Why Membership Structure Matters
A GLMM may involve several sources of grouping, such as schools and classrooms, or patients and clinicians. Before deciding how those sources relate, ask a concrete question: which units belong to which other units? If every classroom belongs to exactly one school, classrooms are nested within schools. If the same items are evaluated by several raters, items and raters are crossed: neither classification is simply a set of units contained in one level of the other.
These are descriptions of the design’s membership structure, not conclusions about which effects should be random. The same classification can be treated differently for different scientific purposes. Correctly distinguishing nesting from crossing identifies which units are shared across groups and which have a single parent. It is an important design description before model notation or covariance structure is considered.
A Membership Test
Start by specifying the units in each classification. For example, a “classroom” is a unit, while “school” is another classification of units. A lower-level unit’s identity must be precise enough to determine whether it remains the same unit across observations. If classroom 2 exists in both School A and School B, those may be two different classrooms with locally reused labels, not one classroom shared across schools.
The nesting test is direct. For each unit in the proposed lower classification, list its parent. If each has exactly one parent, the lower classification is nested within the higher one. Several lower-level units can share the same parent; nesting does not mean one-to-one matching. It means each lower-level unit has one parent. If a unit has no parent or more than one parent, the simple nested relationship does not hold as stated.
For two classifications to be crossed, look for units that occur across levels of the other classification. A rater who evaluates items from several categories, and items evaluated by several raters, do not form a one-parent hierarchy. By contrast, the fact that each observation has both a classroom label and a school label does not by itself make classrooms and schools crossed: if each classroom is in just one school, the classifications are nested.
The membership relation gives a compact justification. Let \(L\) denote the set of lower-level units and \(H\) the set of potential parents. Nesting means there is a function \(p:L\to H\) assigning exactly one parent to every lower-level unit. If one unit \(l\in L\) belongs to two distinct parents \(h_1,h_2\in H\), then a single-valued parent function cannot represent that relationship. Conversely, if every \(l\in L\) has exactly one parent, assigning that parent defines \(p\), which is precisely the nested structure. For crossed classifications, units from both classifications must be represented across levels of the other; there is no one-way parent assignment that captures their relationship.
| Question | Evidence of nesting | Evidence of crossing |
|---|---|---|
| Does each lower-level unit have a unique parent? | Yes; the parent determines its higher-level membership. | No one-parent hierarchy describes the relationship. |
| Are units shared across levels? | A lower-level unit remains within one parent. | Units of each classification occur across levels of the other. |
| Does every possible pair need to be observed? | Not relevant to the unique-parent criterion. | Only for a complete crossing; crossing is about shared structure, not equal cell counts. |
Worked Classifications
Worked Example: Classrooms Nested Within Schools
Suppose a hypothetical district records 71 students from four classrooms. Classroom N1 has 18 students and belongs to North School; N2 has 16 students and also belongs to North School. Classroom S1 has 20 students and belongs to South School; S2 has 17 students and also belongs to South School. Classify classrooms relative to schools.
Step 1: State the units. The lower-level units are the four classrooms, and the potential parents are the two schools. The enrollment total is \(18+16+20+17=71\), consistent with the stated number of students.
Step 2: Check parent membership. N1 and N2 each have North School as their one parent. S1 and S2 each have South School as their one parent. Thus, each classroom has exactly one school parent.
Conclusion. Classrooms are nested within schools. The fact that two classrooms share a school does not make them crossed with schools; both remain distinct lower-level units under the same parent. The students’ counts describe the example, but the classification follows from classroom-to-school membership, not from those counts.
Worked Example: Instruments Crossed With Laboratories
In a hypothetical measurement exercise, three instruments, I1, I2, and I3, are each used once in both Lab A and Lab B. The recorded measurements are: I1 gives 10 and 12 units in Labs A and B; I2 gives 11 and 13 units; I3 gives 9 and 10 units. Classify instrument and laboratory.
Step 1: List the combinations. The six observations are (I1, A), (I1, B), (I2, A), (I2, B), (I3, A), and (I3, B). There are \(3\times2=6\) instrument–laboratory combinations, all observed.
Step 2: Check membership across levels. Each instrument occurs in both laboratories, and each laboratory contains observations from all three instruments. No instrument has one unique laboratory parent, and no laboratory has one unique instrument parent.
Conclusion. Instruments and laboratories are completely crossed in this hypothetical exercise. The measurements themselves are not what establish crossing: the same conclusion follows from the six pairings even if the recorded values were different. Here, all six possible combinations are present, which makes the crossing complete.
Worked Example: Items and Raters
Suppose two hypothetical raters score three short responses on a 0-to-5 scale. Rater A assigns scores 4, 3, and 5 to Items 1, 2, and 3; Rater B assigns scores 3, 4, and 4. Classify items relative to raters, and raters relative to items.
Step 1: Count and identify the units. There are three item units and two rater units. Each item has two scores, one from each rater, and each rater scores three items. The total number of scores is \(3\times2=6\), matching the six listed values.
Step 2: Apply the membership test in both directions. No item belongs to just one rater, because both raters score it. No rater is associated with just one item, because each scores all three. The relationship is not “items nested within raters” or “raters nested within items.”
Conclusion. Items and raters are completely crossed in this example. Although every score is attached to both an item and a rater, the decisive fact is that the same items are shared across raters and the same raters are shared across items.
Reading the Design Carefully
A classification should follow the units in the design, not a shorthand label in a data file. Suppose each school numbers its classrooms 1, 2, and 3. A classroom recorded as “2” in North School is not necessarily the same unit as a classroom recorded as “2” in South School. If the identity is actually School-plus-classroom, those are different classroom units, each with one school parent. Treating the repeated label “2” as one shared classroom would create a false appearance of crossing.
Likewise, a single observed combination does not establish a relationship on its own. The design or unit definitions must tell you whether a unit is genuinely shared across levels or simply observed once. When classifications might change membership over time, define the unit and the time window before deciding. A clinician’s affiliation with a clinic, for instance, may vary across periods; a classification that ignores that change can make a relationship look more hierarchical or more shared than it really is.
The membership test is about classification, not balance. A complete crossing has every possible combination represented, but equal numbers of observations per combination are not part of the definition. Conversely, unequal counts do not turn a nested relationship into a crossed one. For a more complex or incomplete design, describe the actual memberships rather than forcing the words “nested” or “crossed” to imply that all combinations are present.
Why the Distinction Matters
The structural difference is whether units are shared. In a nested arrangement, two classrooms in the same school are distinct classrooms that share a parent. In a crossed arrangement, an item’s observations can be linked to several raters, and a rater’s observations can be linked to several items. That sharing affects how a GLMM must represent the design: treating shared units as if they belonged to separate, unrelated groups would discard information about which observations have the same item or rater.
But the classification alone does not dictate a model. “Crossed” does not mean that both classifications must be random effects, and “nested” does not mean that a particular random-effects specification is automatically appropriate. Whether an effect is fixed or random depends on the scientific question and inferential target. The earlier material on generalized linear models provides relevant background on modeling responses, but the present distinction is specifically about membership among units.
Common Mistakes and Limitations
- Calling any two labels on one observation crossed. An observation can have both a classroom and a school label even when classrooms are nested within schools. Check whether units are shared across levels.
- Reversing nesting. If each classroom has one school parent, classrooms are nested within schools—not schools within classrooms. State the direction explicitly.
- Confusing repeated labels with repeated units. Local labels can be reused. Establish whether the labels identify one shared unit or distinct units before interpreting membership.
- Equating crossing with a balanced table. Complete crossing means all combinations occur; equal counts are not required. Unequal cell sizes do not change the parent relationship.
- Assuming the relationship determines fixed or random status. Nested and crossed describe structure. The choice of fixed or random effects is a separate modeling decision.
The simple definitions also have limits. Some units can have multiple genuine affiliations, or their affiliation can change with time; these cases may not fit a single-parent hierarchy. Some designs contain only a subset of possible combinations. In such settings, document the observed and intended memberships carefully. A clear description of the units is more informative than forcing a complicated structure into a simple label.
Check Your Understanding
For each question, decide what the membership structure implies. Solutions follow each question.
- A hypothetical company has three branches. Each of its six teams belongs to exactly one branch. Are teams nested within branches or crossed with branches?
Solution: Teams are nested within branches. Each team has exactly one branch parent; several teams may share a branch. - Two hypothetical assessors each evaluate the same four reports. Are assessors nested within reports, reports nested within assessors, or crossed?
Solution: The classifications are crossed. Each assessor occurs across all four reports, and each report is evaluated by both assessors. Neither classification has a one-parent relationship to the other. - A data file uses classroom labels 1 and 2 in each of two schools. Each classroom belongs to its own school, but the labels are reused. Should the two classrooms labeled 1 be treated as one classroom shared across schools?
Solution: No. The description says each classroom belongs to its own school, so “classroom 1” in each school identifies a different unit. Each has one parent; classrooms are nested within schools. - Every item is scored by every rater, but one rater has 12 scores and another has 10 because some scores are missing. Does this unequal count alone make the classifications nested?
Solution: No. Unequal observed counts do not create a one-parent relationship. If the design assigns raters across the same items and the missing scores are absent observations, the intended structure remains crossed; the missingness should be described separately.