bifactor_indices: Bifactor Model Dimensionality and Reliability Indices

Description

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)

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.

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.

Author

Ken Kelley [email protected]

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. tools:::Rd_expr_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.

Examples

Run this code

# 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)

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