McDonald's omega

OMEGA() has been superseded by efa_reliability(), which is the recommended interface going forward. It remains available and unchanged so existing code keeps working.

This function finds omega total, hierarchical, and subscale, as well as additional model-based indices of interpretive relevance (H index, ECV, PUC) from a Schmid-Leiman (SL) solution or lavaan single factor, second-order (see below), or bifactor solution. The SL-based omegas can either be found from a psych::schmid(), efa_schmid_leiman(), or, in a more flexible way, by leaving model = NULL and specifying additional arguments. By setting the type argument, results from psych::omega() can be reproduced.

Usage

OMEGA(
  model = NULL,
  type = c("EFAtools", "psych"),
  g_name = "g",
  group_names = NULL,
  add_ind = TRUE,
  factor_corres = NULL,
  var_names = NULL,
  fac_names = NULL,
  g_load = NULL,
  s_load = NULL,
  u2 = NULL,
  cormat = NULL,
  pattern = NULL,
  Phi = NULL,
  variance = c("correlation", "sums_load")
)

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.

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.

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—A Festschrift in honor of Karl Jöreskog (pp. 195–216). Lincolnwood, IL: Scientific Software International.

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.

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.

Gignac, G. E. (2014). On the Inappropriateness of Using Items to Calculate Total Scale Score Reliability via Coefficient Alpha for Multidimensional Scales. European Journal of Psychological Assessment, 30, 130-139.

Arguments

model: class efa_schmid_leiman(), class schmid, or class lavaan object. That is, an output object from efa_schmid_leiman() or psych::schmid(), or a lavaan fit object with a single factor, second-order, or bifactor solution. If of class lavaan, only g_name needs to be specified additionally. If of class efa_schmid_leiman() or schmid, only the arguments factor_corres and cormat need to be specified additionally.
type: character. Either "EFAtools" (default) or "psych" (see details)
g_name: character. The name of the general factor from the lavaan solution. This needs only be specified if model is a lavaan second-order or bifactor solution. Default is "g".
group_names: character. An optional vector of group names. The length must correspond to the number of groups for which the lavaan model was fitted.
add_ind: logical. Whether additional indices (H index, ECV, PUC) should be calculated or not (see details for these indices). If FALSE, only omegas are returned. Default is TRUE.
factor_corres: matrix. A logical matrix or a numeric matrix containing 0's and 1's that indicates which variable corresponds to which group factor. Must have the same dimensions as the matrix of group factor loadings from the SL solution. Cross-loadings are allowed here. See examples for use.
var_names: character. A vector with subtest names in the order of the rows from the SL solution. This needs only be specified if model is left NULL.
fac_names: character. An optional vector of group factor names in the order of the columns of the SL solution. If left NULL, names of the group factors from the entered solution are taken.
g_load: numeric. A vector of general factor loadings from an SL solution. This needs only be specified if model is left NULL.
s_load: matrix. A matrix of group factor loadings from an SL solution. This needs only be specified if model is left NULL.
u2: numeric. A vector of uniquenesses from an SL solution. This needs only be specified if model is left NULL.
cormat: matrix. A correlation matrix to be used when variance = "correlation". If left NULL and an efa_schmid_leiman() output is entered in model, the correlation matrix is taken from the output. If left NULL and a psych::schmid() output is entered, the correlation matrix will be found based on the pattern matrix and Phi from the psych::schmid() output using psych::factor.model(). If left NULL and model is also left NULL, the correlation matrix is found based on the pattern matrix and Phi entered. However, if the correlation matrix is available, cormat should be specified instead of Phi and pattern.
pattern: matrix. Pattern coefficients from an oblique factor solution. This needs only be specified if model is left NULL, variance = "correlation" and cormat is also left NULL.
Phi: matrix. Factor intercorrelations from an oblique factor solution. This needs only be specified if model is left NULL, variance = "correlation" and cormat is also left NULL.
variance: character. If "correlation" (default), then total variances for the whole scale as well as for the subscale composites are calculated based on the correlation matrix. If "sums_load", then total variances are calculated using the squared sums of general factor loadings and group factor loadings and the sum of uniquenesses (see details).

Value

If found for an SL or lavaan second-order or bifactor solution without multiple groups: A matrix with omegas for the whole scale and for the subscales and (only if add_ind = TRUE) with the H index, ECV, and PUC.

tot: Omega total.
hier: Omega hierarchical.
sub: Omega subscale.
H: H index.
ECV: Explained common variance.
PUC: Percent of uncontaminated correlations.

If found for a lavaan single factor solution without multiple groups: A (named) vector with omega total and (if add_ind = TRUE) the H index for the single factor.

If found for a lavaan output from a multiple group analysis: A list containing the output described above for each group.

Details

What this function does

This function calculates McDonald's omegas (McDonald, 1978, 1985, 1999), the H index (Hancock & Mueller, 2001), the explained common variance (ECV; Rodriguez et al., 2016a, 2016b), and the percent of uncontaminated correlations (PUC; Bonifay et al., 2015; Reise et al., 2013).

All types of omegas (total, hierarchical, and subscale) are calculated for the general factor as well as for the subscales / group factors (see, e.g., Gignac, 2014; Rodriguez et al., 2016a, 2016b). Omegas refer to the correlation between a factor and a unit-weighted composite score and thus the true score variance in a unit-weighted composite based on the respective indicators. Omega total is the total true score variance in a composite. Omega hierarchical is the true score variance in a composite that is attributable to the general factor, and omega subscale is the true score variance in a composite attributable to all subscales / group factors (for the whole scale) or to the specific subscale / group factor (for subscale composites).

Accordingly, on a subscale row the hier column reports the share of that subscale's composite variance due to the general factor and the sub column the share due to the subscale-specific factor; the latter corresponds to the omega hierarchical subscale of Rodriguez et al. (2016a, 2016b).

The H index (also construct reliability or replicability index) is the correlation between an optimally-weighted composite score and a factor (Hancock & Mueller, 2001; Rodriguez et al., 2016a, 2016b). It, too, can be calculated for the whole scale / general factor as well as for the subscales / group factors. Low values indicate that a latent variable is not well defined by its indicators.

The ECV (Rodriguez et al., 2016a, 2016b) is the ratio of the variance explained by the general factor and the variance explained by the general factor and the group factors.

The PUC (Bonifay et al., 2015; Reise et al., 2013, Rodriguez et al., 2016a, 2016b) refers to the proportion of correlations in the underlying correlation matrix that is not contaminated by variance of both the general factor and the group factors (i.e., correlations between indicators from different group factors, which reflect only general factor variance). The higher the PUC, the more similar a general factor from a multidimensional model will be to the single factor from a unidimensional model.

How to use this function

If model is a lavaan second-order or bifactor solution, only the name of the general factor from the lavaan model needs to be specified additionally with the g_name argument. It is then determined whether this general factor is a second-order factor (second-order model with one second-order factor assumed) or a breadth factor (bifactor model assumed). Please note that this function only works for second-order models if they contain no more than one second-order factor. In case of a second-order solution, a Schmid-Leiman transformation is performed on the first- and second-order loadings and omega coefficients are obtained from the transformed (orthogonalized) solution (see efa_schmid_leiman() for more information on Schmid-Leiman transformation). There is also the possibility to enter a lavaan single factor solution. In this case, g_name is not needed. Finally, if a solution from a lavaan multiple group analysis is entered, the indices are computed for each group. For lavaan input the composite variances entering the omegas are model-implied (computed from the fitted loadings and residual variances), so the coefficients coincide with the observed-score versions when the model fits perfectly. The type argument is not evaluated if model is of class lavaan.

If model is of class efa_schmid_leiman() or psych::schmid() only the type and, depending on the type (see below), the factor_corres arguments need to be specified additionally. If model is of class psych::schmid() and variance = "correlation" (default), it is recommended to also provide the original correlation matrix in cormat to get more accurate results. Otherwise, the correlation matrix will be found based on the pattern matrix and Phi from the psych::schmid() output using the psych::factor.model() function.

If model = NULL, the arguments type, factor_corres (depending on the type, see below), var_names, g_load, s_load, and u2 and either cormat (recommended) or Phi and pattern need to be specified. If Phi and pattern are specified instead of cormat, the correlation matrix is found using the psych::factor.model() function.

The only difference between type = "EFAtools" and type = "psych" is the determination of variable-to-factor correspondences. type = "psych" reproduces the psych::omega() results, where variable-to-factor correspondences are found by taking the highest group factor loading for each variable as the relevant group factor loading. To do this, factor_corres must be left NULL.

The variance argument controls how the total variances (for the whole scale and the subscale composites) are computed. "correlation" (default) finds them from the correlation matrix – the observed-variance form of omega, which reproduces psych::omega(). "sums_load" instead uses the model-implied variances from the squared sums of the general and group factor loadings and the sum of the uniquenesses, so the whole-scale omega total is McDonald's model-implied total and partitions exactly into omega hierarchical plus omega subscale. The two settings agree on the whole-scale omega total and omega hierarchical up to model misfit, and differ mainly in the whole-scale omega subscale, which counts all group-factor variance under "sums_load" but only the assigned subscale composites under "correlation". On the subscale rows the two conventions agree when simple structure is well-achieved.

Examples

# \donttest{
## Use with lavaan outputs
if (requireNamespace("lavaan", quietly = TRUE)) {

# Create and fit bifactor model in lavaan (assume all variables have SDs of 1)
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_bi <- lavaan::cfa(mod, sample.cov = test_models$baseline$cormat,
                      sample.nobs = 500, estimator = "ml", orthogonal = TRUE)

# Compute omegas and additional indices for bifactor solution
OMEGA(fit_bi, g_name = "g")

# Compute only omegas
OMEGA(fit_bi, g_name = "g", add_ind = FALSE)

# Create and fit second-order model in lavaan (assume all variables have SDs of 1)
mod <- 'F1 =~ V1 + V2 + V3 + V4 + V5 + V6
        F2 =~ V7 + V8 + V9 + V10 + V11 + V12
        F3 =~ V13 + V14 + V15 + V16 + V17 + V18
        g =~ F1 + F2 + F3'
fit_ho <- lavaan::cfa(mod, sample.cov = test_models$baseline$cormat,
                      sample.nobs = 500, estimator = "ml")

# Compute omegas and additional indices for second-order solution
OMEGA(fit_ho, g_name = "g")
}
#> ℹ The specified general factor is a second-order factor; omegas are computed on
#>   the Schmid-Leiman transformed second-order solution.
#> Omega total, omega hierarchical, omega subscale, H index, explained common
#> variance (ECV), and percent of uncontaminated correlations (PUC) for the
#> general factor (top row) and omegas and H index for the group factors:
#> 
#>      tot  hier   sub     H   ECV   PUC
#> g  0.882 0.764 0.118 0.848 0.684 0.706
#> F1 0.744 0.549 0.195 0.361            
#> F2 0.764 0.504 0.259 0.448            
#> F3 0.768 0.505 0.263 0.454            
# }

## Use with an output from the SL function, with type EFAtools
efa_mod <- efa_fit(test_models$baseline$cormat, N = 500, n_factors = 3,
                   estimator = "PAF", rotation = "promax")
sl_mod <- efa_schmid_leiman(efa_mod, estimator = "PAF")

# Two examples how to specify the indicator-to-factor correspondences:

# Based on a specific salience threshold for the loadings (here: .20):
factor_corres_1 <- sl_mod$sl[, c("F1", "F2", "F3")] >= .2

# Or more flexibly (could also be TRUE and FALSE instead of 0 and 1):
factor_corres_2 <- matrix(c(rep(0, 12), rep(1, 6), rep(0, 6), rep(1, 6),
                         rep(0, 6), rep(1, 6), rep(0, 12)), ncol = 3,
                         byrow = FALSE)

OMEGA(sl_mod, type = "EFAtools", factor_corres = factor_corres_1)
#> Omega total, omega hierarchical, omega subscale, H index, explained common
#> variance (ECV), and percent of uncontaminated correlations (PUC) for the
#> general factor (top row) and omegas and H index for the group factors:
#> 
#>      tot  hier   sub     H   ECV   PUC
#> g  0.883 0.740 0.125 0.842 0.652 0.706
#> F1 0.769 0.500 0.269 0.463            
#> F2 0.763 0.494 0.270 0.472            
#> F3 0.744 0.519 0.225 0.408            

## Use with an output from the psych::schmid function, with type psych for
## OMEGA
schmid_mod <- psych::schmid(test_models$baseline$cormat, nfactors = 3,
                            n.obs = 500, fm = "pa", rotate = "Promax")
#> Loading required namespace: GPArotation
# Find correlation matrix from phi and pattern matrix from psych::schmid output
OMEGA(schmid_mod, type = "psych")
#> Omega total, omega hierarchical, omega subscale, H index, explained common
#> variance (ECV), and percent of uncontaminated correlations (PUC) for the
#> general factor (top row) and omegas and H index for the group factors:
#> 
#>       tot  hier   sub     H   ECV   PUC
#> g   0.883 0.750 0.122 0.845 0.661 0.706
#> F1* 0.769 0.498 0.272 0.466            
#> F2* 0.765 0.494 0.271 0.473            
#> F3* 0.745 0.543 0.202 0.379            
# Use specified correlation matrix
OMEGA(schmid_mod, type = "psych", cormat = test_models$baseline$cormat)
#> Omega total, omega hierarchical, omega subscale, H index, explained common
#> variance (ECV), and percent of uncontaminated correlations (PUC) for the
#> general factor (top row) and omegas and H index for the group factors:
#> 
#>       tot  hier   sub     H   ECV   PUC
#> g   0.883 0.750 0.122 0.845 0.661 0.706
#> F1* 0.769 0.498 0.272 0.466            
#> F2* 0.764 0.494 0.270 0.473            
#> F3* 0.745 0.543 0.202 0.379            

## Manually specify components (useful if omegas should be computed for a SL
## or bifactor solution found with another program)
## As an example, we extract the elements from an SL output here. This gives
## the same results as in the second example above.

efa_mod <- efa_fit(test_models$baseline$cormat, N = 500, n_factors = 3,
                   estimator = "PAF", rotation = "promax")
sl_mod <- efa_schmid_leiman(efa_mod, estimator = "PAF")

factor_corres <- matrix(c(rep(0, 12), rep(1, 6), rep(0, 6), rep(1, 6),
                        rep(0, 6), rep(1, 6), rep(0, 12)), ncol = 3,
                        byrow = FALSE)

OMEGA(model = NULL, type = "EFAtools", var_names = rownames(sl_mod$sl),
      g_load = sl_mod$sl[, "g"], s_load = sl_mod$sl[, c("F1", "F2", "F3")],
      u2 = sl_mod$sl[, "u2"], cormat = test_models$baseline$cormat,
      factor_corres = factor_corres)
#> Omega total, omega hierarchical, omega subscale, H index, explained common
#> variance (ECV), and percent of uncontaminated correlations (PUC) for the
#> general factor (top row) and omegas and H index for the group factors:
#> 
#>     tot  hier   sub     H   ECV   PUC
#> g 0.883 0.740 0.125 0.842 0.652 0.706
#> 1 0.769 0.500 0.269 0.463            
#> 2 0.763 0.494 0.270 0.472            
#> 3 0.744 0.519 0.225 0.408