GLM SEM

Description

This shows the output of Structural Equation Modeling (SEM) and Confirmatory Factor Analysis (CFA) functions from the package rwf. The functions include a path diagram plotter that does not need semPlot, a full CFA report (fit indices, parameter estimates, modification indices, predicted scores), and a sample-size planning simulation that tracks how model fit changes as sample size grows.

Installation instructions for rwf can be found here

The code can be found here

Theoretical Background

Confirmatory Factor Analysis

CFA specifies, in advance, which observed indicators load onto which latent factors (unlike EFA, which discovers the structure). For a single-indicator relationship,

\[x_i = \lambda_i \eta + \epsilon_i\]

where \(\eta\) is the latent factor, \(\lambda_i\) is the factor loading (how strongly indicator \(i\) reflects the factor), and \(\epsilon_i\) is measurement error. The model is fit by finding parameter values that best reproduce the observed covariance matrix among indicators.

Model fit indices

Every index in the report is listed with its reference in Fit indices in the report.

Modification indices

A modification index estimates how much \(\chi^2\) would drop if a currently-fixed parameter (typically a residual covariance, or a cross-loading) were freed and re-estimated. Large indices flag places where the model doesn’t match the data — but freeing parameters purely to improve fit, without substantive justification, risks overfitting to sample-specific noise rather than reflecting genuine structure.

Sample size and power for SEM

SEM fit indices and standard errors are sample-size dependent: small samples can fail to reject a poorly-fitting model (low power) while very large samples can flag trivial misfit as statistically significant. simulate_cfa_fit directly visualizes how fit indices stabilize as sample size grows, supporting sample-size planning before data collection.

---

Model Diagrams

plot_cfa_gg draws a path diagram directly in ggplot2 (no semPlot/Rgraphviz system dependency), with latent variables as filled rectangles, observed variables as light rectangles, and edges labeled with loadings.

model_syntax <- "LATENT1=~X1+X2+X3
                 LATENT2=~X4+X5+X6"
df_cfa <- lavaan::simulateData(model = model_syntax, model.type = "cfa", return.type = "data.frame", sample.nobs = 300)
fit <- lavaan::cfa(model_syntax, data = df_cfa)

plot_cfa_gg(fit, what = "std", layout = "tree")

plot_cfa_gg(fit, what = "est", layout = "circle")

what = "std" shows standardized loadings (comparable across indicators regardless of their original scale); what = "est" shows raw, unstandardized estimates (in the original metric of each indicator). Three layout options (tree, circle, spring) control node placement via igraph.

Equality constraints

what = "eq" labels each loading with its parameter label, so loadings constrained to be equal (sharing the same label) are easy to spot; loadings without a label show their estimate. It needs a model with labelled parameters. Here the loadings of X2 and X3 are constrained to be equal (a), and so are those of X5 and X6 (b):

model_eq <- "LATENT1=~X1+a*X2+a*X3
             LATENT2=~X4+b*X5+b*X6"
fit_eq <- lavaan::cfa(model_eq, data = df_cfa)
plot_cfa_gg(fit_eq, what = "eq", layout = "spring")

Styling

label_size and edge_label_size set the text size of the node and path labels; color_latent and color_observed set the fill colour of the latent and observed nodes:

plot_cfa_gg(fit, what = "std", layout = "tree",
            label_size = 4, edge_label_size = 3.5,
            color_latent = "#2c3e50", color_observed = "#ecf0f1")

All layouts at once

plot_cfa is a batch wrapper that runs every combination of layout × what in one call and returns them as a named list — useful for quickly picking the clearest arrangement for a report. The three eq diagrams are only produced for models with labelled parameters, so fit gives 6 diagrams and fit_eq gives all 9:

all_plots <- plot_cfa(fit)
all_plots_eq <- plot_cfa(fit_eq)

names(all_plots)

## [1] "circle_estimates"          "circle_standard_estimates" "tree_estimates"            "tree_standard_estimates"   "spring_estimates"          "spring_standard_estimates"

names(all_plots_eq)

## [1] "circle_estimates"                           "circle_standard_estimates"                  "circle_parameters_wih_equality_constraints" "tree_estimates"                             "tree_standard_estimates"                    "tree_parameters_wih_equality_constraints"   "spring_estimates"                           "spring_standard_estimates"                  "spring_parameters_wih_equality_constraints"

Any diagram can be taken from the list by name:

all_plots$spring_standard_estimates

Full CFA Report

report_cfa bundles fit indices, R², parameter estimates (unstandardized and standardized), modification indices, sample covariance, model-implied predicted scores, and the path diagrams into one structured report.

result <- report_cfa(model = fit)

result$fit_indices

##                                 fit
## npar                   1.300000e+01
## fmin                   9.795390e-03
## chisq                  5.877234e+00
## df                     8.000000e+00
## pvalue                 6.609813e-01
## baseline.chisq         4.499905e+02
## baseline.df            1.500000e+01
## baseline.pvalue        0.000000e+00
## cfi                    1.000000e+00
## tli                    1.009150e+00
## nnfi                   1.009150e+00
## rfi                    9.755110e-01
## nfi                    9.869392e-01
## pnfi                   5.263676e-01
## ifi                    1.004803e+00
## rni                    1.004880e+00
## logl                  -2.969640e+03
## unrestricted.logl     -2.966701e+03
## aic                    5.965279e+03
## bic                    6.013429e+03
## ntotal                 3.000000e+02
## bic2                   5.972200e+03
## rmsea                  0.000000e+00
## rmsea.ci.lower         0.000000e+00
## rmsea.ci.upper         5.458030e-02
## rmsea.ci.level         9.000000e-01
## rmsea.pvalue           9.305991e-01
## rmsea.close.h0         5.000000e-02
## rmsea.notclose.pvalue  4.125356e-03
## rmsea.notclose.h0      8.000000e-02
## rmr                    4.938356e-02
## rmr_nomean             4.938356e-02
## srmr                   2.429525e-02
## srmr_bentler           2.429525e-02
## srmr_bentler_nomean    2.429525e-02
## crmr                   2.874653e-02
## crmr_nomean            2.874653e-02
## srmr_mplus             2.429525e-02
## srmr_mplus_nomean      2.429525e-02
## gfi                    1.000000e+00
## gfi.ci.lower           9.921773e-01
## gfi.ci.upper           1.000000e+00
## gfi.ci.level           9.000000e-01
## cn_05                  7.925618e+02
## cn_01                  1.026494e+03
## gfi_lisrel             9.935428e-01
## agfi_lisrel            9.830498e-01
## pgfi                   3.784925e-01
## mfi                    1.003544e+00
## ecvi                   1.062574e-01

result$r_squared

##    r_squared
## X1 0.5067546
## X2 0.5413438
## X3 0.4589552
## X4 0.4565517
## X5 0.6738854
## X6 0.4511722

result$parameters

##        lhs op     rhs exo        est         se        z       pvalue    ci.lower  ci.upper     std.lv    std.all
## 1  LATENT1 =~      X1   0 1.00000000 0.00000000       NA           NA  1.00000000 1.0000000 1.02017714 0.71186696
## 2  LATENT1 =~      X2   0 1.06492177 0.12227716 8.709081 0.000000e+00  0.82526293 1.3045806 1.08640884 0.73576073
## 3  LATENT1 =~      X3   0 0.94115894 0.10824747 8.694512 0.000000e+00  0.72899779 1.1533201 0.96014883 0.67746232
## 4  LATENT2 =~      X4   0 1.00000000 0.00000000       NA           NA  1.00000000 1.0000000 0.91991042 0.67568608
## 5  LATENT2 =~      X5   0 1.33314945 0.14801629 9.006775 0.000000e+00  1.04304286 1.6232560 1.22637806 0.82090520
## 6  LATENT2 =~      X6   0 1.00262247 0.10935857 9.168212 0.000000e+00  0.78828362 1.2169613 0.92232285 0.67169352
## 7       X1 ~~      X1   0 1.01301661 0.13237075 7.652874 1.976197e-14  0.75357470 1.2724585 1.01301661 0.49324543
## 8       X2 ~~      X2   0 1.00000138 0.14281684 7.001985 2.523537e-12  0.72008552 1.2799172 1.00000138 0.45865615
## 9       X3 ~~      X3   0 1.08677603 0.12752028 8.522378 0.000000e+00  0.83684087 1.3367112 1.08677603 0.54104480
## 10      X4 ~~      X4   0 1.00730129 0.11390706 8.843186 0.000000e+00  0.78404756 1.2305550 1.00730129 0.54344833
## 11      X5 ~~      X5   0 0.72783516 0.15201154 4.788026 1.684302e-06  0.42989801 1.0257723 0.72783516 0.32611464
## 12      X6 ~~      X6   0 1.03480792 0.11582487 8.934246 0.000000e+00  0.80779534 1.2618205 1.03480792 0.54882782
## 13 LATENT1 ~~ LATENT1   0 1.04076139 0.17877176 5.821733 5.824062e-09  0.69037519 1.3911476 1.00000000 1.00000000
## 14 LATENT2 ~~ LATENT2   0 0.84623517 0.14949935 5.660461 1.509673e-08  0.55322183 1.1392485 1.00000000 1.00000000
## 15 LATENT1 ~~ LATENT2   0 0.07126109 0.07078279 1.006757 3.140514e-01 -0.06747063 0.2099928 0.07593314 0.07593314
## 16      X1 r2      X1   0 0.50675457         NA       NA           NA          NA        NA         NA         NA
## 17      X2 r2      X2   0 0.54134385         NA       NA           NA          NA        NA         NA         NA
## 18      X3 r2      X3   0 0.45895520         NA       NA           NA          NA        NA         NA         NA
## 19      X4 r2      X4   0 0.45655167         NA       NA           NA          NA        NA         NA         NA
## 20      X5 r2      X5   0 0.67388536         NA       NA           NA          NA        NA         NA         NA
## 21      X6 r2      X6   0 0.45117218         NA       NA           NA          NA        NA         NA         NA

result$modification_indices[order(-result$modification_indices$mi), ]

##        lhs op     rhs           mi           epc      sepc.all delta         ncp      power decision
## 33      X3 ~~      X6 2.726710e+00  1.253792e-01  1.182294e-01   0.1 1.734555534 0.26065653      (i)
## 23      X1 ~~      X3 2.472947e+00 -1.550222e+00 -1.477459e+00   0.1 0.010290277 0.05117960      (i)
## 20 LATENT2 =~      X2 2.472924e+00 -1.427474e-01 -8.893177e-02   0.1 1.213596945 0.19645587      (i)
## 30      X2 ~~      X6 1.065747e+00 -7.952232e-02 -7.817337e-02   0.1 1.685295436 0.25461851      (i)
## 25      X1 ~~      X5 7.257519e-01  6.591286e-02  7.676184e-02   0.1 1.670505270 0.25280369      (i)
## 19 LATENT2 =~      X1 7.091670e-01  7.396125e-02  4.747588e-02   0.1 1.296402951 0.20669092      (i)
## 27      X2 ~~      X3 7.091517e-01  9.108748e-01  8.737523e-01   0.1 0.008547159 0.05097966      (i)
## 22      X1 ~~      X2 6.732885e-01  9.943729e-01  9.879630e-01   0.1 0.006809303 0.05078038      (i)
## 21 LATENT2 =~      X3 6.732650e-01  7.151687e-02  4.641952e-02   0.1 1.316342813 0.20915513      (i)
## 26      X1 ~~      X6 5.947840e-01 -5.822505e-02 -5.686849e-02   0.1 1.754444255 0.26309143      (i)
## 28      X2 ~~      X4 4.287168e-01 -4.988266e-02 -4.970151e-02   0.1 1.722944721 0.25923426      (i)
## 35      X4 ~~      X6 3.989898e-01  4.897609e-01  4.797054e-01   0.1 0.016633875 0.05190762      (i)
## 17 LATENT1 =~      X5 3.989833e-01  5.299606e-02  3.618994e-02   0.1 1.420585978 0.22203085      (i)
## 24      X1 ~~      X4 2.700138e-01  3.879869e-02  3.840865e-02   0.1 1.793708048 0.26789305      (i)
## 16 LATENT1 =~      X4 1.923742e-01 -3.278983e-02 -2.457051e-02   0.1 1.789239312 0.26734693      (i)
## 36      X5 ~~      X6 1.923695e-01 -5.385531e-01 -6.205577e-01   0.1 0.006632527 0.05076011      (i)
## 32      X3 ~~      X5 1.625584e-01 -3.131018e-02 -3.520457e-02   0.1 1.658204966 0.25129374      (i)
## 34      X4 ~~      X5 9.867952e-02 -3.876797e-01 -4.527691e-01   0.1 0.006565700 0.05075245      (i)
## 18 LATENT1 =~      X6 9.867904e-02 -2.372751e-02 -1.762852e-02   0.1 1.752752963 0.26288444      (i)
## 31      X3 ~~      X4 7.694563e-02 -2.082969e-02 -1.990825e-02   0.1 1.773447327 0.26541624      (i)
## 29      X2 ~~      X5 7.310666e-03  6.762547e-03  7.926719e-03   0.1 1.598587304 0.24396746      (i)
## 15 LATENT1 ~~ LATENT2 6.286205e-12  1.753432e-07  1.868391e-07   0.1 2.044611177 0.29838308      (i)
## 6  LATENT2 =~      X6 2.140021e-12  1.125656e-07  7.541182e-08   0.1 1.688910195 0.25506193      (i)
## 10      X4 ~~      X4 1.967364e-12 -1.402558e-07 -5.434483e-01   0.1 1.000099314 0.17008729      (i)
## 5  LATENT2 =~      X5 1.333117e-12  8.439483e-08  5.196728e-08   0.1 1.871701659 0.27740859      (i)
## 14 LATENT2 ~~ LATENT2 1.234090e-12  9.700989e-08  1.000000e+00   0.1 1.311338154 0.20853667      (i)
## 3  LATENT1 =~      X3 9.918481e-13 -7.300700e-08 -5.255169e-08   0.1 1.860870770 0.27608905      (i)
## 13 LATENT1 ~~ LATENT1 5.391582e-13 -8.253152e-08 -1.000000e+00   0.1 0.791546572 0.14443613      (i)
## 11      X5 ~~      X5 2.483790e-13  5.256892e-08  3.261146e-01   0.1 0.898786988 0.15760680      (i)
## 7       X1 ~~      X1 2.305844e-13 -5.317121e-08 -4.932454e-01   0.1 0.815599435 0.14738587      (i)
## 12      X6 ~~      X6 1.254611e-13  3.621811e-08  5.488278e-01   0.1 0.956439431 0.16470518      (i)
## 8       X2 ~~      X2 4.188849e-14 -2.355895e-08 -4.586562e-01   0.1 0.754714689 0.13992469      (i)
## 2  LATENT1 =~      X2 3.103615e-14  1.280506e-08  8.847094e-09   0.1 1.892798485 0.27997702      (i)
## 9       X3 ~~      X3 2.509295e-15  5.618800e-09  5.410448e-01   0.1 0.794812364 0.14483647      (i)

• fit_indices: the full lavaan fit table (CFI, TLI, RMSEA + CI, SRMR, AIC, BIC, etc.).
• r_squared: proportion of each indicator’s variance explained by its latent factor.
• parameters: every estimated parameter (loadings, variances, covariances) with SE, z, p, and confidence intervals.
• modification_indices: sorted here by size — the largest values are the parameters most likely to improve fit if freed (again, only worth freeing with a substantive rationale).

The rest of the report holds the input data and model matrices:

result$sample_covariance

##          X1          X2         X3          X4         X5         X6
## 1 2.0537779  1.11061581 0.97366776  0.10761073 0.18349180  0.0538701
## 2 1.1106158  2.18028551 1.04593443 -0.03169393 0.04625917 -0.0433154
## 3 0.9736678  1.04593443 2.00866169  0.06915178 0.13837788  0.1694607
## 4 0.1076107 -0.03169393 0.06915178  1.85353633 1.12753597  0.8512851
## 5 0.1834918  0.04625917 0.13837788  1.12753597 2.23183857  1.1302200
## 6 0.0538701 -0.04331540 0.16946071  0.85128506 1.13021997  1.8854876

result$standardized_estimates

## $lambda
##      LATENT1   LATENT2
## X1 0.7118670 0.0000000
## X2 0.7357607 0.0000000
## X3 0.6774623 0.0000000
## X4 0.0000000 0.6756861
## X5 0.0000000 0.8209052
## X6 0.0000000 0.6716935
## attr(,"class")
## [1] "lavaan.matrix" "matrix"       
## 
## $theta
##           X1        X2        X3        X4        X5        X6
## X1 0.4932454 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## X2 0.0000000 0.4586562 0.0000000 0.0000000 0.0000000 0.0000000
## X3 0.0000000 0.0000000 0.5410448 0.0000000 0.0000000 0.0000000
## X4 0.0000000 0.0000000 0.0000000 0.5434483 0.0000000 0.0000000
## X5 0.0000000 0.0000000 0.0000000 0.0000000 0.3261146 0.0000000
## X6 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.5488278
## attr(,"class")
## [1] "lavaan.matrix.symmetric" "matrix"                 
## 
## $psi
##            LATENT1    LATENT2
## LATENT1 1.00000000 0.07593314
## LATENT2 0.07593314 1.00000000
## attr(,"class")
## [1] "lavaan.matrix.symmetric" "matrix"                 
## 
## attr(,"class")
## [1] "lavaan.list" "list"

head(result$predict)

##           X1         X2         X3        X4         X5         X6     LATENT1     LATENT2
## 1 -0.9426779 -2.1028008  0.1912442  3.276463  2.9851827  2.7977557 -0.67595179  1.98070180
## 2  0.8063454  0.4591551 -2.2035808  1.981226  3.7663648  0.9178366 -0.07195081  1.68983705
## 3  2.1002916  0.7484383  0.5253228  2.128261 -0.5622396 -0.3917101  0.90638917  0.08619589
## 4  0.3554615  0.4818293 -0.1334990  1.355596  0.3903519 -2.0340665  0.24361619 -0.03310564
## 5 -1.8511984 -0.9792688 -1.7832304 -2.503643  0.4967461  0.1956658 -1.08521488 -0.31626076
## 6  0.4841860 -0.3228088 -0.6920372  1.614947  5.5834426  1.4969346 -0.01824073  2.32081883

result$call

##                                                                call
## 1   lavaan::lavaan(model=model_syntax,data=df_cfa,model.type="cfa",
## 2 cmd="cfa",int.ov.free=TRUE,int.lv.free=FALSE,auto.fix.first=TRUE,
## 3            auto.fix.single=TRUE,auto.var=TRUE,auto.cov.lv.x=TRUE,
## 4       auto.cov.y=TRUE,auto.th=TRUE,auto.delta=TRUE,auto.efa=TRUE)

• sample_covariance: the observed covariance matrix of the indicators that the model tries to reproduce.
• unstandardized_estimates / standardized_estimates: the model matrices — lambda (loadings), theta (residual variances and covariances) and psi (factor variances and covariances).
• predict: the observed data with each case’s predicted factor scores added as extra columns.
• call: the lavaan call that fitted the model.

With file, the report is also saved: report_cfa(fit, file = "cfa") writes every table to cfa.xlsx and all diagrams to cfa_diagram.pdf, with w and h setting the PDF page size in inches.

Fit indices in the report

result$fit_indices holds every fit measure lavaan computes (Rosseel, 2012). Cut-offs are given only where the literature gives one; “lower is better” indices are for comparing models fitted to the same data. The _nomean variants ignore the mean structure, so they equal the plain index when the model has no means, as here.

Row in fit_indices	Index	Good fit	Reference
npar	Number of free parameters	—	—
fmin	Minimum of the fit function, \(\chi^2/(2N)\)	—	Jöreskog (1969)
chisq, df, pvalue	\(\chi^2\) test of exact fit	\(p > .05\)	Jöreskog (1969)
baseline.chisq, baseline.df, baseline.pvalue	\(\chi^2\) test of the baseline (independence) model, used by the incremental indices	—	Bentler & Bonett (1980)
cfi	Comparative Fit Index	\(\geq .95\)	Bentler (1990); Hu & Bentler (1999)
tli, nnfi	Tucker-Lewis Index / Non-Normed Fit Index (same value)	\(\geq .95\)	Tucker & Lewis (1973); Bentler & Bonett (1980); Hu & Bentler (1999)
rfi	Relative Fit Index	close to 1	Bollen (1986)
nfi	Normed Fit Index	\(\geq .90\)	Bentler & Bonett (1980)
pnfi	Parsimony Normed Fit Index	higher is better	James et al. (1982)
ifi	Incremental Fit Index	close to 1	Bollen (1989a)
rni	Relative Noncentrality Index	\(\geq .95\)	McDonald & Marsh (1990); Hu & Bentler (1999)
logl, unrestricted.logl	Log-likelihood of the model and of the saturated model	—	Jöreskog (1969)
aic	Akaike Information Criterion	lower is better	Akaike (1974)
bic	Bayesian Information Criterion	lower is better	Schwarz (1978)
ntotal	Total sample size	—	—
bic2	Sample-size adjusted BIC	lower is better	Sclove (1987)
rmsea	Root Mean Square Error of Approximation	\(\leq .06\)	Steiger & Lind (1980); Steiger (1990); Hu & Bentler (1999)
rmsea.ci.lower, rmsea.ci.upper, rmsea.ci.level	90% confidence interval of RMSEA	—	Browne & Cudeck (1992)
rmsea.pvalue, rmsea.close.h0	Test of close fit, \(H_0\): RMSEA \(\leq .05\)	\(p > .05\)	Browne & Cudeck (1992)
rmsea.notclose.pvalue, rmsea.notclose.h0	Test of not-close fit, \(H_0\): RMSEA \(\geq .08\)	\(p < .05\)	MacCallum et al. (1996)
rmr, rmr_nomean	Root Mean Square Residual (unstandardized, depends on the scale of the variables)	—	Jöreskog & Sörbom (1981)
srmr, srmr_bentler, srmr_bentler_nomean	Standardized Root Mean Square Residual	\(\leq .08\)	Bentler (1995); Hu & Bentler (1999)
crmr, crmr_nomean	Correlation Root Mean Square Residual (residual correlations, diagonal excluded)	—	Bollen (1989b)
srmr_mplus, srmr_mplus_nomean	SRMR as computed by Mplus	\(\leq .08\)	Asparouhov & Muthén (2018)
gfi, gfi.ci.lower, gfi.ci.upper, gfi.ci.level	Goodness of Fit Index, with confidence interval	close to 1	Jöreskog & Sörbom (1981)
cn_05, cn_01	Hoelter’s critical N at \(\alpha = .05\) and \(.01\)	\(> 200\)	Hoelter (1983)
gfi_lisrel, agfi_lisrel	GFI and Adjusted GFI as computed by LISREL	close to 1	Jöreskog & Sörbom (1981)
pgfi	Parsimony Goodness of Fit Index	higher is better	Mulaik et al. (1989)
mfi	McDonald’s Fit Index	close to 1	McDonald (1989)
ecvi	Expected Cross-Validation Index	lower is better	Browne & Cudeck (1989)

Multi-group models

When the fitted model has more than one group, report_cfa additionally returns a group summary table (group labels, sample sizes per group, and total N) — omitted here since the example above is single-group.

Sample Size Planning

simulate_cfa_fit refits a CFA model repeatedly across a range of sample sizes, in parallel, and returns fit indices at each size — either generating data from known population parameters (model_sim) or resampling from an observed dataset’s correlation structure (df).

model_sim <- "LATENT =~ 1*X1 + 0.5*X2 + 1.5*X3 + 1.5*X4 + X5"
model_est <- "LATENT =~ X1 + X2 + X3 + X4 + X5"
result_sim <- simulate_cfa_fit(model_sim = model_sim, model = model_est, minnobs = 50, maxnobs = 10000, stepping = 100)

The first element is a table with one row per sample size and one column per fit index; the second is a named list of plots, one per fit index:

head(result_sim[[1]][, c("observations", "cfi", "tli", "rmsea", "srmr")])

##   observations       cfi       tli      rmsea        srmr
## 1           50 0.9831898 0.9663796 0.08100305 0.051864543
## 2          150 1.0000000 1.0198220 0.00000000 0.017967412
## 3          250 1.0000000 1.0083660 0.00000000 0.017219490
## 4          350 1.0000000 1.0092316 0.00000000 0.009174356
## 5          450 1.0000000 1.0008744 0.00000000 0.014185840
## 6          550 1.0000000 1.0009186 0.00000000 0.011635798

plot_multiplot(plotlist = result_sim[[2]][c("observations_cfi", "observations_tli",
                                            "observations_rmsea", "observations_srmr")],
               cols = 2)

## [[1]]

Each panel tracks one fit index against sample size. Indices like RMSEA and SRMR typically stabilize and tighten (less sample-to-sample variability) as \(n\) grows — the point at which a given index’s variability becomes acceptably small for your purposes is a data-driven answer to “how large a sample do I need?” for this specific model.

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Conclusion

plot_cfa_gg/plot_cfa visualize a fitted measurement model’s structure without requiring semPlot’s system dependencies; report_cfa collects every quantity needed to write up a CFA (fit, parameters, R², modification indices) in one call; and simulate_cfa_fit turns “what sample size do I need” into an empirical, visualizable question rather than a rule of thumb.

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