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The generalized multiple membership multilevel model

Let y_{i}^{(2)} be an outcome at level 2, where subscript i indexes level 2 units and superscript (2) indicates that the outcome is located at the second level. Using the multiple membership multilevel model (MMMM) with endogenized weights, we can model this outcome in terms of an aggregated level 1 effect \theta_{i}^{(1)} and a level 2 effect \theta_{i}^{(2)}:

(1)   \begin{equation*}y_{i}^{(2)}=\theta_{i}^{(1)}+\theta_{i}^{(2)} \end{equation*}

The level 2 effect in equation (1) is determined by a systematic component of observed explanatory variables at level 2 x_{i}^{(2)} with regression coefficients \beta^{(2)} and a random component u_{i}^{(2)}:

(2)   \begin{equation*}\theta_{i}^{(2)}=x_{i}^{\intercal (2)}\beta^{(2)}+u_{i}^{(2)} \end{equation*}

The random component at this level is a disturbance term, which is assumed to be normally distributed with a mean of zero and constant variance u_{i}^{(2)}\sim N(0,\sigma_{u^{(2)}}^{2}). This part of the model represents the conventional single-level model structure.

The aggregated level 1 effect \theta_{i}^{(1)} in equation (1) models the aggregation of the effects of level 1 units to the second level. It is determined by a weighted sum of the effect of each level 1 unit j on the level 2 outcome in the set of level 1 units z(i) embedded in level 2 unit i:

(3)   \begin{equation*}\theta_{i}^{(1)}=\sum_{j \in z(i)}w_{ij}\zeta_{ij}\end{equation*}

That is, subscript j=1,...,J  indexes level 1 units and the indexing function z(i) returns all level 1 units that are members of level 2 unit i. The multiple membership construct aggregates individual level 1 effects \zeta_{ij} by taking their weighted sum with weights w_{ij}=w_{ij}^{*} for all parties j \in z(i) and w_{ij}=0 for all parties j \notin z(i).

The individual-level 1 effects \zeta_{ij} are determined by a systematic component of observed explanatory variables at level 1 x_{i}^{(1)} with regression coefficients \beta^{(1)} and a random component u_{i}^{(1)}, representing the joint impact of unobserved variables:

(4)   \begin{equation*}\zeta_{ij}=x_{i}^{\intercal (1)}\beta^{(1)}+u_{i}^{(1)}\end{equation*}

The random component at this level is also assumed to be normally distributed with a mean of zero and constant variance u_{i}^{(1)}\sim N(0,\sigma_{u^{(1)}}^{2}).

To examine how the effects of level 1 units propagates to the second level, we endogenize the weights instead of assigning fixed values to each unit:

(5)   \begin{equation*}\begin{split}w_{ij}=\frac{1}{n_{i}^{exp(-(x_{ij}^{\intercal W}\beta^{W}))}} \\s.t. \sum_{i} \sum_{j} w_{ij}=N\end{split}\end{equation*}

where x_{ij}^{W} are explanatory variables with regression coefficients \beta^{W}, n_{i}  is the number of members level 2 unit i, and N equals the total number of observations in the dataset. In this form, the weights are bounded by [0,1].

The weight regression coefficients estimate the impact of explanatory variables on the specific weight of a level 1 unit in its effect on the level 2 outcome. If the weight variables have no impact on the aggregation process, i.e. \beta^{W}=0, the weights reduce to w_{ij}=\frac{1}{n_{i}} (mean aggregation). If \beta^{W} \neq 0, the weights reveal a more complex interplay of level 1 units in their effect on the level 2 outcome. That is, weights will deviate from \frac{1}{n_{i}} and are no longer constant within and between level 2 units. Instead, they depend on x_{ij}^{W}.

The generalized multiple-membership multilevel model thus is a multiple membership model with endogenized weights.