Theory-Guided Specification Search for CFA Models

Theory-aware extension of specification_search. The classic specification search (MacCallum, 1986) is driven purely by fit: its loss only looks at RMSEA/CFI/SRMR, so it "capitalizes on chance" and can drift toward models that fit but break the intended theory (moving items to the wrong factor, or dropping core items of the construct). This function adds a theory-congruence term to the loss that penalizes departures from the theoretical item-to-factor assignment and penalizes dropping theoretical items. A single parameter theory_weight grades how much theory counts relative to fit (0 reproduces the fit-only search; larger values make the search progressively more conservative).

Usage

specification_search_theory(
  data,
  items,
  theory,
  theory_weight = 0.5,
  tw_move = 0.6,
  tw_drop = 0.4,
  tw_k = 0,
  seeds = NULL,
  max_factors = 4,
  min_items_factor = 2,
  cfi_target = 0.95,
  rmsea_target = 0.08,
  srmr_target = 0.08,
  max_iter_per_config = 40,
  max_covs = 5,
  max_items_removed = 6,
  try_bifactor = TRUE,
  operations = c("move", "drop", "cov"),
  estimator = "WLSMV",
  ordered = TRUE,
  std.lv = TRUE,
  mi_min = 10,
  mi_top = 3,
  only_within_factor = FALSE,
  loss_weights = c(rmsea = 0.5, cfi = 0.3, srmr = 0.2),
  patience = 8,
  early_stop_after_meet = 3,
  n_cores = 1,
  verbose = TRUE
)

Arguments

data

Data frame with the observed item responses.

items

Character vector with the item names to search over (must all exist in data).

theory

Named list encoding the theoretical structure, of the form list(FactorA = c("it1","it2",...), FactorB = c(...)). Items not listed are treated as having no theoretical factor (neither rewarded nor penalized).

theory_weight

Non-negative scalar weighting the theory term against the fit term. 0 reproduces the fit-only search; typical values 0.5-1 recover theory-congruent models with equivalent fit. Default 0.5.

tw_move, tw_drop, tw_k

Internal weights of the theory loss: penalty for misassigned items, for dropped theoretical items, and for deviating from the theoretical number of factors. Defaults 0.6, 0.4, 0.0.

seeds, max_factors, min_items_factor, cfi_target, rmsea_target, srmr_target

As in specification_search. If seeds = NULL, the theoretical structure plus block-partition seeds are generated.

max_iter_per_config, max_covs, max_items_removed, try_bifactor, operations

As in specification_search.

estimator, ordered, std.lv, mi_min, mi_top, loss_weights, patience, early_stop_after_meet, verbose

As in specification_search.

only_within_factor

Restrict the cov operation to pairs of items belonging to the same factor. Default FALSE, which reproduces the behaviour of earlier versions.

A residual correlation between items of different factors is rarely what it looks like. It is usually the signature of an omitted cross-loading, a misassigned item or a missing factor, and it absorbs covariance that belongs at the factor level: the estimated interfactor correlation drops and discriminant validity looks better than it is. Measured on a two-factor scale, freeing two cross-factor residuals moved the factor correlation from .956 to .919 and pulled the upper bound of its confidence interval from 1.031 down to .999, which is the difference between two factors that cannot be told apart and two that can.

TRUE is the conservative choice. It does not make cross-factor residuals wrong in principle, since item adjacency and shared method can justify them, but it forces those to be added deliberately rather than by modification index.

n_cores

Number of CPU cores for parallel evaluation of the candidate models within each greedy iteration (which are independent). Default 1 (sequential). Values > 1 use a PSOCK cluster (works on Windows, where fork is unavailable); results are identical to the sequential run, only faster. A practical choice is parallel::detectCores() - 1.

Value

An object of class specification_search (so print.specification_search applies), a list with table (one row per evaluated configuration, including the extra columns congruence and theory_loss), successful, best (the lowest-total-loss model, with its congruence and theory_loss), results, theory_weight and call.

Details

The total loss minimized for each candidate model is $$L = L_{fit} + \code{theory\_weight} \cdot L_{theory}$$ where $L_{fit}$ is the composite fit loss of specification_search $$L_{fit} = w_{rmsea}\max(0,(RMSEA-t_{rmsea})/0.03) + w_{cfi}\max(0,(t_{cfi}-CFI)/0.03) + w_{srmr}\max(0,(SRMR-t_{srmr})/0.03)$$ and the theory loss is $$L_{theory} = \code{tw\_move}\cdot p_{mis} + \code{tw\_drop}\cdot p_{drop} + \code{tw\_k}\cdot |k - k_{theory}|/k_{theory}$$ with $p_{mis}$ the proportion of retained items assigned to a factor that does not match their theoretical dimension, and $p_{drop}$ the proportion of theoretical items dropped from the model. Empirical factors are aligned to theoretical dimensions greedily by maximum item overlap (each factor inherits the most frequent theoretical label among its items), the same criterion used by the project's loading maps. The reported congruence is $1 - p_{mis}$ (1 = the model fully respects the theory). Default seeds include the theoretical structure itself as a starting point.

Warning. Like the fit-only search, this remains an EXPLORATORY device; cross-validate the chosen model on an independent sample.

References

MacCallum, R. C. (1986). Specification searches in covariance structure modeling. Psychological Bulletin, 100(1), 107–120.

Examples