Skip to contents

Bifactor Statistical Indices

Source: R/bifactor_indices.R
bifactor_indices.Rd

Computes the auxiliary bifactor statistical indices from a fitted bifactor lavaan model: ECV (explained common variance), PUC (percent of uncontaminated correlations), omega, omega hierarchical (\(\omega_H\)), omega hierarchical subscale (\(\omega_{HS}\)), the H construct-replicability index, and the item-level ECV (I-ECV). These indices help decide whether a multidimensional scale can be treated as essentially unidimensional and scored with a single total score (Rodriguez, Reise & Haviland, 2016).

Usage

bifactor_indices(fit, general = NULL, verbose = TRUE)

Arguments

fit

A fitted bifactor lavaan object (general factor + orthogonal specific factors).

general

Name of the general factor. If NULL (default) it is auto-detected as the latent variable that loads on all items.

verbose

Logical. If TRUE (default), prints the rounded indices to the console.

Value

A list with: overall (a one-row data frame with ECV, PUC, omega, omega_H and H_general), by_factor (per specific factor: ECV, omega_S, omega_HS, H) and by_item (general/specific loadings and I-ECV). Printed rounded to three decimals.

Details

The model must be a bifactor structure with one general factor loading on all items plus orthogonal specific factors. Standardized loadings are used. With general loadings \(g_i\), specific loadings \(s_i\) and errors \(e_i = 1 - g_i^2 - s_i^2\):

  • ECV = \(\sum g_i^2 / \sum (g_i^2 + s_i^2)\).

  • omega_H = \((\sum g_i)^2 / Var_{total}\), and omega adds the grouped specific variance to the numerator.

  • omega_HS (per specific factor) = \((\sum s_i)^2 / Var_{sub}\).

  • H = \(1/(1 + 1/\sum (\lambda_i^2/(1-\lambda_i^2)))\).

  • PUC = proportion of item correlations that are between items of different specific factors.

  • IECV_i = \(g_i^2 / (g_i^2 + s_i^2)\) (item purity toward G).

References

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

Examples

# Simulate data from a bifactor population (G + three specific factors)
pop <- '
G  =~ 0.6*x1 + 0.6*x2 + 0.6*x3 + 0.6*x4 + 0.6*x5 + 0.6*x6 +
      0.6*x7 + 0.6*x8 + 0.6*x9
S1 =~ 0.5*x1 + 0.5*x2 + 0.5*x3
S2 =~ 0.5*x4 + 0.5*x5 + 0.5*x6
S3 =~ 0.5*x7 + 0.5*x8 + 0.5*x9
'
set.seed(1)
dat <- lavaan::simulateData(pop, sample.nobs = 500, orthogonal = TRUE)

mod <- '
G  =~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9
S1 =~ x1 + x2 + x3
S2 =~ x4 + x5 + x6
S3 =~ x7 + x8 + x9
'
fit <- lavaan::cfa(mod, data = dat, orthogonal = TRUE, std.lv = TRUE)
bi <- bifactor_indices(fit)
#> Bifactor statistical indices (general factor: G )
#> Overall:
#>    ECV  PUC omega omega_H H_general
#>  0.577 0.75 0.793   0.637     0.717
#> 
#> By specific factor:
#>  Factor   ECV omega_S omega_HS     H
#>      S1 0.124   0.631    0.243 0.329
#>      S2 0.194   0.651    0.372 0.457
#>      S3 0.105   0.644    0.195 0.294
bi$by_item
#>   Item Factor   General  Specific     I_ECV
#> 1   x1     S1 0.4440820 0.3937189 0.5598970
#> 2   x2     S1 0.4915281 0.3610763 0.6495039
#> 3   x3     S1 0.4801611 0.3674213 0.6307009
#> 4   x4     S2 0.4163543 0.4813720 0.4279531
#> 5   x5     S2 0.3463718 0.4806573 0.3417999
#> 6   x6     S2 0.4511023 0.4407101 0.5116513
#> 7   x7     S3 0.5179359 0.2764372 0.7782912
#> 8   x8     S3 0.5476438 0.2910210 0.7797927
#> 9   x9     S3 0.4651366 0.4406203 0.5270474