McDonald's omega total

Formula: omega = (sum lambda_i)^2 / Var(T) using factor loadings

Usage

Ctomeg(X, factor_loadings)

Arguments

X

Either an n x p data matrix or a p x p item covariance matrix (detected by squareness and symmetry).

factor_loadings

Loading of each item on the general factor.

Value

List with estimate (omega), omega, alpha, var_total, uniquenesses, p, method.

Details

Var(T) is the variance of the total score, i.e. the sum of every entry of the item covariance matrix. Equivalently omega = (sum lambda)^2 / ((sum lambda)^2 + sum theta) with theta_i = S_ii - lambda_i^2 the unique variances. Under tau-equivalence (all loadings equal) omega coincides with Cronbach's alpha; otherwise omega is the larger of the two.

References

McDonald (1999), Test Theory: A Unified Treatment, Erlbaum, ch. 6.

Examples

X <- cbind(c(1, 2, 3, 4, 5, 6), c(2, 2, 4, 4, 6, 7), c(1, 3, 3, 5, 5, 6))
Ctomeg(X, factor_loadings = c(0.8, 0.9, 0.7))
#> $estimate
#> [1] 0.1828571
#> 
#> $omega
#> [1] 0.1828571
#> 
#> $alpha
#> [1] 0.9746032
#> 
#> $var_total
#> [1] 31.5
#> 
#> $uniquenesses
#> [1] 2.860000 3.356667 2.876667
#> 
#> $p
#> [1] 3
#> 
#> $method
#> [1] "McDonald omega total"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"