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Bifactor Model Dimensionality and Reliability Indices

Source: R/bifactor_indices.R
bifactor_indices.Rd

Computes the indices used to judge whether a multidimensional scale is nonetheless unidimensional enough to score as a single total (Rodriguez, Reise, and Haviland, 2016): the explained common variance (ECV), coefficient omega and omega hierarchical (omega_H) for the general factor, omega hierarchical subscale (omega_HS) for each group factor, the percentage of uncontaminated correlations (PUC), and coefficient H, the construct reliability (maximal reliability) of Hancock and Mueller (2001). The input is a fitted bifactor model in which one general factor loads on every item and each item loads on exactly one orthogonal group factor.

Usage

bifactor_indices(fit, general = NULL)

Arguments

fit

A fitted bifactor lavaan model: one general factor on all items plus orthogonal group factors, each item on one group factor. The factors must be orthogonal (the bifactor specification).

general

Optional name of the general factor. When NULL (default) the general factor is detected as the one that loads on every item.

Value

A data.frame (class dmar_tbl) with one row per factor (the general factor first, then each group factor) and columns factor, ECV, omega, omega_H, omega_HS, PUC, and H. Quantities that do not apply to a row are NA (for example omega_H and PUC on a group factor). The "improper" attribute flags a Heywood solution.

Details

Let \(\lambda^g_i\) be item \(i\)'s standardized loading on the general factor, \(\lambda^s_i\) its loading on its group factor, and \(\theta_i\) its standardized residual variance. With orthogonal factors the overall ECV is \(\sum_i (\lambda^g_i)^2 / \sum_i [(\lambda^g_i)^2 + (\lambda^s_i)^2]\); omega and omega_H share the total-score variance \((\sum_i \lambda^g_i)^2 + \sum_g (\sum_{i \in g} \lambda^s_i)^2 + \sum_i \theta_i\) as denominator, with the general part \((\sum_i \lambda^g_i)^2\) in the numerator of omega_H. Each group factor's omega_HS uses the analogous numerator and that subscale's own total variance. PUC is one minus the share of item pairs that fall within the same group factor. Coefficient H is computed from \(\sum \lambda^2 / (1 - \lambda^2)\) on the relevant loadings (see reliability_H). High ECV and PUC with a high omega_H support scoring a single total; substantial subscale omega_HS argues for reporting subscales as well.

A bifactor model that is over-parameterized for the data frequently yields an improper (Heywood) solution – a standardized loading outside \([-1, 1]\) or a negative residual variance – whose indices are not trustworthy. When that happens the indices are still returned but a warning is issued and the "improper" attribute is set to TRUE.

References

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). Scientific Software International.

Reise, S. P. (2012). The rediscovery of bifactor measurement models. Multivariate Behavioral Research, 47(5), 667–696. doi:10.1080/00273171.2012.715555

Rodriguez, A., Reise, S. P., & Haviland, M. G. (2016). Evaluating bifactor models: Calculating and interpreting statistical indices. Psychological Methods, 21(2), 137–150.

See also

reliability_omega, reliability_H.

Other multivariate and latent variable methods: average_variance_extracted(), cfa_1(), cfa_2(), cfa_k(), ci_eigenvalue(), common_method_marker(), common_method_single_factor(), dmacs(), ecvi(), htmt(), irt_grm(), irt_information(), measurement_alignment(), measurement_invariance(), procrustes_phi(), simple_structure()

Author

Ken Kelley kkelley@nd.edu

Examples

# Nine items: one general factor and three orthogonal group factors.
set.seed(113)
n <- 600
g <- rnorm(n); grp <- list(rnorm(n), rnorm(n), rnorm(n))
X <- vapply(1:9, function(i)
  0.5 * g + 0.5 * grp[[ceiling(i / 3)]] + sqrt(0.5) * rnorm(n), numeric(n))
colnames(X) <- paste0("x", 1:9)
model <- "g  =~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9
          f1 =~ x1 + x2 + x3
          f2 =~ x4 + x5 + x6
          f3 =~ x7 + x8 + x9"
fit <- lavaan::cfa(model, data = as.data.frame(X),
                   orthogonal = TRUE, std.lv = TRUE)
bifactor_indices(fit)
#>  factor ECV   omega omega_H omega_HS PUC  H    
#>  g      0.547 0.858 0.673   <NA>     0.75 0.769
#>  f1     0.609 0.727 <NA>    0.28     <NA> 0.412
#>  f2     0.543 0.756 <NA>    0.345    <NA> 0.477
#>  f3     0.49  0.743 <NA>    0.377    <NA> 0.506