Fit Path Models to a Correlation Matrix
model_fit.RdFit Path Models to a Correlation Matrix
Arguments
- model_input
Model input as a character string. Multiple models need to be on their own line. Regression paths use
~, latent measurement paths use=~, and residual variances or covariances use~~.- R
A named correlation matrix, typically the pooled correlation matrix extracted from a fitted
marsobject.- method_null
Null-model method for observed-variable path models. Currently
"sem"uses an independence model with all off-diagonal correlations set to zero.- N
Sample size used to scale the discrepancy function into chi-square fit statistics.
Value
A list containing path coefficients, chi-square fit statistics, CFI, TLI, RMSEA, SRMR, raw fit-index values, and flags indicating whether bounded display values were clipped.
Details
model_fit() chooses between two fitting paths based on the syntax in
model_input.
For observed-variable path models that use only ~, coefficients are
computed algebraically from the supplied correlation matrix. For each
regression equation, standardized coefficients are
\(R_{xx}^{-1}R_{xy}\). The implied residual covariance matrix is assembled
from the observed exogenous correlations and \(1 - R^2\) residual
variances, and model fit is evaluated with the package's GLS-style
discrepancy between the observed and implied correlation matrices. The
SEM-style null model is an independence model with the diagonal of
R retained.
For latent-variable or covariance models using =~ or ~~, the
function builds RAM-style directed-effect (A) and residual covariance
(S) matrices and minimizes the normal-theory SEM discrepancy
$$
F(\theta) = \log|\Sigma(\theta)| +
\mathrm{tr}\{R\Sigma(\theta)^{-1}\} - \log|R| - p,
$$
where \(\Sigma(\theta)\) is the model-implied correlation matrix for the
observed variables. Optimization is performed with
optim using method = "BFGS", up to 2000
iterations, and a numerical Hessian is requested for downstream uncertainty
calculations. Variance parameters are optimized on the log scale to keep
them positive. Fixed parameters can be supplied as
value * variable; otherwise the first loading for each unscaled latent
factor is fixed to one for marker-variable identification.
Chi-square statistics use (N - 1) * F. Incremental and residual fit
indices are derived from the fitted and null discrepancies. CFI, TLI, and
SRMR are bounded for display, with raw values returned in
fit_index_raw and clipping indicators returned in
fit_index_flags.
Examples
Br <- matrix(c(1.00000000, -0.09773331, -0.1755029, 0.3186775,
-0.09773331, 1.00000000, 0.5271873, -0.4175596,
-0.17550292, 0.5271872, 1.0000000, -0.4006848,
0.31867753, -0.41755963, -0.4006848, 1.0000000),
nrow = 4, byrow = TRUE)
colnames(Br) <- c("Performance", "Self_confidence", "Cognitive", "Somatic" )
rownames(Br) <- colnames(Br)
## Proposed path model
model <- "## Regression paths
Performance ~ Self_confidence + Cognitive + Somatic
Self_confidence ~ Cognitive + Somatic "
N <- 573
model_fit(model_input = model, R = Br,
method_null = "sem", N)
#> $path_coefficients
#> $path_coefficients[[1]]
#> Self_confidence -> Performance Cognitive -> Performance
#> 0.08316153 -0.09261001
#> Somatic -> Performance
#> 0.31629500
#>
#> $path_coefficients[[2]]
#> Cognitive -> Self_confidence Somatic -> Self_confidence
#> 0.4287053 -0.2457839
#>
#>
#> $Model
#> Chi2 df pvalue
#> 59.1102 2.0000 0.0000
#>
#> $NullModel
#> Chi2Null dfNull
#> 643.3373 6.0000
#>
#> $CFI
#> [1] 0.9103925
#>
#> $TLI
#> [1] 0.7311776
#>
#> $RMSEA
#> [1] 0.2234312
#>
#> $RMSEA_CI
#> [1] 0.1671203 0.2835352
#>
#> $SRMR
#> [1] 0.2672185
#>
#> $fit_index_raw
#> CFI TLI RMSEA SRMR
#> 0.9103925 0.7311776 0.2234312 0.2672185
#>
#> $fit_index_flags
#> CFI_clipped TLI_clipped RMSEA_adjusted SRMR_clipped
#> FALSE FALSE FALSE FALSE
#>
model_fit(model_input = model, R = Br,
method_null = "sem", N )
#> $path_coefficients
#> $path_coefficients[[1]]
#> Self_confidence -> Performance Cognitive -> Performance
#> 0.08316153 -0.09261001
#> Somatic -> Performance
#> 0.31629500
#>
#> $path_coefficients[[2]]
#> Cognitive -> Self_confidence Somatic -> Self_confidence
#> 0.4287053 -0.2457839
#>
#>
#> $Model
#> Chi2 df pvalue
#> 59.1102 2.0000 0.0000
#>
#> $NullModel
#> Chi2Null dfNull
#> 643.3373 6.0000
#>
#> $CFI
#> [1] 0.9103925
#>
#> $TLI
#> [1] 0.7311776
#>
#> $RMSEA
#> [1] 0.2234312
#>
#> $RMSEA_CI
#> [1] 0.1671203 0.2835352
#>
#> $SRMR
#> [1] 0.2672185
#>
#> $fit_index_raw
#> CFI TLI RMSEA SRMR
#> 0.9103925 0.7311776 0.2234312 0.2672185
#>
#> $fit_index_flags
#> CFI_clipped TLI_clipped RMSEA_adjusted SRMR_clipped
#> FALSE FALSE FALSE FALSE
#>