Define a multiple-membership block (the micro-to-macro aggregation)

Specifies a multiple-membership level where group-level units (e.g., occupations) are composed of member-level units (e.g., tasks). Each block aggregates the weighted member records into one or more group-level features whose coefficients are estimated by the main model. The components map onto the framework's notation \(\theta^{micro,f}(M_{it})\):

id = id(mmid, mainid): the membership structure \(S_{it}\)
vars = vars(x): the member attributes \(x_{kt}\)
w = w(~ ...): the weights \(w_{ikt}\) (who counts)
fn = fn("..."): the aggregation function \(f\) (how contributions combine); fn("sum") is the additive case, other types are emergent features (variance, concentration, thresholds, ...)
RE = re(...): member random effects, aggregated by the weights (\(\sum_k w_{ikt} u_{0k}\), plus random slopes via re(1 + x))
name: block name for referencing the feature in main-formula interactions (auto-generated when omitted)

Usage

mm(
  id,
  vars = NULL,
  w = NULL,
  fn = NULL,
  RE = NULL,
  FE = NULL,
  name = NULL,
  ...
)

Arguments

id: An id object: id(mmid, mainid).
vars: A vars object with member attributes, or NULL for weights-only blocks (RE-only, or fn("hhi")-family).
w: A w object specifying the weights. Default: w(~ 1/n, scale = TRUE) (equal weights).
fn: A fn object selecting the aggregation function. Default: fn("sum") (the additive weighted mean).
RE: Random effects: TRUE (shorthand for re(1)), a re object, or NULL/FALSE for none. Only available with fn("sum"): member random effects do not compose with dispersion-type features.
FE: Fixed effects: a fe object, or NULL. Mutually exclusive with RE. With many members this is weakly identified — prefer RE (partial pooling) at the member level.
name: Optional block name (unquoted or string) used to reference the block's feature in main-formula interactions (e.g. Ax:education). Auto-generated from the feature when omitted (e.g. A_x, V_x). Must not collide with a data column.
...: Not used; catches removed arguments (c =, and ar =, which moved into the effects grammar: RE = re(1, ar = TRUE)) with a migration message.

Value

A bml_mm object.

Details

One mechanism. Every block emits group-level feature(s) whose coefficients are main-model coefficients (class "b", labeled by the feature name, e.g. A_x). fn("sum") is the additive case \(\theta = \beta A_x + \sum_k w_k u_{0k}\): the feature \(A_x\) plus the optional RE. There is no separate "sum mode".

Multiple blocks can be combined with + to stack features (mean + variance + concentration, ...). RE can be specified for one block per member-id group.

References

Rosche, B. (2026). A Multilevel Model for Coalition Governments: Uncovering Party-Level Dependencies Within and Between Governments. Political Analysis.

Examples

if (FALSE) { # \dontrun{
# Additive aggregation (weighted mean effect)
mm(id = id(task, occ), vars = vars(x), w = w(~ importance, scale = TRUE), fn = fn("sum"))

# With member random intercepts
mm(id = id(task, occ), vars = vars(x), w = w(~ 1/n), fn = fn("sum"), RE = TRUE)

# Random intercept + slope (residual effect heterogeneity)
mm(id = id(task, occ), vars = vars(x), w = w(~ 1/n), fn = fn("sum"), RE = re(1 + x))

# Emergent features
mm(id = id(task, occ), vars = vars(x), w = w(~ 1/n), fn = fn("var"))
mm(id = id(task, occ), w = w(~ importance), fn = fn("hhi"))
mm(id = id(task, occ), vars = vars(x), w = w(~ 1/n),
   fn = fn("threshold", c = est(), kappa = 10))
mm(id = id(task, occ), vars = vars(x), w = w(~ 1/n), fn = fn("smax", kappa = est()))

# Named block, referenced in a cross-level interaction
bml(Y ~ education + Ax:education +
      mm(name = Ax, id = id(task, occ), vars = vars(x), w = w(~ 1/n), fn = fn("sum")),
    data = dat)
} # }