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Reliability and common-variance coefficients for a factor solution

Source: R/efa_reliability.R
efa_reliability.Rd

Computes model-based reliability coefficients (McDonald's omega total, hierarchical, and subscale; standardized Cronbach's alpha; and the H index) together with the bifactor common-variance indices (explained common variance, ECV, and percent of uncontaminated correlations, PUC) for the general factor and each group factor, and returns them as a tidy, long-format table. The coefficients can be obtained from a Schmid-Leiman solution (efa_schmid_leiman() or psych::schmid()), an oblique efa_fit() (correlated-factors) solution, a lavaan single-factor, second-order, or bifactor fit, a raw bifactor loading matrix, or manually supplied components.

Usage

efa_reliability(
  model = NULL,
  coefficients = NULL,
  g_name = "g",
  group_names = NULL,
  factor_map = NULL,
  variance = c("correlation", "sums_load"),
  var_names = NULL,
  fac_names = NULL,
  g_load = NULL,
  s_load = NULL,
  u2 = NULL,
  cormat = NULL,
  pattern = NULL,
  Phi = NULL
)

Source

McDonald, R. P. (1978). Generalizability in factorable domains: Domain validity and generalizability. Educational and Psychological Measurement, 38, 75-79.

McDonald, R. P. (1985). Factor analysis and related methods. Hillsdale, NJ: Erlbaum.

McDonald, R. P. (1999). Test theory: A unified treatment. Mahwah, NJ: Erlbaum.

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-334.

Hancock, G. R., & Mueller, R. O. (2001). Rethinking construct reliability within latent variable systems. In R. Cudeck, S. du Toit, & D. Sörbom (Eds.), Structural equation modeling: Present and future (pp. 195-216). Lincolnwood, IL: Scientific Software International.

Bonifay, W. E., Reise, S. P., Scheines, R., & Meijer, R. R. (2015). When are multidimensional data unidimensional enough for structural equation modeling? An evaluation of the DETECT multidimensionality index. Structural Equation Modeling, 22, 504-516.

Reise, S. P., Scheines, R., Widaman, K. F., & Haviland, M. G. (2013). Multidimensionality and structural coefficient bias in structural equation modeling: A bifactor perspective. Educational and Psychological Measurement, 73, 5-26.

Rodriguez, A., Reise, S. P., & Haviland, M. G. (2016a). Applying bifactor statistical indices in the evaluation of psychological measures. Journal of Personality Assessment, 98, 223-237.

Rodriguez, A., Reise, S. P., & Haviland, M. G. (2016b). Evaluating bifactor models: Calculating and interpreting statistical indices. Psychological Methods, 21, 137-150.

Arguments

model

an efa_schmid_leiman(), schmid (psych::schmid()), efa_fit() (oblique), or lavaan object; a raw bifactor loading matrix (general factor first); or NULL to supply the components manually via g_load, s_load, u2, and var_names.

coefficients

character. An optional subset of the coefficients to return, any of "omega_total", "omega_hierarchical", "omega_subscale", "alpha", "H", "ECV", and "PUC". Default NULL returns all of them.

g_name

character. The name of the general factor in the lavaan solution. Only needed for a lavaan second-order or bifactor fit. Default is "g".

group_names

character. An optional vector of group names for a lavaan multiple-group fit. Its length must match the number of groups.

factor_map

matrix. A logical or 0/1 matrix indicating which variable corresponds to which group factor, with the same dimensions as the group loading matrix (cross-loadings are allowed). If NULL (default), each variable is assigned to the group factor on which it loads most strongly. Not used for lavaan input.

variance

character. The total-variance denominator for the coefficients: "correlation" (default) takes it from the correlation matrix (the observed-score omega, as in psych::omega()); "sums_load" uses the model-implied composite variance from the squared loading sums and the uniquenesses, which needs no correlation matrix and so is the way to score a bare loading matrix or manual components given without one. Some inputs fix the convention: lavaan is always model-implied and an oblique efa_fit() is always correlation-based.

var_names

character. Subtest names in the row order of the loadings. Only needed when model is NULL.

fac_names

character. An optional vector of group-factor names in the column order of the loadings. Taken from the input if NULL.

g_load

numeric. General-factor loadings. Only needed when model is NULL.

s_load

matrix. Group-factor loadings. Only needed when model is NULL.

u2

numeric. Uniquenesses. Only needed when model is NULL (or to override the communality-based default for a raw bifactor matrix).

cormat

matrix. A correlation matrix used when variance = "correlation". If NULL, it is taken from the input object or reconstructed from pattern and Phi where possible.

pattern

matrix. Pattern coefficients from an oblique solution, used with Phi to reconstruct cormat when model is NULL and no cormat is given.

Phi

matrix. Factor intercorrelations from an oblique solution, used with pattern.

Value

An object of class efa_reliability: a long-format data frame with one row per computed coefficient, with columns

coefficient

the coefficient name (e.g. "omega_total").

level

"general" for the general-factor row, "group" otherwise.

factor

the factor label ("g" for the general factor).

group

the group label, or NA for a single ungrouped solution.

value

the coefficient value.

Structurally undefined cells (for example ECV and PUC on a group factor) are omitted. The object carries a settings attribute (the total-variance convention used) and a kind attribute tagging each coefficient as a reliability coefficient or a common-variance index, and has a print() method.

Details

Coefficients

The reliability coefficients are McDonald's omegas (McDonald, 1978, 1985, 1999), standardized Cronbach's alpha (Cronbach, 1951), and the H index (construct replicability; Hancock & Mueller, 2001). The omegas give the share of true score variance in a unit-weighted composite: omega total the share due to all factors together, omega hierarchical the share due to the general factor only, and omega subscale the share due to the group factors (for the whole scale) or to the specific group factor (for a subscale composite). Alpha is computed from the standardized correlation matrix. The H index is the reliability of an optimally weighted composite; low values indicate a factor that is not well defined by its indicators.

The common-variance indices ECV and PUC (Bonifay et al., 2015; Reise et al., 2013; Rodriguez et al., 2016a, 2016b) describe the general factor and so are reported for the general factor only. The ECV is the share of the common variance that is explained by the general factor. The PUC is the proportion of correlations that reflect general-factor variance alone (those between indicators of different group factors); the higher it is, the more similar the general factor is to the single factor of a unidimensional model.

Input

An oblique efa_fit() object is scored as the correlated-factors model it is (having no general factor, it omits the bifactor indices – omega hierarchical, ECV, and PUC), and a bare loading matrix is read as a raw bifactor solution (general factor in the first column). For a correlated-factors efa_fit() solution variance is always "correlation". The indicator-to-factor correspondences come from factor_map when it is supplied; otherwise each variable is assigned to the group factor on which it loads most strongly. For lavaan input the composite variances are model-implied (variance is not used), and the coefficients are computed per group.

See also

efa_fit() for the solution these are computed from, and OMEGA(), the superseded function that returns these same coefficients in a wide, per-factor layout.

Other reliability coefficients: print.efa_reliability()

Examples

## From an oblique EFA (correlated-factors) solution. With no factor_map, each
## item is auto-assigned to its highest-loading factor.
efa_mod <- efa_fit(test_models$baseline$cormat, N = 500, n_factors = 3,
                   estimator = "PAF", rotation = "promax")
efa_reliability(efa_mod)
#> 
#> Total variance from the correlation matrix.
#> 
#> ── Reliability coefficients ────────────────────────────────────────────────────
#> 
#>      tot   sub  alpha    H
#> g   .883         .868
#> F1  .734  .734   .768  .760
#> F2  .680  .680   .763  .753
#> F3  .667  .667   .743  .738

## From a Schmid-Leiman solution, with an explicit indicator-to-factor map.
sl_mod <- efa_schmid_leiman(efa_mod, estimator = "PAF")
fc <- sl_mod$sl[, c("F1", "F2", "F3")] >= .2
efa_reliability(sl_mod, factor_map = fc)
#> 
#> Total variance from the correlation matrix.
#> 
#> ── Reliability coefficients ────────────────────────────────────────────────────
#> 
#>      tot  hier   sub  alpha    H
#> g   .883  .740  .125   .868  .842
#> F1  .769  .500  .269   .768  .463
#> F2  .763  .494  .270   .763  .472
#> F3  .744  .519  .225   .743  .408
#> 
#> ── Common-variance indices ─────────────────────────────────────────────────────
#> 
#>     ECV   PUC
#> g  .652  .706

## Request a subset of the coefficients only.
efa_reliability(sl_mod, factor_map = fc,
                coefficients = c("omega_total", "alpha"))
#> 
#> Total variance from the correlation matrix.
#> 
#> ── Reliability coefficients ────────────────────────────────────────────────────
#> 
#>      tot  alpha
#> g   .883   .868
#> F1  .769   .768
#> F2  .763   .763
#> F3  .744   .743

## From a lavaan bifactor solution.
# \donttest{
if (requireNamespace("lavaan", quietly = TRUE)) {
mod <- 'F1 =~ V1 + V2 + V3 + V4 + V5 + V6
        F2 =~ V7 + V8 + V9 + V10 + V11 + V12
        F3 =~ V13 + V14 + V15 + V16 + V17 + V18
        g =~ V1 + V2 + V3 + V4 + V5 + V6 + V7 + V8 + V9 + V10 + V11 + V12 +
             V13 + V14 + V15 + V16 + V17 + V18'
fit <- lavaan::cfa(mod, sample.cov = test_models$baseline$cormat,
                   sample.nobs = 500, estimator = "ml", orthogonal = TRUE)
efa_reliability(fit, g_name = "g")
}
#> 
#> Total variance from the model-implied composite variance.
#> 
#> ── Reliability coefficients ────────────────────────────────────────────────────
#> 
#>      tot  hier   sub  alpha    H
#> g   .883  .765  .118   .868  .849
#> F1  .748  .570  .177   .742  .379
#> F2  .767  .501  .265   .762  .482
#> F3  .769  .494  .274   .768  .473
#> 
#> ── Common-variance indices ─────────────────────────────────────────────────────
#> 
#>     ECV   PUC
#> g  .672  .706
# }