Disaggregating level-specific effects in cross-classified multilevel models

Abstract

In psychology and other fields, data often have a cross-classified structure, whereby observations are nested within multiple types of non-hierarchical clusters (e.g., repeated measures cross-classified by persons and stimuli). This paper discusses ways that, in cross-classified multilevel models, slopes of lower-level predictors can implicitly reflect an ambiguous blend of multiple effects (for instance, a purely observation-level effect as well as a unique between-cluster effect for each type of cluster). The possibility of conflating multiple effects of lower-level predictors is well recognized for non-cross-classified multilevel models, but has not been fully discussed or clarified for cross-classified contexts. Consequently, in published cross-classified modeling applications, this possibility is almost always ignored, and researchers routinely specify models that conflate multiple effects. In this paper, we show why this common practice can be problematic, and show how to disaggregate level-specific effects in cross-classified models. We provide a novel suite of options that include fully cluster-mean-centered, partially cluster-mean-centered, and contextual effect models, each of which provides a unique interpretation of model parameters. We further clarify how to avoid both fixed and random conflation, the latter of which is widely misunderstood even in non-cross-classified models. We provide simulation results showing the possible deleterious impact of such conflation in cross-classified models, and walk through pedagogical examples to illustrate the disaggregation of level-specific effects. We conclude by considering additional model complexities that can arise with cross-classification, providing guidance for researchers in choosing among model specifications, and describing newly available software to aid researchers who wish to disaggregate effects in practice.

Appendices

Appendix A: Interpretation of coefficients in fixed-slope fully cluster-mean-centered, partially cluster-mean-centered, and contextual effect cross-classified models

In Appendix A, we provide mathematical derivations to show how to interpret the fixed coefficients in the cross-classified model specifications listed in Table 1, and, in particular, show which coefficients are unconflated, which are conflated, and, of the latter, the effects of which they are implicitly a weighted average. This can be determined by re-expressing each model as a regression of ${y}_{ijk}$ on ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$, ${x}_{\cdot j\cdot }$, and ${x}_{\cdot \cdot k}$. When writing the model this way, the slopes represent the model-implied within-cluster, between-J-cluster, and between-K-cluster effect, respectively.

Uncentered x (or centered-by-a-constant x) models

The first uncentered x model noted in Table 1 includes only ${x}_{ijk}$ and yields a slope that is a conflated blend of ${\gamma }_{w}$, ${\gamma }_{b(J)}$, and ${\gamma }_{b(K)}$; this was shown in Eq. (6) by re-expressing the model as a regression of ${y}_{ijk}$ on ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$, ${x}_{\cdot j\cdot }$, and ${x}_{\cdot \cdot k}$, and clarifying that the model-implied within-cluster, between-J-cluster, and between-K-cluster effects were equivalent. Following a similar procedure, we can consider the model in row 2 of Table 1 that includes as predictors both ${x}_{ijk}$ and ${x}_{\cdot j\cdot }$:

$$\begin{array}{l}{y}_{ijk}={\gamma }_{00}+{\gamma }_{c(K)}{x}_{ijk}+{\gamma }_{bcc(J)}{x}_{\cdot j\cdot }+{u}_{0j}+{u}_{0k}+{e}_{ijk}\\ \quad\ \ ={\gamma }_{00}+{\gamma }_{c(K)}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{\gamma }_{c(K)}({x}_{\cdot j\cdot }+{x}_{\cdot \cdot k})+{\gamma }_{bcc(J)}{x}_{\cdot j\cdot }+{u}_{0j}+{u}_{0k}+{e}_{ijk}\\ \quad\,\,\,\,\,={\gamma }_{00}+{\gamma }_{c(K)}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({\gamma }_{bcc(J)}+{\gamma }_{c(K)}){x}_{\cdot j\cdot }+{\gamma }_{c(K)}{x}_{\cdot \cdot k}+{u}_{0j}+{u}_{0k}+{e}_{ijk}\end{array}$$

(23)

This re-expression in Eq. 23) shows that the within-cluster effect is constrained equal to the between-K-cluster effect, as both are given as ${\gamma }_{c(K)}$ (i.e., the re-expressed/model-implied slope of ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$ = the re-expressed/model-implied slope of ${x}_{\cdot \cdot k}$ = ${\gamma }_{c(K)}$). Hence, this ${\gamma }_{c(K)}$ coefficient is a conflated blend/weighted average of the underlying ${\gamma }_{w}$ and ${\gamma }_{b(K)}$ defined in the general unconflated model in Eq. (4). The between-J-cluster effect is given as ${\gamma }_{bcc(J)}+{\gamma }_{c(K)}$. Note that, since there is no implicit constraint on the directly estimated slope of ${x}_{\cdot j\cdot }$ (i.e., ${\gamma }_{bcc(J)}$) in this model, the between-cluster effect is appropriately recovered, i.e., ${\gamma }_{bcc(J)}+{\gamma }_{c(K)}={\gamma }_{b(J)}$. However, ${\gamma }_{bcc(J)}$ is itself of questionable utility, as it doesn’t represent the true underlying contextual effect for J clusters (i.e., the difference in the between-J-cluster effect and the within-cluster effect, ${\gamma }_{b(J)}-{\gamma }_{w}$), and instead is a conflated contextual effect, representing the difference in the between-J-cluster effect and the conflated effect ${\gamma }_{c(K)}$. The degree to which ${\gamma }_{bcc(J)}$ differs from the true contextual effect, ${\gamma }_{bc(J)}$, is equal to the difference in ${\gamma }_{w}$ and ${\gamma }_{c(K)}$, as shown here:

$$\begin{array}{l}{\gamma }_{bcc(J)}={\gamma }_{b(J)}-{\gamma }_{c(K)}\\ \qquad\ \ ={\gamma }_{bc(J)}+({\gamma }_{w}-{\gamma }_{c(K)})\end{array}$$

(24)

These same general ideas apply to the model in row 3 of Table 1 that instead includes slopes of ${x}_{ijk}$ and ${x}_{\cdot \cdot k}$, which can be re-expressed as:

$$\begin{array}{l}{y}_{ijk}={\gamma }_{00}+{\gamma }_{c(J)}{x}_{ijk}+{\gamma }_{bcc(K)}{x}_{\cdot \cdot k}+{u}_{0j}+{u}_{0k}+{e}_{ijk}\\\quad\ \ ={\gamma }_{00}+{\gamma }_{c(J)}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{\gamma }_{c(J)}({x}_{\cdot j\cdot }+ x _{\cdot\cdot k})+ {\gamma }_{bcc(K)} x_{\cdot\cdot k} +{u}_{0j}+{u}_{0k}+{e}_{ijk}\\ \quad\,\,\,\,={\gamma }_{00}+{\gamma }_{c(J)}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{\gamma }_{c(J)}{x}_{\cdot j\cdot }+({\gamma }_{bcc(K)}+{\gamma }_{c(J)}){x}_{\cdot \cdot k}+{u}_{0j}+{u}_{0k}+{e}_{ijk}\end{array}$$

(25)

Here, ${\gamma }_{c(J)}$ is a conflated blend of ${\gamma }_{w}$ and ${\gamma }_{b(J)}$, and ${\gamma }_{bcc(K)}$ is a conflated contextual effect, i.e., ${\gamma }_{bcc(K)}={\gamma }_{b(K)}-{\gamma }_{c(J)}$, which can also be written as ${\gamma }_{bcc(K)}={\gamma }_{bc(K)}+({\gamma }_{w}-{\gamma }_{c(J)})$.

The last uncentered x model in Table 1, i.e., the contextual effect model with both cluster means, can be re-expressed as:

$$\begin{array}{l}{y}_{ijk}={\gamma }_{00}+{\gamma }_{w}{x}_{ijk}+{\gamma }_{bc(J)}{x}_{\cdot j\cdot }+{\gamma }_{bc(K)}{x}_{\cdot \cdot k}+{u}_{0j}+{u}_{0k}+{e}_{ijk}\\ \quad\ \ ={\gamma }_{00}+{\gamma }_{w}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{\gamma }_{w}({x}_{\cdot j\cdot }+{x}_{\cdot \cdot k})+{\gamma }_{bc(J)}{x}_{\cdot j\cdot }+{\gamma }_{bc(K)}{x}_{\cdot \cdot k}+{u}_{0j}+{u}_{0k}+{e}_{ijk}\\ \quad\,\,\,\,={\gamma }_{00}+{\gamma }_{w}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({\gamma }_{bc(J)}+{\gamma }_{w}){x}_{\cdot j\cdot }+({\gamma }_{bc(K)}+{\gamma }_{w}){x}_{\cdot \cdot k}+{u}_{0j}+{u}_{0k}+{e}_{ijk}\end{array}$$

(26)

In this model, there are no implicit constraints placed across the level-specific effects, and as such, it accurately recovers each of these effects and is likelihood-equivalent to the fully cluster-mean-centered model presented in Eq. (4). This model is a reparameterization of the fully cluster-mean-centered model in which ${\gamma }_{bc(J)}+{\gamma }_{w}$ and ${\gamma }_{bc(K)}+{\gamma }_{w}$ are equal to ${\gamma }_{b(J)}$ and ${\gamma }_{b(K)}$, respectively, and thus ${\gamma }_{bc(J)}$ is the unconflated contextual effect for J clusters (${\gamma }_{bc(J)}={\gamma }_{b(J)}-{\gamma }_{w}$) and ${\gamma }_{bc(K)}$ is the unconflated contextual effect for K clusters (${\gamma }_{bc(K)}={\gamma }_{b(K)}-{\gamma }_{w}$).

Partially cluster-mean-centered models

In rows 5–8 of Table 1, we present models with predictor ${x}_{ijk}$ cluster-mean-centered by only the J-cluster means (${x}_{\cdot j\cdot }$ s). The first such partially cluster-mean-centered model in row 5 of Table 1 includes ${x}_{ijk}-{x}_{\cdot j\cdot }$ as the only predictor, which can be re-expressed as:

$$\begin{array}{l}{\gamma }_{ijk}={\gamma }_{00}+{\gamma }_{c(K)}({x}_{ijk}-{x}_{\cdot j\cdot })+{u}_{0j}+{u}_{0k}+{e}_{ijk}\\ \quad\ \ ={\gamma }_{00}+{\gamma }_{c(K)}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{\gamma }_{c(K)}{x}_{\cdot \cdot k}+{u}_{0j}+{u}_{0k}+{e}_{ijk}\end{array}$$

(27)

In this model, the slope of ${x}_{ijk}-{x}_{\cdot j\cdot }$, ${\gamma }_{c(K)}$, is a conflated blend of the underlying ${\gamma }_{w}$ and ${\gamma }_{b(K)}$, given that the within-effect and between-K-cluster effect are constrained to be the equivalent to ${\gamma }_{c(K)}$, whereas the between-J-cluster effect is implicitly constrained to be 0.

In row 6 of Table 1, both ${x}_{ijk}-{x}_{\cdot j\cdot }$ and ${x}_{\cdot j\cdot }$ are included as predictors in the model, which can be re-expressed as:

$$\begin{array}{l}{\gamma }_{ijk}={\gamma }_{00}+{\gamma }_{c(K)}({x}_{ijk}-{x}_{\cdot j\cdot })+{\gamma }_{b(J)}{x}_{\cdot j\cdot }+{u}_{0j}+{u}_{0k}+{e}_{ijk}\\ \quad\ \ ={\gamma }_{00}+{\gamma }_{c(K)}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{\gamma }_{c(K)}{x}_{\cdot \cdot k}+{\gamma }_{b(J)}{x}_{\cdot j\cdot }+{u}_{0j}+{u}_{0k}+{e}_{ijk}\end{array}$$

(28)

Here, ${\gamma }_{c(K)}$ is still a conflated blend of the underlying ${\gamma }_{w}$ and ${\gamma }_{b(K)}$, as the within-effect and between-K-cluster effect are still constrained to be equivalent. The slope of the J-cluster mean, ${x}_{\cdot j\cdot }$, accurately represents the between-J-cluster effect, ${\gamma }_{b(J)}$.

Including K-cluster mean (${x}_{\cdot \cdot k}$) instead of J-cluster mean (${x}_{\cdot j\cdot }$) yields the model in row 7 of Table 1, which can be re-expressed as:

$$\begin{array}{l}{\gamma }_{ijk}={\gamma }_{00}+{\gamma }_{w}({x}_{ijk}-{x}_{\cdot j\cdot })+{\gamma }_{bc(K)}{x}_{\cdot \cdot k}+{u}_{0j}+{u}_{0k}+{e}_{ijk}\\ \quad \ \ ={\gamma }_{00}+{\gamma }_{w}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{\gamma }_{w}{x}_{\cdot \cdot k}+{\gamma }_{bc(K)}{x}_{\cdot \cdot k}+{u}_{0j}+{u}_{0k}+{e}_{ijk}\\ \quad\,\,\,\, ={\gamma }_{00}+{\gamma }_{w}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({\gamma }_{w}+{\gamma }_{bc(K)}){x}_{\cdot \cdot k}+{u}_{0j}+{u}_{0k}+{e}_{ijk}\end{array}$$

(29)

This model does not place implicit constraints on the within-cluster effect nor the between-K-cluster effect, and the former is represented by the slope of ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$ and the latter is equal to ${\gamma }_{bc(K)}+{\gamma }_{w}$, where ${\gamma }_{bc(K)}$ is the K-cluster contextual effect. The between-J-cluster effect is implicitly constrained to be zero.

The last partially cluster-mean-centered model in Table 1 includes x cluster-mean-centered by J clusters (i.e., ${x}_{ijk}-{x}_{\cdot j\cdot }$) and the two cluster means as predictors, which can be re-expressed as:

$$\begin{array}{l}{\gamma }_{ijk}={\gamma }_{00}+{\gamma }_{w}({x}_{ijk}-{x}_{\cdot j\cdot })+{\gamma }_{b(J)}{x}_{\cdot j\cdot }+{\gamma }_{bc(K)}{x}_{\cdot \cdot k}+{u}_{0j}+{u}_{0k}+{e}_{ijk}\\ \quad \ \ ={\gamma }_{00}+{\gamma }_{w}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{\gamma }_{w}{x}_{\cdot \cdot k}+{\gamma }_{b(J)}{x}_{\cdot j\cdot }+{\gamma }_{bc(K)}{x}_{\cdot \cdot k}+{u}_{0j}+{u}_{0k}+{e}_{ijk}\\ \quad \,\,\,\, ={\gamma }_{00}+{\gamma }_{w}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{\gamma }_{b(J)}{x}_{\cdot j\cdot }+({\gamma }_{w}+{\gamma }_{bc(K)}){x}_{\cdot \cdot k}+{u}_{0j}+{u}_{0k}+{e}_{ijk}\end{array}$$

(30)

This model places no implicit constraints across level-specific effects and is a reparameterization of the fully cluster-mean-centered model represented in Eq. (4). The slope of ${x}_{ijk}-{x}_{\cdot j\cdot }$, ${x}_{\cdot j\cdot }$, and ${x}_{\cdot \cdot k}$ denote the within-cluster effect, between-J-cluster effect, and K-cluster contextual effect, respectively.

Note that, in Table 1, we do not include models that are cluster-mean-centered by K-cluster means, as these would yield the same analytic results as those shown for the cluster-mean-centered by J-cluster means models (just switching all K to J and all k to j).

Fully cluster-mean-centered models

Fully cluster-mean-centered models include the purely within-cluster component of x, ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$, as the level 1 predictor. As such, there are no constraints placed across the within-cluster, between-J-cluster, and between-K-cluster effects, as shown in rows 9–12 of Table 1. The first fully cluster-mean-centered model depicted in row 9 of Table 1 simply includes ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$ as the only predictor:

$${y}_{ijk}={\gamma }_{00}+{\gamma }_{w}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{u}_{0j}+{u}_{0k}+{e}_{ijk}$$

(31)

Here the within-cluster effect is represented by the slope of ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$, and both between-cluster effects are implicitly constrained to be zero. Adding the J-cluster mean of x to Eq. (31) yields the model in row 10 of Table 1:

$${\gamma }_{ijk}={\gamma }_{00}+{\gamma }_{w}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{\gamma }_{b(J)}{x}_{\cdot j\cdot }+{u}_{0j}+{u}_{0k}+{e}_{ijk}$$

(32)

This allows for a non-zero between-J-cluster effect, given as the slope of ${x}_{\cdot j\cdot }$. Similarly, adding instead the K-cluster mean of x yields the model in row 11 of Table 1:

$${\gamma }_{ijk}={\gamma }_{00}+{\gamma }_{w}({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{\gamma }_{b(K)}{x}_{\cdot \cdot k}+{u}_{0j}+{u}_{0k}+{e}_{ijk}$$

(33)

This allows for a non-zero between-K-cluster effect, given as the slope of ${x}_{\cdot \cdot k}$. Last, including both cluster means yields the model presented in Eq. (4), which can directly estimate each of the three possible effects of x.

Appendix B Interpretation of random slopes in fully cluster-mean-centered, partially cluster-mean-centered, and contextual effect cross-classified models

In Appendix B, we provide mathematical derivations to show how to interpret the random slopes in various cross-classified model specifications, similar to what was shown in Appendix A for fixed-slope models. Here, by re-expressing the random portion of the model as a regression of ${y}_{ijk}$ on ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$, ${x}_{\cdot j\cdot }$, and ${x}_{\cdot \cdot k}$, we can see which random components are unconflated and which are conflated.

We first define what can be termed the homoscedastic maximal random effect structure that includes J-cluster and K-cluster random effects for ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$, a K-cluster random effect for ${x}_{\cdot j\cdot }$, and a J-cluster random effect for ${x}_{\cdot \cdot k}$. The random effect portion of the model (i.e., the cluster-level error terms) are thus given as:

$$\begin{array}{c}RE={u}_{0j}+{u}_{0k}+({u}_{wj}+{u}_{wk})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{u}_{bk}{x}_{\cdot j\cdot }+{u}_{bj}{x}_{\cdot \cdot k}\\ \left[\begin{array}{c}{u}_{0j}\\ {u}_{wj}\\ {u}_{bj}\\ {u}_{0k}\\ {u}_{wk}\\ {u}_{bk}\end{array}\right]\sim MVN\left(\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{cccccc}{\tau }_{00(J)}& & & & & \\ {\tau }_{0w(J)}& {\tau }_{ww(J)}& & & & \\ {\tau }_{0b(J)}& {\tau }_{wb(J)}& {\tau }_{bb(J)}& & & \\ 0& 0& 0& {\tau }_{00(K)}& & \\ 0& 0& 0& {\tau }_{0w(K)}& {\tau }_{ww(K)}& \\ 0& 0& 0& {\tau }_{0b(K)}& {\tau }_{wb(K)}& {\tau }_{bb(K)}\end{array}\right]\right)\end{array}$$

(34)

Here we use ${u}_{w}$ terms to represent cluster-specific deviations in the effect of the fully cluster-mean-centered portion of x, and ${u}_{b}$ terms to represent cluster-specific deviations in the effect of the cluster-mean portion of x, noting that the effect of the J-cluster mean can vary across K clusters, and vice versa.

We also define the heteroscedastic maximal random effect structure that adds a J-cluster random effect for ${x}_{\cdot j\cdot }$ and a K-cluster random effect for ${x}_{\cdot \cdot k}$, given as:

$$\begin{array}{c}RE={u}_{0j}+{u}_{0k}+({u}_{wj}+{u}_{wk})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({u}_{bk}+{u}_{hj}){x}_{\cdot j\cdot }+({u}_{bj}+{u}_{hk}){x}_{\cdot \cdot k}\\ \left[\begin{array}{c}{u}_{0j}\\ {u}_{wj}\\ {u}_{bj}\\ {u}_{hj}\\ {u}_{0k}\\ {u}_{wk}\\ {u}_{bk}\\ {u}_{hk}\end{array}\right]\sim MVN\left(\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{cccccccc}{\tau }_{00(J)}& & & & & & & \\ {\tau }_{0w(J)}& {\tau }_{ww(J)}& & & & & & \\ {\tau }_{0b(J)}& {\tau }_{wb(J)}& {\tau }_{bb(J)}& & & & & \\ {\tau }_{0h(J)}& {\tau }_{wh(J)}& {\tau }_{bh(J)}& {\tau }_{hh(J)}& & & & \\ 0& 0& 0& 0& {\tau }_{00(K)}& & & \\ 0& 0& 0& 0& {\tau }_{0w(K)}& {\tau }_{ww(K)}& & \\ 0& 0& 0& 0& {\tau }_{0b(K)}& {\tau }_{wb(K)}& {\tau }_{bb(K)}& \\ 0& 0& 0& 0& {\tau }_{0h(K)}& {\tau }_{wh(K)}& {\tau }_{bh(K)}& {\tau }_{hh(K)}\end{array}\right]\right)\end{array}$$

(35)

Note that this specification assumes that there is heteroscedasticity at level 2, in that the variance of cluster means of y are allowed to follow a quadratic function of the cluster means of x. Specifically:

$$\begin{array}{l}\mathrm{var}({y}_{\cdot j\cdot }|{x}_{\cdot j\cdot })=\mathrm{var}({u}_{0j}+{u}_{hj}{x}_{\cdot j\cdot }|{x}_{\cdot j\cdot })\\ \qquad \qquad \quad\,\,={\tau }_{00(J)}+{x}_{\cdot j\cdot }{\tau }_{0h(J)}+{x}_{\cdot j\cdot }^{2}{\tau }_{hh(J)}\end{array}$$

(36)

and

$$\begin{array}{l}\mathrm{var}({y}_{\cdot \cdot k}|{x}_{\cdot \cdot k})=\mathrm{var}({u}_{0k}+{u}_{hk}{x}_{\cdot \cdot k}|{x}_{\cdot \cdot k})\\ \qquad \qquad\quad\,\,\,\,={\tau }_{00(K)}+{x}_{\cdot \cdot k}{\tau }_{0h(K)}+{x}_{\cdot\cdot k }^{2}{\tau }_{hh(K)}\end{array}$$

(37)

(note that, in both Eqs. (36) and (37), every term in the model other than the two random effect terms would be constant values and hence can be excluded from the variance expression). Thus, the random effects of ${u}_{hj}$ and ${u}_{hk}$¹⁶ (where the “h” stands for “heteroscedasticity”) can be seen as mechanisms for modeling heteroscedasticity at level 2, and they thus have a very different interpretation than a typical random effect (see Rights & Sterba, 2023, for more details in a hierarchical context). Such random effect terms are highly atypical, even in hierarchical contexts, but it is necessary to consider this specification to explicate some of the issues of random conflation in certain models, as certain random effect structures are constrained versions of this heteroscedastic structure.

Uncentered x (or centered-by-a-constant x) models

We start with the most basic and standard random slope specification, that is, the one in which there is a J-cluster and K-cluster random effect for the uncentered x:

$$\begin{array}{l}RE={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k}){x}_{ijk}\\ \quad \,\,\,\,={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({u}_{1j}+{u}_{1k})({x}_{\cdot j\cdot }+{x}_{\cdot \cdot k})\\ \quad \,\,\,={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({u}_{1j}+{u}_{1k}){x}_{\cdot j\cdot }+({u}_{1j}+{u}_{1k}){x}_{\cdot \cdot k}\end{array}$$

(38)

Comparing this re-expression to the more general heteroscedastic maximal specification in Eq. (35) clarifies that the former is a constrained version of the latter, and makes the implicit constraints that each possible J-cluster random slope term is exactly equal (i.e., ${u}_{wj}={u}_{bj}={u}_{hj}$, each represented by ${u}_{1j}$) and that each possible K-cluster random slope term is exactly equal (i.e., ${u}_{wk}={u}_{bk}={u}_{hk}$, each represented by ${u}_{1k}$). In terms of constraints on the actual model parameters (noting that the error terms are not themselves parameters), these are more apparent when considering the random effect covariance matrix of this uncentered x model:

$$\left[\begin{array}{c}{u}_{0j}\\ {u}_{1j}\\ {u}_{0k}\\ {u}_{1k}\end{array}\right]\sim MVN\left(\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{cccc}{\tau }_{00(J)}& & & \\ {\tau }_{01(J)}& {\tau }_{11(J)}& & \\ 0& 0& {\tau }_{00(K)}& \\ 0& 0& {\tau }_{01(K)}& {\tau }_{11(K)}\end{array}\right]\right)$$

(39)

Here we can see that the variance of each of the three possible J-cluster random slope terms are given by the same parameter (${\tau }_{11(J)}$), as is the variance of each of the K-cluster random slope terms (${\tau }_{11(K)}$) and hence this model places the four model constraints of $\mathrm{var}({u}_{wj})=\mathrm{var}({u}_{bj})$, $\mathrm{var}({u}_{wj})=\mathrm{var}({u}_{hj})$, $\mathrm{var}({u}_{wk})=\mathrm{var}({u}_{bk})$, and $\mathrm{var}({u}_{wk})=\mathrm{var}({u}_{hk})$. It also assumes that the three pairwise combinations of the J-cluster terms have correlation of 1, as do the three pairwise combinations of K-cluster terms, and hence the model places six additional constraints that each of ${\mathrm{corr}}({u}_{wj},{u}_{bj})$, ${\mathrm{corr}}({u}_{wj},{u}_{hj})$, ${\mathrm{corr}}({u}_{bj},{u}_{hj})$, ${\mathrm{corr}}({u}_{wk},{u}_{bk})$, ${\mathrm{corr}}({u}_{wk},{u}_{hk})$, and ${\mathrm{corr}}({u}_{bk},{u}_{hk})$ are equal to 1 (and since all error terms by definition have the same mean of 0, these constraints hold if and only if ${u}_{wj}={u}_{bj}={u}_{hj}$ and ${u}_{wk}={u}_{bk}={u}_{hk}$). Last, as there is only one intercept-slope covariance term for J clusters and one for K clusters in Eq. 39, this uncentered x model also places the four additional constraints that ${\mathrm{corr}}({u}_{0j},{u}_{wj})={\mathrm{corr}}({u}_{0j},{u}_{bj})$, ${\mathrm{corr}}({u}_{0j},{u}_{wj})={\mathrm{corr}}({u}_{0j},{u}_{hj})$, ${\mathrm{corr}}({u}_{0k},{u}_{wk})={\mathrm{corr}}({u}_{0k},{u}_{bk})$, and ${\mathrm{corr}}({u}_{0k},{u}_{wk})={\mathrm{corr}}({u}_{0k},{u}_{hk})$. Thus, there are 14 constraints in total that, when placed on the general heteroscedastic random effect structure in Eq. (35), yield the uncentered x random effect specification in Eqs. (38-39).

In terms of interpreting the random portion of this uncentered x model, the random slope term for each cluster type is some weighted average of the ${u}_{w}$, ${u}_{b}$, and ${u}_{h}$ terms, and hence implicitly reflects some combination of the across-cluster slope variability in the within-cluster effect and in the between-cluster effect of x, as well as possible heteroscedasticity at level 2. That is, the ${\tau }_{11(J)}$ term defines the heterogeneity in the effect of both ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$ and ${x}_{\cdot \cdot k}$ and is also involved in the model-implied variance of the cluster means of y conditional on the cluster means of x, i.e.,

$$\begin{array}{l}\mathrm{var}({y}_{\cdot j\cdot }|{x}_{\cdot j\cdot })=\mathrm{var}({u}_{0j}+{u}_{1j}{x}_{\cdot j\cdot }|{x}_{\cdot j\cdot })\\ \qquad \qquad \quad \,\,\, ={\tau }_{00(J)}+{x}_{\cdot j\cdot }{\tau }_{01(J)}+{x}_{\cdot j\cdot }^{2}{\tau }_{11(J)}\end{array}$$

(40)

Similarly, the ${\tau }_{11(K)}$ term defines the heterogeneity in the effect of both ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$ and ${x}_{\cdot j\cdot }$ and is also involved in the model-implied variance of the cluster means of y conditional on the cluster means of x, i.e.,

$$\begin{array}{l}\mathrm{var}({y}_{\cdot \cdot k}|{x}_{\cdot \cdot k})=\mathrm{var}({u}_{0k}+{u}_{1k}{x}_{\cdot \cdot k}|{x}_{\cdot \cdot k})\\ \qquad \qquad \quad \,\,\,\,={\tau }_{00(K)}+{x}_{\cdot \cdot k}{\tau }_{01(K)}+{x}_{\cdot \cdot k}^{2}{\tau }_{11(K)}\end{array}$$

(41)

We next consider an uncentered x random slope specification that additionally adds J-cluster random slopes for ${x}_{\cdot \cdot k}$ and K-cluster random slopes for ${x}_{\cdot j\cdot }$, and is thus given as:

$$\begin{array}{l}RE={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k}){x}_{ijk}+{u}_{2k}{x}_{\cdot j\cdot }+{u}_{2j}{x}_{\cdot \cdot k}\\ \quad\ \ ={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({u}_{1j}+{u}_{1k})({x}_{\cdot j\cdot }+{x}_{\cdot \cdot k})+{u}_{2k}{x}_{\cdot j\cdot }+{u}_{2j}{x}_{\cdot \cdot k}\\ \quad \ \ ={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({u}_{1j}+{u}_{1k}+{u}_{2k}){x}_{\cdot j\cdot }+({u}_{1j}+{u}_{2j}+{u}_{1k}){x}_{\cdot \cdot k}\end{array}$$

(42)

This re-expression clarifies what each random slope term represents in relation to the more general heteroscedastic random effect specification, that is:

$$\begin{array}{l}{u}_{wj}={u}_{1j}\\ {u}_{bj}\ ={u}_{1j}+{u}_{2j}\\ {u}_{hj}\,={u}_{1j}\\ {u}_{wk}={u}_{1k}\\ {u}_{bk}\,={u}_{1k}+{u}_{2k}\\ {u}_{hk}\,={u}_{1k} \end{array}$$

(43)

Hence, this model still places the constraint that the ${u}_{w}$ terms are equal to the corresponding ${u}_{h}$ terms, each given by ${u}_{1}$. However, because there is no implicit constraint that involves the random slope terms for ${x}_{\cdot j\cdot }$ and ${x}_{\cdot \cdot k}$ (${u}_{2k}$ and ${u}_{2j}$, respectively), the ${u}_{b}$ terms are adequately recovered in this model. The ${u}_{2k}$ term then represents a sort of conflated “contextual effect” error term in that it denotes the K-cluster-specific difference in the underlying ${u}_{bk}$ and the (conflated) ${u}_{1k}$ term (i.e., ${u}_{2k}={u}_{bk}-{u}_{1k}$) and likewise the ${u}_{2j}$ denotes the cluster-J-specific difference in the (conflated) ${u}_{1j}$ term and the underlying ${u}_{bj}$ (i.e., ${u}_{2j}={u}_{bj}-{u}_{1j}$).

In terms of the specific constraints on the model parameters, we can consider the random effect covariance matrix of this model:

$$\left[\begin{array}{c}{u}_{0j}\\ {u}_{1j}\\ {u}_{2j}\\ {u}_{0k}\\ {u}_{1k}\\ {u}_{2k}\end{array}\right]\sim MVN\left(\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{cccccc}{\tau }_{00(J)}& & & & & \\ {\tau }_{01(J)}& {\tau }_{11(J)}& & & & \\ {\tau }_{02(J)}& {\tau }_{12(J)}& {\tau }_{22(J)}& & & \\ 0& 0& 0& {\tau }_{00(K)}& & \\ 0& 0& 0& {\tau }_{01(K)}& {\tau }_{11(K)}& \\ 0& 0& 0& {\tau }_{02(K)}& {\tau }_{12(K)}& {\tau }_{22(K)}\end{array}\right]\right)$$

(44)

This model places eight constraints total on the general heteroscedastic specification in Eq. (35), namely $\mathrm{var}({u}_{wj})=\mathrm{var}({u}_{hj})$, $\mathrm{var}({u}_{wk})=\mathrm{var}({u}_{hk})$, ${\mathrm{corr}}({u}_{wj},{u}_{hj})=1$, ${\mathrm{corr}}({u}_{wk},{u}_{hk})=1$, ${\mathrm{corr}}({u}_{0j},{u}_{wj})={\mathrm{corr}}({u}_{0j},{u}_{hj})$, ${\mathrm{corr}}({u}_{0k},{u}_{wk})={\mathrm{corr}}({u}_{0k},{u}_{hk})$, ${\mathrm{corr}}({u}_{bj},{u}_{wj})={\mathrm{corr}}({u}_{bj},{u}_{hj})$, and ${\mathrm{corr}}({u}_{bk},{u}_{wk})={\mathrm{corr}}({u}_{bk},{u}_{hk})$. In interpreting the random effect variances, the random slope term for ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$ for each cluster type represents some weighted average of the ${u}_{w}$ and ${u}_{h}$ terms and hence the ${\tau }_{11}$ variances implicitly reflect some combination of the across-cluster slope variability in the within-cluster effect of x as well as possible heteroscedasticity at level 2 (noting that the model-implied variances of the cluster means of y conditional on the cluster means of x are equal to the expressions given in Eqs. 40 and 41). The random slope variance of the cluster means (${\tau }_{22(J)}$ and ${\tau }_{22(K)}$) reflect variability in the conflated contextual effect terms (${u}_{2j}$ and ${u}_{2k}$) defined above.

Last, we consider the uncentered x model that adds to the expression in Eq. (42) a J-cluster random slope term for ${x}_{\cdot j\cdot }$ and a K-cluster random slope term for ${x}_{\cdot \cdot k}$:

$$\begin{array}{l}RE={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k}){x}_{ijk}+({u}_{2k}+{u}_{3j}){x}_{\cdot j\cdot }+({u}_{2j}+{u}_{3k}){x}_{\cdot \cdot k}\\ \quad\ \ ={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({u}_{1j}+{u}_{1k})({x}_{\cdot j\cdot }+{x}_{\cdot \cdot k})+({u}_{2k}+{u}_{3j}){x}_{\cdot j\cdot }+({u}_{2j}+{u}_{3k}){x}_{\cdot \cdot k}\\ \quad\,\,\,\, ={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({u}_{1j}+{u}_{3j}+{u}_{1k}+{u}_{2k}){x}_{\cdot j\cdot }+({u}_{1j}+{u}_{2j}+{u}_{1k}+{u}_{3k}){x}_{\cdot \cdot k}\end{array}$$

(45)

This re-expression clarifies what each random slope term represents in relation to the more general heteroscedastic random effect specification, that is:

$$\begin{array}{l}{u}_{wj}={u}_{1j}\\ {u}_{bj}\, ={u}_{1j}+{u}_{2j}\\ {u}_{hj}\, ={u}_{1j}+{u}_{3j}\\ {u}_{wk}\,={u}_{1k}\\ {u}_{bk}\, ={u}_{1k}+{u}_{2k}\\ {u}_{hk}\,={u}_{1k}+{u}_{3k} \end{array}$$

(46)

Here, there are no implicit constraints in relation to the heteroscedastic maximal specification, and thus each of the ${u}_{w}$ , ${u}_{b}$, and ${u}_{h}$ terms are adequately recovered in this model. The ${u}_{2}$ terms represent a sort of unconflated “contextual effect” error term in that they denote the cluster-specific difference in the underlying ${u}_{b}$ and ${u}_{w}$ term (i.e., ${u}_{2}={u}_{b}-{u}_{w}$). The random effect covariance matrix is given as:

$$\left[\begin{array}{c}{u}_{0j}\\ {u}_{1j}\\ {u}_{2j}\\ {u}_{3j}\\ {u}_{0k}\\ {u}_{1k}\\ {u}_{2k}\\ {u}_{3k}\end{array}\right]\sim MVN\left(\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{cccccccc}{\tau }_{00(J)}& & & & & & & \\ {\tau }_{01(J)}& {\tau }_{11(J)}& & & & & & \\ {\tau }_{02(J)}& {\tau }_{12(J)}& {\tau }_{22(J)}& & & & & \\ {\tau }_{03(J)}& {\tau }_{13(J)}& {\tau }_{23(J)}& {\tau }_{33(J)}& & & & \\ 0& 0& 0& 0& {\tau }_{00(K)}& & & \\ 0& 0& 0& 0& {\tau }_{01(K)}& {\tau }_{11(K)}& & \\ 0& 0& 0& 0& {\tau }_{02(K)}& {\tau }_{12(K)}& {\tau }_{22(K)}& \\ 0& 0& 0& 0& {\tau }_{03(K)}& {\tau }_{13(K)}& {\tau }_{23(K)}& {\tau }_{33(K)}\end{array}\right]\right)$$

(47)

This specification is likelihood-equivalent to the general heteroscedastic specification in Eq. (35), and the random slope variances of ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$ have the same interpretation across both parameterizations. The random slope variances of the level 2 predictors in Eq. (47), however, have a unique interpretation here in that they represent the variance in the contextual effect terms mentioned above.

Partially cluster-mean-centered models

The most basic partially cluster-mean-centered model includes random effects of both cluster types for the J-cluster-mean-centered x, without any random effects for the cluster means:

$$\begin{array}{l}RE={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot })\\ \quad\ \ ={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({u}_{1j}+{u}_{1k}){x}_{\cdot \cdot k} \end{array}$$

(48)

The re-expression shows what each random slope term represents in relation to the general heteroscedastic random effect model shown in Eq. (35):

$$\begin{array}{l}{u}_{wj}={u}_{1j}\\ {u}_{bj}\,={u}_{1j}\\ {u}_{hj}\,=0\\ {u}_{wk}={u}_{1k}\\ {u}_{bk}\,=0\\ {u}_{hk}={u}_{1k} \end{array}$$

(49)

Hence, this model places the constraints that ${u}_{wj}$ equals ${u}_{bj}$ (both given by ${u}_{1j}$) and that ${u}_{wk}$ equals ${u}_{hk}$ (both given as ${u}_{1k}$). Additionally, the J-cluster heteroscedastic component, ${u}_{hj}$, is constrained to be zero, as is the K-cluster between-cluster effect component, ${u}_{bk}$. The random effect covariance matrix of this model is given as:

$$\left[\begin{array}{c}{u}_{0j}\\ {u}_{1j}\\ {u}_{0k}\\ {u}_{1k}\end{array}\right]\sim MVN\left(\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{cccc}{\tau }_{00(J)}& & & \\ {\tau }_{01(J)}& {\tau }_{11(J)}& & \\ 0& 0& {\tau }_{00(K)}& \\ 0& 0& {\tau }_{01(K)}& {\tau }_{11(K)}\end{array}\right]\right)$$

(50)

This model places a total of 14 constraints on the general heteroscedastic specification in Eq. (35), namely: $\mathrm{var}({u}_{wj})=\mathrm{var}({u}_{bj})$, $\mathrm{var}({u}_{wk})=\mathrm{var}({u}_{hk})$, $\mathrm{var}({u}_{hj})=0$, $\mathrm{var}({u}_{bk})=0$, ${\mathrm{corr}}({u}_{wj},{u}_{bj})=1$, ${\mathrm{corr}}({u}_{wk},{u}_{bk})=0$, ${\mathrm{corr}}({u}_{wj,}{u}_{hj})={0}$, ${\mathrm{corr}}({u}_{wk},{u}_{hk})=1$, ${\mathrm{corr}}({u}_{bj},{u}_{hj})={0}$, ${\mathrm{corr}}({u}_{bk},{u}_{hk})=0$, ${\mathrm{corr}}\mathrm{(}{u}_{0j},{u}_{wj}\mathrm{)}=\mathrm{corr(}{u}_{0j},{u}_{bj}\mathrm{)}$, ${\mathrm{corr}}\mathrm{(}{u}_{0k},{u}_{bk})=0$, ${\mathrm{corr}}\mathrm{(}{u}_{0j},{u}_{hj}\mathrm{)}=0$, ${\mathrm{corr}}\mathrm{(}{u}_{0k},{u}_{wk}\mathrm{)}=\mathrm{corr(}{u}_{0k},{u}_{hk}\mathrm{)}$. In interpreting the random effect variances, the ${\tau }_{11(J)}$ implicitly reflects some combination of the variability in the within-cluster and between-J-cluster effects of x, and ${\tau }_{11(K)}$ reflects some combination of the variability in the within-cluster effect as well as possible K-cluster heteroscedasticity at level 2.

We next consider the partially cluster-mean-centered model that additionally adds J-cluster random slopes for ${x}_{\cdot \cdot k}$ and K-cluster random slopes for ${x}_{\cdot j\cdot }$, and is thus given as:

$$\begin{array}{l}RE={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot })+{u}_{2k}{x}_{\cdot j\cdot }+{u}_{2j}{x}_{\cdot \cdot k}\\\quad \ \ ={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({u}_{1j}+{u}_{1k}){x}_{\cdot \cdot k}+{u}_{2k}{x}_{\cdot j\cdot }+{u}_{2j}{x}_{\cdot \cdot k}\\ \quad \ \ ={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{u}_{2k}{x}_{\cdot j\cdot }+({u}_{1j}+{u}_{1k}+{u}_{2j}){x}_{\cdot \cdot k}\end{array}$$

(51)

What each random slope term represents in relation to general heteroscedastic random effect specification is shown as follows:

$$\begin{array}{l}{u}_{wj}={u}_{1j}\\ {u}_{bj}\,={u}_{1j}+{u}_{2j}\\ {u}_{hj}\,=0\\ {u}_{wk}={u}_{1k}\\ {u}_{bk}\,={u}_{2k}\\ {u}_{hk}={u}_{1k} \end{array}$$

(52)

Hence, the model still places the constraint that ${u}_{wk}$ is equal to ${u}_{hk}$ (both given as ${u}_{1k}$), and that ${u}_{hj}$ equals zero. However, there is no implicit constraint involving the random slope terms for either ${x}_{\cdot j\cdot }$ or ${x}_{\cdot \cdot k}$ (${u}_{2k}$ and ${u}_{2j}$, respectively), and thus the ${u}_{b}$ terms are recovered. Specifically, ${u}_{2k}={u}_{bk}$, and the ${u}_{2j}$ term represents the previously defined conflated contextual effect, i.e., the J-cluster-specific difference in the underlying ${u}_{bk}$ and the conflated ${u}_{1j}$ term (i.e., ${u}_{2j}={u}_{bj}-{u}_{1j}$). The random effect covariance matrix of this model is given as:

$$\left[\begin{array}{c}{u}_{0j}\\ {u}_{1j}\\ {u}_{2j}\\ {u}_{0k}\\ {u}_{1k}\\ {u}_{2k}\end{array}\right]\sim MVN\left(\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{cccccc}{\tau }_{00(J)}& & & & & \\ {\tau }_{01(J)}& {\tau }_{11(J)}& & & & \\ {\tau }_{02(J)}& {\tau }_{12(J)}& {\tau }_{22(J)}& & & \\ 0& 0& 0& {\tau }_{00(K)}& & \\ 0& 0& 0& {\tau }_{01(K)}& {\tau }_{11(K)}& \\ 0& 0& 0& {\tau }_{02(K)}& {\tau }_{12(K)}& {\tau }_{22(K)}\end{array}\right]\right)$$

(53)

This model places eight constraints on the general heteroscedastic specification in Eq. (35), namely $\mathrm{var}({u}_{hj})=0$, $\mathrm{var}({u}_{wk})=\mathrm{var}({u}_{hk})$, ${\mathrm{corr}}({u}_{wj},{u}_{hj})={0}$, ${\mathrm{corr}}({u}_{wk},{u}_{hk})=1$, ${\mathrm{corr}}({u}_{bj},{u}_{hj})={0}$, ${\mathrm{corr}}({u}_{wk},{u}_{bk})={\mathrm{corr}}({u}_{bk},{u}_{hk})$, ${\mathrm{corr}}({u}_{0j},{u}_{hj})={0}$, and ${\mathrm{corr}}({u}_{0k},{u}_{wk})={\mathrm{corr}}({u}_{0k},{u}_{hk})$. The two random slope variances for the level 1 predictor ${x}_{ijk}-{x}_{\cdot j\cdot }$ have different interpretations: since ${u}_{1j}$ represents ${u}_{wj}$, the ${\tau }_{11(J)}$ variance reflects across-J-cluster variability in the within-cluster effect of x; since ${u}_{1k}$, on the other hand, represents some weighted average of ${u}_{wk}$ and ${u}_{hk}$ , the ${\tau }_{11(K)}$ variance reflects across-K-cluster variability in some combination of the within-cluster effect as well as possible K-cluster heteroscedasticity at level 2. The random slope variance for the J-cluster mean (${\tau }_{22(K)}$) reflects across-K-cluster variability in the between-J-cluster effect, whereas the random slope variance for K-cluster mean (${\tau }_{22(J)}$) reflects variability in the previously defined conflated contextual effect term (${u}_{2j}$).

Last, we consider adding the heteroscedastic components for both cluster types, and the model is given as:

$$\begin{array}{l}RE={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot })+({u}_{2k}+{u}_{3j}){x}_{\cdot j\cdot }+({u}_{2j}+{u}_{3k}){x}_{\cdot \cdot k}\\ \quad \ \ ={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({u}_{1j}+{u}_{1k}){x}_{\cdot \cdot k}+({u}_{2k}+{u}_{3j}){x}_{\cdot j\cdot }+({u}_{2j}+{u}_{3k}){x}_{\cdot \cdot k}\\ \quad \ \ ={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({u}_{2k}+{u}_{3j}){x}_{\cdot j\cdot }+({u}_{1j}+{u}_{1k}+{u}_{2j}+{u}_{3k}){x}_{\cdot \cdot k}\end{array}$$

(54)

Again, this re-expression clarifies how each specification is related to the general heteroscedastic random effect model, that is:

$$\begin{array}{l}{u}_{wj}\,={u}_{1j}\\ {u}_{bj}\,\,={u}_{1j}+{u}_{2j}\\ {u}_{hj}\,={u}_{3j}\\ {u}_{wk}={u}_{1k}\\ {u}_{bk}\,={u}_{2k}\\ {u}_{hk}\,={u}_{1k}+{u}_{3k} \end{array}$$

(55)

There is no implicit constraint involving any of the random slope terms for ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$, ${x}_{\cdot j\cdot }$, or ${x}_{\cdot \cdot k}$, and thus ${u}_{w}$, ${u}_{b}$ and ${u}_{h}$ are all recovered from this model. The ${u}_{2j}$ represents the unconflated “contextual effect” error term, i.e., the underlying difference in the J-cluster random effects for the between-K-cluster and within-cluster effects (i.e., ${u}_{2j}={u}_{bj}-{u}_{wj}$). The ${u}_{3k}$ represents a different type of unconflated “contextual effect,” namely, the difference in the K-cluster random effects for the within-cluster effects and the K-cluster heteroscedastic component (i.e., ${u}_{3k}={u}_{hk}-{u}_{1k}$). The random effect covariance matrix is given as:

$$\left[\begin{array}{c}{u}_{0j}\\ {u}_{1j}\\ {u}_{2j}\\ {u}_{3j}\\ {u}_{0k}\\ {u}_{1k}\\ {u}_{2k}\\ {u}_{3k}\end{array}\right]\sim MVN\left(\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{cccccccc}{\tau }_{00(J)}& & & & & & & \\ {\tau }_{01(J)}& {\tau }_{11(J)}& & & & & & \\ {\tau }_{02(J)}& {\tau }_{12(J)}& {\tau }_{22(J)}& & & & & \\ {\tau }_{03(J)}& {\tau }_{13(J)}& {\tau }_{23(J)}& {\tau }_{33(J)}& & & & \\ 0& 0& 0& 0& {\tau }_{00(K)}& & & \\ 0& 0& 0& 0& {\tau }_{01(K)}& {\tau }_{11(K)}& & \\ 0& 0& 0& 0& {\tau }_{02(K)}& {\tau }_{12(K)}& {\tau }_{22(K)}& \\ 0& 0& 0& 0& {\tau }_{03(K)}& {\tau }_{13(K)}& {\tau }_{23(K)}& {\tau }_{33(K)}\end{array}\right]\right)$$

(56)

This specification is likelihood-equivalent to the general heteroscedastic specification in Eq. (35). In terms of interpretation, both of the random slope variances for ${x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k}$, ${\tau }_{11(J)}$ and ${\tau }_{11(K)}$, and both random slope variances for ${x}_{\cdot j\cdot }$, ${\tau }_{22(K)}$ and ${\tau }_{33(J)}$, have the same interpretation as the general heteroscedastic specification. The random slope variances for ${x}_{\cdot \cdot k}$, however, have unique interpretations, as ${\tau }_{22(J)}$ and ${\tau }_{33(K)}$ represent the variances in the aforementioned contextual effect residuals.

Fully cluster-mean-centered models

Last, we consider fully cluster-mean-centered models. The most basic of these models includes only the fully cluster-mean-centered x as the predictor:

$$RE={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})$$

(57)

Here, each ${u}_{w}$ is represented by ${u}_{1}$, so that the interpretation of ${u}_{1}$ is the same as that of the general heteroscedastic specification in Eq. (35); however, both of the between-cluster random effects and the level 2 heteroscedasticity components are constrained to be zero. The random effect covariance matrix is then:

$$\left(\begin{array}{c}{u}_{0j}\\ {u}_{1j}\\ {u}_{0k}\\ {u}_{1k}\end{array}\right)\sim MVN\left(\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{cccc}{\tau }_{00(J)}& & & \\ {\tau }_{01(J)}& {\tau }_{11(J)}& & \\ 0& 0& {\tau }_{00(K)}& \\ 0& 0& {\tau }_{01(K)}& {\tau }_{11(K)}\end{array}\right]\right)$$

(58)

Here, each variance has the same interpretation as that from the general heteroscedastic expression in Eq. (35). In relation to the general heteroscedastic expression, this model places a total of 14 constraints, i.e., that all variances and covariances shown in Eq. (35) but not B25 are equal to zero.

Expanding upon this model, one could also add a J-cluster random component for ${x}_{\cdot \cdot k}$ and a K-cluster random component for ${x}_{\cdot j\cdot }$, which would yield the following random effect specification:

$$RE={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+{u}_{2k}{x}_{\cdot j\cdot }+{u}_{2j}{x}_{\cdot \cdot k}$$

(59)

In this model, ${u}_{1}$ still represents ${u}_{w}$ as specified in general heteroscedastic model in Eq. (35), whereas ${u}_{2k}$ represents ${u}_{bk}$ and ${u}_{2j}$ represents ${u}_{bj}$. The random effect covariance matrix is then as follows:

$$\left(\begin{array}{c}{u}_{0j}\\ {u}_{1j}\\ {u}_{2j}\\ {u}_{0k}\\ {u}_{1k}\\ {u}_{2k}\end{array}\right)\sim MVN\left(\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{cccccc}{\tau }_{00(J)}& & & & & \\ {\tau }_{01(J)}& {\tau }_{11(J)}& & & & \\ {\tau }_{02(J)}& {\tau }_{12(J)}& {\tau }_{22(J)}& & & \\ 0& 0& 0& {\tau }_{00(K)}& & \\ 0& 0& 0& {\tau }_{01(K)}& {\tau }_{11(K)}& \\ 0& 0& 0& {\tau }_{02(K)}& {\tau }_{12(K)}& {\tau }_{22(K)}\end{array}\right]\right)$$

(60)

Each variance again has the same interpretation as the corresponding variance from the general heteroscedastic expression in Eq. (35), but this constrains all other terms from Eq. (35) to zero.

Last, if one were to additionally add the level 2 heteroscedasticity component for both cluster types, the random effects would be given as:

$$RE={u}_{0j}+{u}_{0k}+({u}_{1j}+{u}_{1k})({x}_{ijk}-{x}_{\cdot j\cdot }-{x}_{\cdot \cdot k})+({u}_{2k}+{u}_{3j}){x}_{\cdot j\cdot }+({u}_{2j}+{u}_{3k}){x}_{\cdot \cdot k}$$

(61)

And the random effect covariance matrix would be:

$$\left[\begin{array}{c}{u}_{0j}\\ {u}_{1j}\\ {u}_{2j}\\ {u}_{3j}\\ {u}_{0k}\\ {u}_{1k}\\ {u}_{2k}\\ {u}_{3k}\end{array}\right]\sim MVN\left(\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right],\left[\begin{array}{cccccccc}{\tau }_{00(J)}& & & & & & & \\ {\tau }_{01(J)}& {\tau }_{11(J)}& & & & & & \\ {\tau }_{02(J)}& {\tau }_{12(J)}& {\tau }_{22(J)}& & & & & \\ {\tau }_{03(J)}& {\tau }_{13(J)}& {\tau }_{23(J)}& {\tau }_{33(J)}& & & & \\ 0& 0& 0& 0& {\tau }_{00(K)}& & & \\ 0& 0& 0& 0& {\tau }_{01(K)}& {\tau }_{11(K)}& & \\ 0& 0& 0& 0& {\tau }_{02(K)}& {\tau }_{12(K)}& {\tau }_{22(K)}& \\ 0& 0& 0& 0& {\tau }_{03(K)}& {\tau }_{13(K)}& {\tau }_{23(K)}& {\tau }_{33(K)}\end{array}\right]\right)$$

(62)

This is equal to the general heteroscedastic model given in Eq. (35).

Appendix C: R function to fully or partially cluster-mean-center level 1 variables and compute cluster means of level 1 variables for cross-classified models

cc.gmc R function Description:

This function reads in a dataset containing level 1 predictors and cluster identifying variables, and outputs the fully cluster-mean-centered level 1 predictor along with each type of cluster mean. It can accommodate any number of cross-classifications and can center multiple level 1 predictors simultaneously. Partially cluster-mean-centered level 1 predictors can be computed by specifying a subset of cluster types, as described below. Using this function, researchers can utilize any of the centering options discussed throughout this paper (see, e.g., Table 1).

This function is a modified version of the gmc function from the rockchalk package (Johnson, 2019).

cc.gmc R function Input:

dframe – Dataset with row denoting observations and columns denoting variables

x – List of level 1 predictors to be centered (E.g., c(“V1”,”V2”))

by – List of variables names indicating higher-level clusters (E.g., c(“Cluster1”,”Cluster2”)). To create partially cluster-mean-center variables, enter only a subset of clusters here.

fulldataframe – Logical argument indicating whether or not to return full data frame. If set to TRUE, full data frame with newly added columns of cluster-mean-centered level 1 predictors and cluster means is returned. If set to FALSE, only cluster-mean-centered level 1 predictors, cluster means, and cluster identification variables will be returned.

cc.gmc R function Code:

cc.gmc R function simulated example:

Code used for models in empirical example section: