above versus not above the threshold” can be a useful outcome in its own right. The binary analysis should still be described as an analysis of that threshold question, rather than as a complete analysis of the ordinal response.
A threshold is less persuasive when it is selected after inspecting which split produces the strongest association, or when the categories on either side represent substantively different states. For instance, combining “moderate” and “severe” may be inappropriate if those levels lead to different decisions, even if their numerical category labels are adjacent. The category labels, ordering, and intended use—not numerical spacing—determine whether a collapse makes sense.
Worked Example: Translating Category Probabilities Into a Threshold Probability
Suppose a hypothetical ordinal GLMM for a four-category symptom rating gives the following fitted probabilities for a participant with specified covariates and random effect \(b_i=0\): \(P(Y_{ij}=1\mid x_{ij},b_i)=0.10\), \(P(Y_{ij}=2\mid x_{ij},b_i)=0.25\), \(P(Y_{ij}=3\mid x_{ij},b_i)=0.40\), and \(P(Y_{ij}=4\mid x_{ij},b_i)=0.25\). Assume the four categories are exhaustive and mutually exclusive, and these probabilities sum to 1.00.
If a service protocol defines ratings 3 or 4 as requiring follow-up, use \(c=2\). The fitted probability of crossing that threshold is
The resulting binary probability, 0.65, answers the protocol question for the stated covariates and random effect. It does not reveal that the fitted probability of the most severe category is 0.25, or distinguish the 0.40 probability of category 3 from that of category 4. A binary model is reasonable if the follow-up threshold is the target; it is inadequate if the separate severity probabilities are also important.
Deriving the Information Loss
The loss can be shown directly from the likelihood. For one observation with ordinal probabilities \(p_1,\ldots,p_K\), the probability of binary value 1 is the sum of the probabilities of every category mapped to 1. This sum follows because those category events are mutually exclusive:
The same reasoning gives \(P(Z=0)=\sum_{k=1}^{c}p_k=1-q\). Once \(Y\) has been replaced by \(Z\), all assignments of categories within a side of the cutoff yield the same binary observation. For instance, with \(c=2\), an observed 0 is compatible with category 1 or category 2. The binary likelihood adds their probabilities together and no longer identifies which occurred. In clustered data this summation is made conditional on the random effect when specifying \(q_{ij}(c)\); the cluster dependence must still be represented in the model.
List the ordered categories and explain which distinctions matter for the scientific or operational question.
Define \(c\) and state in context what \(Z_{ij}=1\) means. Avoid choosing the threshold solely because it produces a preferred result.
Confirm that crossing the cutoff is meaningful and that the GLMM accounts for the clustering relevant to the observations.
State that category-specific distinctions within each side of the cutoff are not modeled by the binary response.
Diagnostics, Alternatives, and Limitations
Before collapsing, inspect the category counts overall and across important covariates or clusters. Sparse categories can make category-specific estimation difficult, but sparsity alone does not establish that merging is scientifically appropriate. Check whether the proposed threshold has a substantive interpretation, whether enough observations occur on both sides to support the intended comparison, and whether the resulting binary outcome still varies across clusters. If all or nearly all observations in some clusters fall on one side, the binary response may provide little information about the threshold probability in those clusters.
Compare the binary analysis with an ordinal analysis when retaining the category distinctions is important and the ordinal model is appropriate for the response process. A binary GLMM focuses on crossing a selected threshold; an ordinal GLMM retains the ordered categories and their category probabilities under its specified model. Treating the categories as nominal is another possible response representation, but it does not use their ordering in the same way. These approaches answer different questions, so comparison is most useful when the estimands and assumptions are made explicit.
A common mistake is to say, “The response has five categories, but the upper two are similar, so combining them loses no information.” Similarity is not identity: unless the two categories truly represent the same state for the analysis, their distinction is discarded by the merge. The correction is to state the intended threshold question, justify why the categories are combined for that question, and acknowledge what the binary result cannot describe. A further limitation is that conclusions can depend on the cutoff; sensitivity analyses using scientifically plausible thresholds can reveal that dependence, but should not be used to select a favorable result after the fact.
Worked Example: Interpreting a Binary GLMM Effect at Its Actual Target
Suppose a hypothetical binary GLMM estimates that, at a fixed random effect and fixed covariates, the probability of a rating above a prespecified threshold is 0.30 for one exposure level and 0.50 for another. The threshold is the event of interest, and the model assumptions include a suitable conditional binary mean model and an adequate representation of within-cluster dependence.
The conditional odds at the first exposure level are \(0.30/(1-0.30)=0.30/0.70=0.4286\). At the second level they are \(0.50/(1-0.50)=1.000\). The conditional odds ratio is therefore \(1.000/0.4286=2.333\). In context, at the stated covariates and random effect, the estimated odds of being above the threshold are 2.333 times as high at the second exposure level as at the first.
This interpretation concerns the binary threshold event only. It does not establish that every ordinal category becomes more likely, nor does it identify how the probabilities of categories on either side changed. It is not a causal interpretation unless the study design and assumptions justify one.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For an ordinal response with categories 1 through 5 and cutoff \(c=3\), which categories map to binary value 1?
- If the category probabilities are \((0.15,0.20,0.25,0.25,0.15)\), calculate the probability of \(Z=1\) at cutoff \(c=2\).
- Two groups have the same number of responses above a cutoff but different category counts within the lower group. What information does the binary response discard?
- Why is a cutoff selected after trying many possible splits a concern?
- A binary GLMM estimates a higher probability of exceeding a severity threshold for one covariate value. What does that result not establish about the separate ordinal categories?
Solutions. (1) Categories 4 and 5 map to 1 because \(Z=I(Y>3)\). (2) \(P(Z=1)=0.25+0.25+0.15=0.65\). (3) It discards which category occurred within the lower group, even though the binary totals agree. (4) Selecting the split based on results makes the target data-dependent and can favor a seemingly strong association; a scientifically justified cutoff should be specified independently of that search. (5) It does not establish how probability changes among individual categories, or that every category moves in the same direction.