The raw residual matrix is the model misfit in the covariance metric,
E = S - Sigma_hat, with S the sample covariance and
Sigma_hat the model-implied covariance. Summaries run over the
lower triangle including the diagonal, p (p + 1) / 2 distinct
elements:
RMR = sqrt(2 / (p (p + 1)) * sum_{i <= j} (s_ij - sigma_ij)^2) and
\(SRMR = sqrt(2 / (p (p + 1)) * sum_{i <= j}
((s_ij - sigma_ij) / sqrt(s_ii s_jj))^2)\).
Value
List with estimate (SRMR), residual, srmr,
rmr, max_abs_residual, max_abs_standardised,
sum_sq_residual, mean_residual, trace_residual,
n_elements, p, srmr_acceptable, method.
Details
SRMR divides each residual by the product of the observed standard
deviations, so it is scale free; Hu & Bentler's conventional cutoff is
SRMR <= 0.08, applied by srmr_acceptable. Bentler & Yuan
show that in small samples the residual-based statistics behave far
better than the likelihood-ratio chi-square, which is why the residual
matrix rather than the chi-square is the object returned here.
Both matrices must be square, of the same order, and symmetric to within 1e-8; a non-symmetric input is an error, not something to silently symmetrise.
References
Bentler, P. M. & Yuan, K.-H. (1999). Structural equation modeling with small samples: test statistics. Multivariate Behavioral Research 34(2), 181-197. doi:10.1207/s15327906mb340203 ; Hu, L. & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis. Structural Equation Modeling 6(1), 1-55. doi:10.1080/10705519909540118
Examples
Semsro(sample_cov = 5L, fitted_cov = 5L)
#> $estimate
#> [1] 0
#>
#> $residual
#> [,1]
#> [1,] 0
#>
#> $srmr
#> [1] 0
#>
#> $rmr
#> [1] 0
#>
#> $max_abs_residual
#> [1] 0
#>
#> $max_abs_standardised
#> [1] 0
#>
#> $sum_sq_residual
#> [1] 0
#>
#> $mean_residual
#> [1] 0
#>
#> $trace_residual
#> [1] 0
#>
#> $n_elements
#> [1] 1
#>
#> $p
#> [1] 1
#>
#> $srmr_acceptable
#> [1] 1
#>
#> $method
#> [1] "SEM residual matrix S - Sigma-hat with RMR/SRMR (Bentler & Yuan 1999; Hu & Bentler 1999)"
#>
#> attr(,"class")
#> [1] "morie_rich_result" "list"