Multigroup exploratory factor analysis

Fit an exploratory factor analysis in each of several groups at a common number of factors and bring the per-group solutions into one shared orientation so their loadings can be compared. Each group is fitted with efa_fit(); the solutions are then aligned either to a symmetric consensus target or to a chosen reference group (see Alignment).

Usage

efa_group(
  x,
  groups = NULL,
  n_factors,
  N = NA,
  reference_group = NULL,
  b_boot = 0L,
  ci = 0.95,
  seed = NULL,
  delta = 0.1,
  invariance = FALSE,
  ...
)

Arguments

x: A data frame or matrix of raw data (with groups), or a named list of per-group data sets – either raw data frames/matrices or correlation matrices (all of one kind).
groups: A vector with one value per row of x, giving each row's group. Only used when x is a single raw data set; leave NULL when x is a list. Rows with a missing group value are dropped with a warning.
n_factors: numeric. The common number of factors extracted in every group.
N: numeric. The number of observations per group, used only for correlation-matrix input: either a single value applied to all groups or one value per group. Ignored for raw data, where N is taken from each group's data. Default is NA.
reference_group: The group to align the others to (a group name or an integer index). If NULL (default), orthogonal and unrotated solutions use the symmetric consensus target; oblique solutions fall back to the first group as reference. Supplying a value forces the reference alignment.
b_boot: numeric. The number of non-parametric bootstrap replicates used to form percentile confidence intervals for the between-group Tucker congruences. 0 (the default) skips the bootstrap and returns the congruence point estimates only. Bootstrapping requires raw data; it is skipped with a warning for correlation-matrix input.
ci: numeric. The confidence level for the bootstrap congruence intervals, a single value in (0, 1). Default is 0.95.
seed: numeric or NULL. An optional seed making the analysis reproducible. It covers the per-group fits – whose rotation may draw random starts – whether or not a bootstrap is run, and additionally makes the bootstrap independent of the number of parallel workers (the replicate fits run with future_lapply(), for which a parallel plan can be set via future::plan()). When supplied, the caller's random-number stream is restored afterwards, leaving no side effect. Default is NULL.
delta: numeric. The salience threshold for the per-item loading-difference flag table: an item's loading on a factor is flagged for a group pair when the groups' aligned loadings differ by at least delta in absolute value. This is a descriptive salience heuristic, not a significance test; common alternatives are 0.15 and 0.20. The threshold applies to whatever loading metric the chosen rotation produces (pattern coefficients for an oblique rotation). 0 flags every cell. Default is 0.1.
invariance: logical. Whether to add an approximate-invariance verdict per factor and group pair from the Lorenzo-Seva and ten Berge (2006) congruence bands (see Value). Default is FALSE.
...: Additional arguments passed to efa_fit() for every group (for example estimator, rotation, cor_method, or an estimate_control() / rotate_control() to select a preset).

Value

An object of class efa_group, a list containing:

loadings: A named list of the aligned per-group loading matrices. Their columns match the columns of target in order and sign.
target: The alignment target: the symmetric consensus target, or the reference group's own loadings.
Phi: A named list of the aligned per-group factor intercorrelations for an oblique rotation; NULL otherwise.
congruence: Tucker congruence between the aligned group loadings, a list with: matrices, a nested list whose [[g]][[h]] element is the factor-by-factor congruence matrix between the aligned loadings of groups g and h; matched, a groups-by-groups-by-factors array of the matched-factor congruences (the diagonal of each pairwise matrix); and degenerate, a groups-by-groups logical matrix flagging pairs whose congruence is undefined (for example, a near-zero factor), for which the corresponding entries are NA. When b_boot > 0 (raw data), three further elements are added: matched_se, the bootstrap standard error of each matched congruence; matched_ci, a list of lower and upper percentile confidence limits (each a groups-by-groups-by-factors array); and n_boot, the number of bootstrap replicates the intervals are based on.
diffs: A data frame with one row per group pair summarising the differences between their aligned loadings: the mean, median, minimum, and maximum absolute difference, the root-mean-square difference (rmse), and n_flagged, the number of loading cells whose absolute difference reaches delta.
flags: A data frame with one row per group pair, item, and factor giving the signed loading difference (diff), its absolute value (abs_diff), and flagged, whether it reaches delta. When a bootstrap was run (b_boot > 0, raw data), ci_lower, ci_upper, and ci_excludes_0 add the percentile confidence interval for the difference and whether it excludes zero; otherwise these are NA.
invariance: When invariance = TRUE, a data frame with one row per group pair and factor giving the matched Tucker congruence (phi), its bootstrap CI lower bound (phi_lower, NA without a bootstrap), and an approximate-invariance verdict based on the Lorenzo-Seva and ten Berge (2006) similarity bands: phi >= 0.95 is "equal" and [0.85, 0.95) is "fair"; congruences < 0.85, below their bands, are labelled "incongruent". The verdict is read from phi_lower when a bootstrap is available (conservative) and from phi otherwise. NULL when invariance = FALSE.
efa: The named list of per-group efa_fit() objects (each retains its own diagnostics, e.g. heywood).
alignment: The alignment result: the consensus object (see efa_procrustes()), or a list with the reference group, the target, and the per-group Procrustes results. On the consensus path this is the raw record of the Generalized Procrustes iteration, so its target and aligned_loadings are in the orientation that iteration converged to, before the gauge described under Alignment is applied; the gauged matrices are target and loadings above.
settings: A list of the settings used, including the per-group N, the alignment method, the group that seeded the consensus frame (alignment_start, NULL on the reference path), the orientation the shared frame was put in (gauge: the rotation's own name, "principal_axes", or "identity" for a single factor; NULL on the reference path), the rotation, the estimator, the input type, and whether a bootstrap is available (can_bootstrap, FALSE for correlation-matrix input).

Details

Input

Groups can be supplied in two ways: raw data together with a grouping vector (x a data frame or matrix, groups one value per row), or a named list of per-group data sets in x (with groups left NULL). The list may hold raw data frames or correlation matrices (supply N), but not a mix of the two. All groups must contain the same items in the same order; a different item set or order is an error rather than being silently reordered.

Every group is fitted at the same n_factors (a common-\(k\) multigroup model). Extra arguments in ... (for example estimator, rotation, cor_method, or an estimate_control() / rotate_control() carrying the tuning knobs) are forwarded unchanged to each efa_fit() call, so the estimator and rotation are common to all groups.

The \(k\)-factor model must be identified for the shared item set: its degrees of freedom \(((p - k)^2 - (p + k)) / 2\) must be non-negative. Unlike a single efa_fit() fit – which only warns on an under-identified model – a multigroup fit aborts, because a shared alignment target across an under-identified group is not interpretable.

Alignment

A factor solution is identified only up to a rotation of its factors, so the per-group solutions must be brought into a common orientation before their loadings can be compared. Two strategies are available and are chosen automatically:

Consensus (the default for orthogonal rotations and for unrotated solutions): a symmetric Generalized Procrustes Analysis target is built across all groups (Gower, 1975), and every group's loadings are rotated to it. The consensus frame is only identified up to a global rotation of its factors, so it is then rotated into a fixed gauge and the same transform is applied to every group. The gauge is the simple structure of the rotation that was requested, evaluated on the consensus target itself, so the shared loadings are in the same kind of frame as the per-group solutions they summarise. Where no criterion identifies a frame – an unrotated solution, and a "bifactorT" request with two factors, whose criterion is identically zero with a single group factor – the principal-axes orientation of the target is used instead. Either way the columns are ordered by decreasing sum of squares and signed by their column sums, and the shared orientation, and hence every reported congruence, difference, and flag, is independent of the order in which the groups are supplied.
Reference: every group's loadings are aligned by Procrustes rotation to one reference group's loadings, which are kept fixed. This path is used when reference_group is given, and is used automatically for oblique rotations because the consensus iteration is not defined for oblique transforms with more than one factor. When an oblique rotation triggers the reference path without an explicit reference_group, the first group is used and a message reports this; the requested rotation is never silently changed.

In both cases the returned per-group loadings share the column order and sign of the returned target.

Comparing the aligned loadings

Because the per-group loadings share one orientation, they can be compared cell by cell. efa_group() reports a per-pair summary of their differences (diffs) and a per-item, per-factor flag table (flags) marking cells whose absolute difference reaches delta. delta is a descriptive salience heuristic, not a significance test; a bootstrap (b_boot > 0) additionally reports whether each difference's confidence interval excludes zero. With invariance = TRUE, a per-factor verdict grades the matched Tucker congruence by the Lorenzo-Seva and ten Berge (2006) similarity bands – "equal" (>= .95) and "fair" ([.85, .95)) – labelling weaker congruences "incongruent"; when a bootstrap is available the verdict uses the congruence CI lower bound, so a factor is judged "equal" only if even the lower bound clears the band.

References

Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman & Hall.

Gower, J. C. (1975). Generalized Procrustes analysis. Psychometrika, 40, 33-51. doi: 10.1007/BF02291478

Lorenzo-Seva, U., and ten Berge, J. M. F. (2006). Tucker's congruence coefficient as a meaningful index of factor similarity. Methodology, 2, 57-64. doi: 10.1027/1614-2241.2.2.57

Examples

# Raw data split by a grouping vector (unrotated, consensus alignment)
g <- rep(c("g1", "g2"), length.out = nrow(GRiPS_raw))
mg <- efa_group(GRiPS_raw, groups = g, n_factors = 1)
#> ℹ `x` is not a correlation matrix; computing correlations from the raw data.
#> ℹ `x` is not a correlation matrix; computing correlations from the raw data.
mg$loadings
#> $g1
#>                  F1
#> fun       0.7742597
#> friends   0.8561209
#> enjoy     0.8635188
#> hurt      0.7286359
#> part      0.8183418
#> commonly  0.8379167
#> chances   0.7772425
#> attracted 0.8410818
#> 
#> $g2
#>                  F1
#> fun       0.8156701
#> friends   0.8437139
#> enjoy     0.8842662
#> hurt      0.8035149
#> part      0.8161558
#> commonly  0.8247201
#> chances   0.8003353
#> attracted 0.8531488
#> 

# Per-pair difference summary and the per-item salience-flag table
mg$diffs
#>   group_1 group_2 mean_abs_diff median_abs_diff min_abs_diff max_abs_diff
#> 1      g1      g2    0.02499828      0.01697205  0.002185992   0.07487899
#>         rmse n_flagged
#> 1 0.03309813         0
mg$flags
#>   group_1 group_2 indicator factor         diff    abs_diff flagged ci_lower
#> 1      g1      g2       fun     F1 -0.041410308 0.041410308   FALSE       NA
#> 2      g1      g2   friends     F1  0.012407057 0.012407057   FALSE       NA
#> 3      g1      g2     enjoy     F1 -0.020747446 0.020747446   FALSE       NA
#> 4      g1      g2      hurt     F1 -0.074878986 0.074878986   FALSE       NA
#> 5      g1      g2      part     F1  0.002185992 0.002185992   FALSE       NA
#> 6      g1      g2  commonly     F1  0.013196661 0.013196661   FALSE       NA
#> 7      g1      g2   chances     F1 -0.023092808 0.023092808   FALSE       NA
#> 8      g1      g2 attracted     F1 -0.012067010 0.012067010   FALSE       NA
#>   ci_upper ci_excludes_0
#> 1       NA            NA
#> 2       NA            NA
#> 3       NA            NA
#> 4       NA            NA
#> 5       NA            NA
#> 6       NA            NA
#> 7       NA            NA
#> 8       NA            NA

# \donttest{
# Percentile bootstrap confidence intervals for the between-group congruences, with an
# approximate-invariance verdict read conservatively off the congruence CI lower bound
mg_ci <- efa_group(GRiPS_raw, groups = g, n_factors = 1, b_boot = 100, seed = 42,
                   invariance = TRUE)
#> ℹ `x` is not a correlation matrix; computing correlations from the raw data.
#> ℹ `x` is not a correlation matrix; computing correlations from the raw data.
mg_ci$congruence$matched_ci
#> $lower
#> , , F1
#> 
#>           g1        g2
#> g1 1.0000000 0.9973991
#> g2 0.9973991 1.0000000
#> 
#> 
#> $upper
#> , , F1
#> 
#>           g1        g2
#> g1 1.0000000 0.9996716
#> g2 0.9996716 1.0000000
#> 
#> 
mg_ci$invariance
#>   group_1 group_2 factor       phi phi_lower verdict
#> 1      g1      g2     F1 0.9994099 0.9973991   equal

# A named list of correlation matrices sharing the same items, common
# three-factor model, orthogonal rotation -> symmetric consensus target
bands <- list(age_6_8 = WJIV_ages_6_8$cormat, age_14_19 = WJIV_ages_14_19$cormat)
Ns <- c(WJIV_ages_6_8$N, WJIV_ages_14_19$N)
efa_group(bands, n_factors = 3, N = Ns, rotation = "varimax")
#> ── Multigroup exploratory factor analysis ──────────────────────────────────────
#> 
#> 2 groups (age_6_8, age_14_19) · 3 factors · PAF extraction · varimax rotation
#> Aligned to a symmetric consensus target.
#> N: age_6_8 = 825, age_14_19 = 1685
#> 
#> ── Factor congruence ───────────────────────────────────────────────────────────
#> 
#> Tucker's congruence between the aligned group loadings (matched factors):
#> 
#>                        F1    F2    F3
#> age_6_8 vs age_14_19  .981  .989  .995
#> 
#> ── Loading differences ─────────────────────────────────────────────────────────
#> 
#> Absolute loading differences by group pair (salience threshold |Δ| ≥ .1):
#> 
#>                       mean  median   min   max  rmse  flagged
#> age_6_8 vs age_14_19  .048    .033  .001  .206  .065   20/141

# An oblique rotation aligns to a reference group (reported via a message)
efa_group(bands, n_factors = 3, N = Ns, rotation = "promax")
#> Oblique rotations are aligned to a reference group, not a symmetric consensus
#> target.
#> ℹ The consensus target is undefined for oblique rotations with more than one
#>   factor.
#> ℹ Group "age_6_8" is used as the reference; set `reference_group` to choose
#>   another.
#> ── Multigroup exploratory factor analysis ──────────────────────────────────────
#> 
#> 2 groups (age_6_8, age_14_19) · 3 factors · PAF extraction · promax rotation
#> Aligned to reference group age_6_8.
#> N: age_6_8 = 825, age_14_19 = 1685
#> 
#> ── Factor congruence ───────────────────────────────────────────────────────────
#> 
#> Tucker's congruence between the aligned group loadings (matched factors):
#> 
#>                        F1    F2    F3
#> age_6_8 vs age_14_19  .976  .978  .981
#> 
#> ── Loading differences ─────────────────────────────────────────────────────────
#> 
#> Absolute loading differences by group pair (salience threshold |Δ| ≥ .1):
#> 
#>                       mean  median   min   max  rmse  flagged
#> age_6_8 vs age_14_19  .066    .058  .000  .197  .082   32/141
# }