Restricted Maximum Likelihood Estimation for Parameters of the Social Relations Model

Abstract

In many areas of research, the round-robin design is used to study interpersonal judgments and behaviors. The resulting data are analyzed with the social relations model (SRM), whereby almost all previously published studies have used ANOVA-based methods or multilevel-based methods to obtain SRM parameter estimates. In this article, the SRM is embedded into the linear mixed model framework, and it is shown how restricted maximum likelihood can be employed to estimate the SRM parameters. It is also described how the effect of covariates on the SRM-specific effects can be estimated. An example is presented to illustrate the approach. We also present the results of a simulation study in which the performance of the proposed approach is compared to the ANOVA method.

Appendices

Appendix 1: Variance–Covariance Matrix of the Judgment Vector y

Equation 5, the basic assumptions outlined above, and some standard results concerning the variance–covariance matrix of a vector of random variables are used to derive the variance–covariance matrix of the judgment vector y. Please note that

$$\begin{aligned}&\begin{pmatrix} Z_{1} &{} Z_{2}\\ Z_{2} &{} Z_{1} \end{pmatrix} \left( \begin{pmatrix} \sigma ^2_{\alpha } &{} \sigma _{\alpha \beta }\\ \sigma _{\alpha \beta } &{} \sigma ^2_{\beta } \end{pmatrix} \otimes I_{mk}\right) \begin{pmatrix} Z_{1}^{'} &{} Z_{2}^{'} \\ Z_{2}^{'} &{} Z_{1}^{'} \end{pmatrix} = \begin{pmatrix} Z_{1} &{} Z_{2}\\ Z_{2} &{} Z_{1} \end{pmatrix} \begin{pmatrix} \sigma ^2_{\alpha } \cdot I_{mk} &{} \sigma _{\alpha \beta } \cdot I_{mk} \\ \sigma _{\alpha \beta } \cdot I_{mk} &{} \sigma ^2_{\beta } \cdot I_{mk} \end{pmatrix} \begin{pmatrix} Z_{1}^{'} &{} Z_{2}^{'} \\ Z_{2}^{'} &{} Z_{1}^{'} \end{pmatrix}\\&\quad = \begin{pmatrix} Z_{1}\sigma ^2_{\alpha }Z_{1}^{'} + Z_{2}\sigma ^2_{\beta }Z_{2}^{'} + Z_{1}\sigma _{\alpha \beta }Z_{2}^{'} + Z_{2}\sigma _{\alpha \beta }Z_{1}^{'} &{} Z_{1}\sigma ^2_{\alpha }Z_{2}^{'} + Z_{2}\sigma ^2_{\beta }Z_{1}^{'} + Z_{1}\sigma _{\alpha \beta }Z_{1}^{'} + Z_{2}\sigma _{\alpha \beta }Z_{2}^{'}\\ Z_{2}\sigma ^2_{\alpha }Z_{1}^{'} + Z_{1}\sigma ^2_{\beta }Z_{2}^{'} + Z_{2}\sigma _{\alpha \beta }Z_{2}^{'} + Z_{1}\sigma _{\alpha \beta }Z_{1}^{'} &{} Z_{2}\sigma ^2_{\alpha }Z_{2}^{'} + Z_{1}\sigma ^2_{\beta }Z_{1}^{'} + Z_{2}\sigma _{\alpha \beta }Z_{1}^{'} + Z_{1}\sigma _{\alpha \beta }Z_{2}^{'} \end{pmatrix}\\&\quad = \begin{pmatrix} Z_{1}\sigma ^2_{\alpha }Z_{1}^{'} &{} Z_{1}\sigma ^2_{\alpha }Z_{2}^{'}\\ Z_{2}\sigma ^2_{\alpha }Z_{1}^{'} &{} Z_{2}\sigma ^2_{\alpha }Z_{2}^{'} \end{pmatrix} + \begin{pmatrix} Z_{2}\sigma ^2_{\beta }Z_{2}^{'} &{} Z_{2}\sigma ^2_{\beta }Z_{1}^{'}\\ Z_{1}\sigma ^2_{\beta }Z_{2}^{'} &{} Z_{1}\sigma ^2_{\beta }Z_{1}^{'} \end{pmatrix} + \begin{pmatrix} \sigma _{\alpha \beta }(Z_{1}Z_{2}^{'} + Z_{2}Z_{1}^{'}) &{} \sigma _{\alpha \beta }(Z_{1}Z_{1}^{'} + Z_{2}Z_{2}^{'})\\ \sigma _{\alpha \beta }(Z_{2}Z_{2}^{'} + Z_{1}Z_{1}^{'}) &{} \sigma _{\alpha \beta }(Z_{2}Z_{1}^{'} + Z_{1}Z_{2}^{'}) \end{pmatrix}\\&\quad = \sigma ^2_{\alpha } \begin{pmatrix} Z_{1}Z_{1}^{'} &{} Z_{1}Z_{2}^{'} \\ Z_{2}Z_{1}^{'} &{} Z_{2}Z_{2}^{'} \end{pmatrix} + \sigma ^2_{\beta } \begin{pmatrix} Z_{2}Z_{2}^{'} &{} Z_{2}Z_{1}^{'}\\ Z_{1}Z_{2}^{'} &{} Z_{1}Z_{1}^{'} \end{pmatrix} + \sigma _{\alpha \beta } \begin{pmatrix} Z_{1}Z_{2}^{'} + Z_{2}Z_{1}^{'} &{} Z_{1}Z_{1}^{'} + Z_{2}Z_{2}^{'} \\ Z_{2}Z_{2}^{'} + Z_{1}Z_{1}^{'} &{} Z_{2}Z_{1}^{'} + Z_{1}Z_{2}^{'} \end{pmatrix}\\&\quad = \sigma ^2_{\alpha } \begin{pmatrix} Z_{1}Z_{1}^{'} &{} Z_{1}Z_{2}^{'} \\ Z_{2}Z_{1}^{'} &{} Z_{2}Z_{2}^{'} \end{pmatrix} + \sigma ^2_{\beta } \begin{pmatrix} Z_{2}Z_{2}^{'} &{} Z_{2}Z_{1}^{'}\\ Z_{1}Z_{2}^{'} &{} Z_{1}Z_{1}^{'} \end{pmatrix} + \sigma _{\alpha \beta } \begin{pmatrix} Z_{1} &{} Z_{2} \\ Z_{2} &{} Z_{1} \end{pmatrix} \begin{pmatrix} Z_{2}^{'} &{} Z_{1}^{'} \\ Z_{1}^{'} &{} Z_{2}^{'} \end{pmatrix}. \end{aligned}$$

Furthermore,

$$\begin{aligned}&\begin{pmatrix} Z_{3} &{} 0\\ 0 &{} Z_{3} \end{pmatrix} \left( \begin{pmatrix} \sigma ^2_{\gamma } &{} \sigma _{\gamma \gamma }\\ \sigma _{\gamma \gamma } &{} \sigma ^2_{\gamma } \end{pmatrix} \otimes I_{m(m-1)k}\right) \begin{pmatrix} Z_{3}^{'} &{} 0\\ 0 &{} Z_{3}^{'} \end{pmatrix}\\&\quad = \begin{pmatrix} Z_{3} &{} 0\\ 0 &{} Z_{3} \end{pmatrix} \begin{pmatrix} \sigma ^2_{\gamma } \cdot I_{m(m-1)k} &{} \sigma _{\gamma \gamma } \cdot I_{m(m-1)k} \\ \sigma _{\gamma \gamma } \cdot I_{m(m-1)k} &{} \sigma ^2_{\gamma } \cdot I_{m(m-1)k} \end{pmatrix} \begin{pmatrix} Z_{3}^{'} &{} 0\\ 0 &{} Z_{3}^{'} \end{pmatrix}\\&\quad = \begin{pmatrix} Z_{3}\sigma ^2_{\gamma }Z_{3}^{'} &{} Z_{3}\sigma _{\gamma \gamma }Z_{3}^{'}\\ Z_{3}\sigma _{\gamma \gamma }Z_{3}^{'} &{} Z_{3}\sigma ^2_{\gamma }Z_{3}^{'} \end{pmatrix}= \sigma ^2_{\gamma } \begin{pmatrix} Z_{3}Z_{3}^{'} &{} 0 \\ 0 &{} Z_{3}Z_{3}^{'} \end{pmatrix} + \sigma _{\gamma \gamma } \begin{pmatrix} 0 &{} Z_{3}Z_{3}^{'}\\ Z_{3}Z_{3}^{'} &{} 0 \end{pmatrix}, \end{aligned}$$

and

$$\begin{aligned}&\begin{pmatrix} I_{N} &{} 0\\ 0 &{} I_{N} \end{pmatrix} \left( \begin{pmatrix} \sigma ^2_{\epsilon } &{} \sigma _{\epsilon \epsilon }\\ \sigma _{\epsilon \epsilon } &{} \sigma ^2_{\epsilon } \end{pmatrix} \otimes I_N\right) \begin{pmatrix} I_{N} &{} 0\\ 0 &{} I_{N} \end{pmatrix} = \begin{pmatrix} I_{N} &{} 0\\ 0 &{} I_{N} \end{pmatrix} \begin{pmatrix} \sigma ^2_{\epsilon } \cdot I_N &{} \sigma _{\epsilon \epsilon } \cdot I_N \\ \sigma _{\epsilon \epsilon } \cdot I_N &{} \sigma ^2_{\epsilon } \cdot I_N \end{pmatrix} \begin{pmatrix} I_{N} &{} 0\\ 0 &{} I_{N} \end{pmatrix}\\&\quad = \begin{pmatrix} I_{N}\sigma ^2_{\epsilon }I_{N} &{} I_{N}\sigma _{\epsilon \epsilon }I_{N}\\ I_{N}\sigma _{\epsilon \epsilon }I_{N} &{} I_{N}\sigma ^2_{\epsilon }I_{N} \end{pmatrix} = \sigma ^2_{\epsilon } \begin{pmatrix} I_{N} &{} 0 \\ 0 &{} I_{N} \end{pmatrix} + \sigma _{\epsilon \epsilon } \begin{pmatrix} 0 &{} I_{N}\\ I_{N} &{} 0 \end{pmatrix}. \end{aligned}$$

The variance–covariance matrix of the judgment vector y then is

$$\begin{aligned} \text {cov} \Bigg (\begin{pmatrix} y_1 \\ y_2 \end{pmatrix}&= \text {cov}\left( \begin{pmatrix} Z_g \\ Z_g \end{pmatrix} g + \begin{pmatrix} Z_1 &{} Z_2 \\ Z_2 &{} Z_1 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} + \begin{pmatrix} Z_3 &{} 0 \\ 0 &{} Z_3 \end{pmatrix} \begin{pmatrix} \gamma _{1} \\ \gamma _{2} \end{pmatrix} + \begin{pmatrix} I_N &{} 0 \\ 0 &{} I_N \end{pmatrix} \begin{pmatrix} \epsilon _{1} \\ \epsilon _{2} \end{pmatrix}\right) \\&= \begin{pmatrix} Z_g \\ Z_g \end{pmatrix} \text {cov}(g) \begin{pmatrix} Z_g^{'} \\ Z_g^{'} \end{pmatrix} + \begin{pmatrix} Z_{1} &{} Z_{2}\\ Z_{2} &{} Z_{1} \end{pmatrix} \text {cov}\left( \begin{pmatrix} \alpha \\ \beta \end{pmatrix}\right) \begin{pmatrix} Z_{1}^{'} &{} Z_{2}^{'}\\ Z_{2}^{'} &{} Z_{1}^{'} \end{pmatrix}\\&\quad + \begin{pmatrix} Z_{3} &{} 0\\ 0 &{} Z_{3} \end{pmatrix} \text {cov}\left( \begin{pmatrix} \gamma _{1} \\ \gamma _{2} \end{pmatrix}\right) \begin{pmatrix} Z_{3}^{'} &{} 0\\ 0 &{} Z_{3}^{'} \end{pmatrix} + \begin{pmatrix} I_N &{} 0\\ 0 &{} I_N \end{pmatrix} \text {cov}\left( \begin{pmatrix} \epsilon _{1} \\ \epsilon _{2} \end{pmatrix}\right) \begin{pmatrix} I_{N}^{'} &{} 0\\ 0 &{} I_{N}^{'} \end{pmatrix}\\&= \begin{pmatrix} Z_g \\ Z_g \end{pmatrix} \begin{pmatrix} \sigma ^{2}_{g} \cdot I_k \\ \sigma ^{2}_{g} \cdot I_k \end{pmatrix} \begin{pmatrix} Z_g^{'} \\ Z_g^{'} \end{pmatrix} + \begin{pmatrix} Z_{1} &{} Z_{2}\\ Z_{2} &{} Z_{1} \end{pmatrix} \left( \begin{pmatrix} \sigma ^2_{\alpha } &{} \sigma _{\alpha \beta }\\ \sigma _{\alpha \beta } &{} \sigma ^2_{\beta } \end{pmatrix} \otimes I_{mk}\right) \begin{pmatrix} Z_{1}^{'} &{} Z_{2}^{'}\\ Z_{2}^{'} &{} Z_{1}^{'} \end{pmatrix}\\&\quad + \begin{pmatrix} Z_{3} &{} 0\\ 0 &{} Z_{3} \end{pmatrix} \left( \begin{pmatrix} \sigma ^2_{\gamma } &{} \sigma _{\gamma \gamma }\\ \sigma _{\gamma \gamma } &{} \sigma ^2_{\gamma } \end{pmatrix} \otimes I_{m(m-1)k}\right) \begin{pmatrix} Z_{3}^{'} &{} 0\\ 0 &{} Z_{3}^{'} \end{pmatrix}\\&\quad + \begin{pmatrix} I_N &{} 0\\ 0 &{} I_N \end{pmatrix} \left( \begin{pmatrix} \sigma ^2_{\epsilon } &{} \sigma _{\epsilon \epsilon }\\ \sigma _{\epsilon \epsilon } &{} \sigma ^2_{\epsilon } \end{pmatrix} \otimes I_{N}\right) \begin{pmatrix} I_{N}^{'} &{} 0\\ 0 &{} I_{N}^{'} \end{pmatrix}\\&= \sigma _{g}^{2} \begin{pmatrix} Z_gZ_g^{'} &{} Z_gZ_g^{'}\\ Z_gZ_g^{'} &{} Z_gZ_g^{'} \end{pmatrix}\\&\quad + \sigma _{\alpha }^{2} \begin{pmatrix} Z_{1} Z_{1}^{'} &{} Z_{1} Z_{2}^{'} \\ Z_{2} Z_{1}^{'} &{} Z_{2} Z_{2}^{'} \end{pmatrix} + \sigma _{\beta }^{2} \begin{pmatrix} Z_{2} Z_{2}^{'} &{} Z_{2} Z_{1}^{'} \\ Z_{1} Z_{2}^{'} &{} Z_{1} Z_{1}^{'} \end{pmatrix} + \sigma _{\alpha \beta } \begin{pmatrix} Z_{1} &{} Z_{2} \\ Z_{2} &{} Z_{1} \end{pmatrix} \begin{pmatrix} Z_{2}^{'} &{} Z_{1}^{'} \\ Z_{1}^{'} &{} Z_{2}^{'} \end{pmatrix}\\&\quad + \sigma _{\gamma }^{2} \begin{pmatrix} Z_{3}Z_{3}^{'} &{} 0 \\ 0 &{} Z_{3}Z_{3}^{'} \end{pmatrix} +\sigma _{\gamma \gamma } \begin{pmatrix} 0 &{} Z_{3}Z_{3}^{'} \\ Z_{3}Z_{3}^{'} &{} 0 \end{pmatrix}\\&\quad + \sigma ^2_{\epsilon } \begin{pmatrix} I_{N} &{} 0 \\ 0 &{} I_{N} \end{pmatrix} + \sigma _{\epsilon \epsilon } \begin{pmatrix} 0 &{} I_{N}\\ I_{N} &{} 0 \end{pmatrix}. \end{aligned}$$

Appendix 2: The Other Social Relation Models

The vector of round-robin judgments y for the basic SRM (see Equation 1) is

$$\begin{aligned} \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = Xb + \begin{pmatrix} Z_1 &{} Z_2 \\ Z_2 &{} Z_1 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} + \begin{pmatrix} Z_3 &{} 0 \\ 0 &{} Z_3 \end{pmatrix} \begin{pmatrix} \gamma _{1} \\ \gamma _{2} \end{pmatrix}, \end{aligned}$$

where $y_{1} = (y_{12}, \ldots , y_{(m-1)m})$ and $y_{2} = (y_{21}, \ldots , y_{m(m-1)})$. The vector of actor effects is $\alpha = (\alpha _{1}, \ldots , \alpha _{m})$ and the vector of partner effects is $\beta = (\beta _{1}, \ldots , \beta _{m})$. $\gamma _1$ and $\gamma _2$ denote the relationship effect vectors. Both are defined analogously to $y_1$ and $y_2$, respectively. b denotes the general mean, X is a matrix that contains 1s only, and $Z_i$, $i = 1, \ldots , 3$, are design matrices containing 0s and 1s (indicating the presence of a random effect). In the basic SRM, relationship effects cannot be separated from error, hence $Z_3 = I_N$ (where N is the number of judgments). The variance–covariance matrix V of the model is

$$\begin{aligned} \text {V}&= \sigma _{\alpha }^{2} \begin{pmatrix} Z_{1} Z_{1}^{'} &{} Z_{1} Z_{2}^{'} \\ Z_{2} Z_{1}^{'} &{} Z_{2} Z_{2}^{'} \end{pmatrix} + \sigma _{\beta }^{2} \begin{pmatrix} Z_{2} Z_{2}^{'} &{} Z_{2} Z_{1}^{'} \\ Z_{1} Z_{2}^{'} &{} Z_{1} Z_{1}^{'} \end{pmatrix} + \sigma _{\alpha \beta } \begin{pmatrix} Z_{1} &{} Z_{2} \\ Z_{2} &{} Z_{1} \end{pmatrix} \begin{pmatrix} Z_{2}^{'} &{} Z_{1}^{'} \\ Z_{1}^{'} &{} Z_{2}^{'} \end{pmatrix}\\&\quad \ + \sigma _{\gamma }^{2} \begin{pmatrix} I_N &{} 0 \\ 0 &{} I_N \end{pmatrix} + \sigma _{\gamma \gamma } \begin{pmatrix} 0 &{} I_N \\ I_N &{} 0 \end{pmatrix}. \end{aligned}$$

For model estimation it has to be assumed that there are at least four group members, $m \ge 4$, and at least one measure per dyad.

The vector of judgments y for the single-measure, multiple group SRM is

$$\begin{aligned} \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = Xb + \begin{pmatrix} Z_g \\ Z_g \end{pmatrix} g + \begin{pmatrix} Z_1 &{} Z_2 \\ Z_2 &{} Z_1 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} + \begin{pmatrix} Z_3 &{} 0 \\ 0 &{} Z_3 \end{pmatrix} \begin{pmatrix} \gamma _{1} \\ \gamma _{2} \end{pmatrix}, \end{aligned}$$

where $y_{1} = (y_{121}, \ldots , y_{(m-1)mk})$ and $y_{2} = (y_{211}, \ldots , y_{m(m-1)k})$. The vector of group effects is $g = (g_1,\ldots ,g_k)$, the actor effect vector is $\alpha = (\alpha _{1k}, \ldots , \alpha _{mk})$, and the vector of partner effects is $\beta = (\beta _{1k}, \ldots , \beta _{mk})$. The relationship effect vectors, $\gamma _1$ and $\gamma _2$, are again defined analogously to $y_1$ and $y_2$, respectively. b contains the general mean, X is a matrix of 1s, and $Z_g$, $Z_i$, $i = 1, \ldots , 3$, are the design matrices. Similar to the basic SRM, relationship effects cannot be separated from error in this SRM, thus $Z_3 = I_N$. The variance–covariance matrix of this SRM is

$$\begin{aligned} \text {V}&= \sigma _{g}^{2} \begin{pmatrix} Z_{g}Z_{g}^{'} &{} Z_{g}Z_{g}^{'}\\ Z_{g}^{'}Z_{g} &{} Z_{g}Z_{g}^{'} \end{pmatrix}\\&\quad \ + \sigma _{\alpha }^{2} \begin{pmatrix} Z_{1} Z_{1}^{'} &{} Z_{1} Z_{2}^{'} \\ Z_{2} Z_{1}^{'} &{} Z_{2} Z_{2}^{'} \end{pmatrix} + \sigma _{\beta }^{2} \begin{pmatrix} Z_{2} Z_{2}^{'} &{} Z_{2} Z_{1}^{'} \\ Z_{1} Z_{2}^{'} &{} Z_{1} Z_{1}^{'} \end{pmatrix} + \sigma _{\alpha \beta } \begin{pmatrix} Z_{1} &{} Z_{2} \\ Z_{2} &{} Z_{1} \end{pmatrix} \begin{pmatrix} Z_{2}^{'} &{} Z_{1}^{'} \\ Z_{1}^{'} &{} Z_{2}^{'} \end{pmatrix}\\&\quad \ + \sigma _{\gamma }^{2} \begin{pmatrix} Z_{3}Z_{3}^{'} &{} 0 \\ 0 &{} Z_{3}Z_{3}^{'} \end{pmatrix} + \sigma _{\gamma \gamma } \begin{pmatrix} 0 &{} Z_{3}Z_{3}^{'} \\ Z_{3}Z_{3}^{'} &{} 0 \end{pmatrix}. \end{aligned}$$

To estimate the model parameters, it has to be assumed that there are at least two groups, $k \ge 2$, with four group members each, $m \ge 4$, and one measure per dyad, $l = 1$.

Finally, the judgment vector y for the multiple-measure, one group SRM (see Equation 3) can be represented by

$$\begin{aligned} \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = Xb + \begin{pmatrix} Z_1 &{} Z_2 \\ Z_2 &{} Z_1 \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} + \begin{pmatrix} Z_3 &{} 0 \\ 0 &{} Z_3 \end{pmatrix} \begin{pmatrix} \gamma _{1} \\ \gamma _{2} \end{pmatrix} + \begin{pmatrix} \epsilon _{1} \\ \epsilon _{2} \end{pmatrix}. \end{aligned}$$

$y_{1} = (y_{121},\ldots ,y_{12l} \ldots , y_{(m-1)ml})$ and $y_{2} = (y_{211}, \ldots ,y_{21l},\ldots , y_{m(m-1)l})$. The actor effect vector is $\alpha = (\alpha _{1}, \ldots , \alpha _{m})$ and the vector of partner effects is $\beta = (\beta _{1} \ldots , \beta _{m})$. The vector $\gamma _1 = (\gamma _{12}, \ldots , \gamma _{(m-1)m})$ and the vector $\gamma _2 = (\gamma _{21}, \ldots , \gamma _{m(m-1)})$ denote the relationship effects. The vector of errors are defined analogously to $y_1$ and $y_2$, respectively. Again, b represents the general mean, X is a matrix containing 1s, and $Z_i$, $i = 1, \ldots , 3$, are design matrices. In this model, relationship effects can be separated from error, hence $Z_3 \ne I_N$. The variance–covariance matrix of this model is

$$\begin{aligned} \text {V}&= \sigma _{\alpha }^{2} \begin{pmatrix} Z_{1} Z_{1}^{'} &{} Z_{1} Z_{2}^{'} \\ Z_{2} Z_{1}^{'} &{} Z_{2} Z_{2}^{'} \end{pmatrix} + \sigma _{\beta }^{2} \begin{pmatrix} Z_{2} Z_{2}^{'} &{} Z_{2} Z_{1}^{'} \\ Z_{1} Z_{2}^{'} &{} Z_{1} Z_{1}^{'} \end{pmatrix} + \sigma _{\alpha \beta } \begin{pmatrix} Z_{1} &{} Z_{2} \\ Z_{2} &{} Z_{1} \end{pmatrix} \begin{pmatrix} Z_{2}^{'} &{} Z_{1}^{'} \\ Z_{1}^{'} &{} Z_{2}^{'} \end{pmatrix}\\&\quad \ + \sigma _{\gamma }^{2} \begin{pmatrix} Z_{3}Z_{3}^{'} &{} 0 \\ 0 &{} Z_{3}Z_{3}^{'} \end{pmatrix} + \sigma _{\gamma \gamma } \begin{pmatrix} 0 &{} Z_{3}Z_{3}^{'} \\ Z_{3}Z_{3}^{'} &{} 0 \end{pmatrix}\\&\quad \ + \sigma _{\epsilon }^{2} \begin{pmatrix} I_N &{} 0 \\ 0 &{} I_N \end{pmatrix} + \sigma _{\epsilon \epsilon } \begin{pmatrix} 0 &{} I_N \\ I_N &{} 0 \end{pmatrix}. \end{aligned}$$

To estimate the model, there have to at least four group members, $m \ge 4$, and two measures per dyad, $l \ge 2$.