testOMF(
.object                = NULL,
.alpha                 = 0.05,
.fit_measures          = FALSE,
.R                     = 499,
.saturated             = FALSE,
.seed                  = NULL,
.verbose               = TRUE,
...
)

## Arguments

.object An R object of class cSEMResults resulting from a call to csem(). An integer or a numeric vector of significance levels. Defaults to 0.05. Logical. (EXPERIMENTAL) Should additional fit measures be included? Defaults to FALSE. Character string. How should inadmissible results be treated? One of "drop", "ignore", or "replace". If "drop", all replications/resamples yielding an inadmissible result will be dropped (i.e. the number of results returned will potentially be less than .R). For "ignore" all results are returned even if all or some of the replications yielded inadmissible results (i.e. number of results returned is equal to .R). For "replace" resampling continues until there are exactly .R admissible solutions. Depending on the frequency of inadmissible solutions this may significantly increase computing time. Defaults to "drop". Integer. The number of bootstrap replications. Defaults to 499. Logical. Should a saturated structural model be used? Defaults to FALSE. Integer or NULL. The random seed to use. Defaults to NULL in which case an arbitrary seed is chosen. Note that the scope of the seed is limited to the body of the function it is used in. Hence, the global seed will not be altered! Logical. Should information (e.g., progress bar) be printed to the console? Defaults to TRUE. Can be used to determine the fitting function used in the calculateGFI function.

## Value

A list of class cSEMTestOMF containing the following list elements:

$Test_statistic The value of the test statistics. $Critical_value

The corresponding critical values obtained by the bootstrap.

$Decision The test decision. One of: FALSE (Reject) or TRUE (Do not reject). $Information

The .R bootstrap values; The number of admissible results; The seed used and the number of total runs.

## Details

Bootstrap-based test for overall model fit originally proposed by Beran and Srivastava (1985) . See also Dijkstra and Henseler (2015) who first suggested the test in the context of PLS-PM.

By default, testOMF() tests the null hypothesis that the population indicator correlation matrix equals the population model-implied indicator correlation matrix. Several discrepancy measures may be used. By default, testOMF() uses four distance measures to assess the distance between the sample indicator correlation matrix and the estimated model-implied indicator correlation matrix, namely the geodesic distance, the squared Euclidean distance, the standardized root mean square residual (SRMR), and the distance based on the maximum likelihood fit function. The reference distribution for each test statistic is obtained by the bootstrap as proposed by Beran and Srivastava (1985) .

It is possible to perform the bootstrap-based test using fit measures such as the CFI, RMSEA or the GFI if .fit_measures = TRUE. This is experimental. To the best of our knowledge the applicability and usefulness of the fit measures for model fit assessment have not been formally (statistically) assessed yet. Theoretically, the logic of the test applies to these fit indices as well. Hence, their applicability is theoretically justified. Only use if you know what you are doing.

If .saturated = TRUE the original structural model is ignored and replaced by a saturated model, i.e., a model in which all constructs are allowed to correlate freely. This is useful to test misspecification of the measurement model in isolation.

## References

Beran R, Srivastava MS (1985). “Bootstrap Tests and Confidence Regions for Functions of a Covariance Matrix.” The Annals of Statistics, 13(1), 95--115. doi: 10.1214/aos/1176346579 , https://doi.org/10.1214/aos/1176346579.

Dijkstra TK, Henseler J (2015). “Consistent and Asymptotically Normal PLS Estimators for Linear Structural Equations.” Computational Statistics & Data Analysis, 81, 10--23.

csem(), calculateSRMR(), calculateDG(), calculateDL(), cSEMResults, testMICOM(), testMGD(), exportToExcel()

## Examples

# ===========================================================================
# Basic usage
# ===========================================================================
model <- "
# Structural model
eta2 ~ eta1
eta3 ~ eta1 + eta2

# (Reflective) measurement model
eta1 =~ y11 + y12 + y13
eta2 =~ y21 + y22 + y23
eta3 =~ y31 + y32 + y33
"

## Estimate
out <- csem(threecommonfactors, model, .approach_weights = "PLS-PM")

## Test
testOMF(out, .R = 50, .verbose = FALSE, .seed = 320)
#> ________________________________________________________________________________
#> --------- Test for overall model fit based on Beran & Srivastava (1985) --------
#>
#> Null hypothesis:
#>
#>        ┌──────────────────────────────────────────────────────────────────┐
#>        │                                                                  │
#>        │   H0: The model-implied indicator covariance matrix equals the   │
#>        │   population indicator covariance matrix.                        │
#>        │                                                                  │
#>        └──────────────────────────────────────────────────────────────────┘
#>
#> Test statistic and critical value:
#>
#> 	                                  	Critical value
#> 	Distance measure    Test statistic	  95%
#> 	dG                      0.0060    	0.0192
#> 	SRMR                    0.0158    	0.0272
#> 	dL                      0.0112    	0.0333
#> 	dML                     0.0320    	0.1027
#>
#>
#> Decision:
#>
#> 	                    	Significance level
#> 	Distance measure    	     95%
#> 	dG                  	Do not reject
#> 	SRMR                	Do not reject
#> 	dL                  	Do not reject
#> 	dML                 	Do not reject
#>