Generate a Population Covariance Matrix From a One-Factor Confirmatory Factor Model

Given a vector of factor loadings ($\lambda$) and the corresponding vector of unique (error) variances ($\psi^2$), this function computes the implied population covariance matrix under a single-factor confirmatory factor model: $$\Sigma = \Lambda \Lambda^\top + \Psi.$$ This builds the model implied covariance for a confirmatory factor model with uncorrelated errors. The unique-variances matrix $\Psi$ is strictly diagonal, so no residual covariances (correlated uniquenesses) are allowed; a model with correlated errors is outside the scope of this function. Because the model is a single common factor, the loadings matrix $\Lambda$ reduces to a column vector $\lambda$ of length $p$ (one per indicator), and the unique-variances matrix $\Psi$ is the diagonal $\mathrm{diag}(\psi^2)$ of a length-$p$ vector. The formals are named in the lowercase vector form (lambda and psi_squared) to reflect this; the matrix-form symbols ($\Lambda$, $\Psi$) remain in the mathematical exposition above.

Usage

covmat_from_cfa(lambda, psi_squared, ...)

Arguments

lambda: A numeric vector of factor loadings (one per indicator). Can also be supplied as a single-row or single-column matrix and will be coerced to a vector.
psi_squared: A numeric vector of unique (error) variances, one per indicator. Recycled to match the length of lambda when a single value is supplied (equal-error-variance model).
...: Optional advanced controls. Currently the only recognized passthrough is tol_det: the tolerance below which the determinant of the implied covariance matrix triggers a positive- definite warning (default 1e-05). The argument is hidden in ... because the default is appropriate for almost every application; users who pass an unrecognized name through ... (for example, a misspelling) are notified with a warning so the typo is not silently ignored.

Value

A list with the single element population_cov: the implied population covariance matrix of the manifest indicators ($p \times p$, symmetric).

Details

Under the single-factor common-factor model each indicator score is $x_i = \lambda_i \xi + \delta_i$, where $\xi$ is the (standardized) latent factor and $\delta_i$ is the indicator-specific residual with variance $\psi_i^2$. The population covariance among the manifest indicators is therefore $$\Sigma = \Lambda \Lambda^\top + \Psi,$$ where $\Lambda$ is the $p \times 1$ column of loadings $(\lambda_1, \ldots, \lambda_p)^\top$ and $\Psi = \mathrm{diag}(\psi_1^2, \ldots, \psi_p^2)$. In code we work with the vectors lambda and psi_squared directly. Because $\Psi$ is built with diag() from a length-$p$ vector, the errors are uncorrelated by construction: there is no way to specify a residual covariance between two indicators. A confirmatory factor model with correlated errors (correlated uniquenesses) requires a more general formulation than this function provides.

The cfa spelling matches the cfa_1 naming.

Author

Ken Kelley kkelley@nd.edu

Examples

# Five indicators with equal loadings and equal error variances
covmat_from_cfa(lambda = rep(0.7, 5), psi_squared = rep(0.51, 5))
#> $population_cov
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,] 1.00 0.49 0.49 0.49 0.49
#> [2,] 0.49 1.00 0.49 0.49 0.49
#> [3,] 0.49 0.49 1.00 0.49 0.49
#> [4,] 0.49 0.49 0.49 1.00 0.49
#> [5,] 0.49 0.49 0.49 0.49 1.00
#> 

# Unequal loadings
covmat_from_cfa(lambda      = c(0.5, 0.6, 0.7, 0.8),
                psi_squared = c(0.75, 0.64, 0.51, 0.36))
#> $population_cov
#>      [,1] [,2] [,3] [,4]
#> [1,] 1.00 0.30 0.35 0.40
#> [2,] 0.30 1.00 0.42 0.48
#> [3,] 0.35 0.42 1.00 0.56
#> [4,] 0.40 0.48 0.56 1.00
#>