OMEGA: McDonald's omega

In EFAtools: Fast and Flexible Implementations of Exploratory Factor Analysis Tools

McDonald's omega

Description

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, SL, 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")
)

Arguments

model	class SL, class schmid, or class lavaan object. That is, an output object from SL 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 SL 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 SL 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).

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; Sijtsma, 2009), 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).

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 / grouup factors. Low values indicate that a latent variable is not well defined by its indicators.

The ECV (Sijtsma, 2009, 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 coefficents are obtained from the transformed (orthogonalized) solution (see SL 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. The type argument is not evaluated if model is of class lavaan.

If model is of class SL 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 calculation of the total variance (for the whole scale as well as the subscale composites) can also be controlled in this function using the variance argument. For both types—"EFAtools" and "psych" —variance is set to "correlation" by default, which means that total variances are found using the correlation matrix. If variance = "sums_load" the total variance is calculated using the squared sums of general loadings and group factor loadings and the sum of the uniquenesses. This will only get comparable results to variance = "correlation" if no cross-loadings are present and simple structure is well-achieved in general with the SL solution (i.e., the uniquenesses should capture almost all of the variance not explained by the general factor and the variable's allocated group factor).

Value

If found for an SL or lavaan second-order of 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.

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.

Sijtsma, K. (2009). On the use, the misuse, and the very limited usefulness of Cronbach’s alpha. Psychometrika, 74, 107–120.

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.

Examples

## Use with lavaan outputs

# 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")


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

## 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")
# Find correlation matrix from phi and pattern matrix from psych::schmid output
OMEGA(schmid_mod, type = "psych")
# Use specified correlation matrix
OMEGA(schmid_mod, type = "psych", cormat = test_models$baseline$cormat)

## 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(test_models$baseline$cormat, N = 500, n_factors = 3,
               type = "EFAtools", method = "PAF", rotation = "promax")
sl_mod <- SL(efa_mod, type = "EFAtools", method = "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)

EFAtools documentation built on Jan. 6, 2023, 5:16 p.m.