### Introduction

In SEM, one of the first steps is to evaluate the model’s global fit.
This is commonly done by presenting multiple fit indices, with some of
the most common being based on the model’s
$\chi^2$.
We have developed Bayesian versions of these indices (Garnier-Villarreal and Jorgensen 2020) that can
be computed with *blavaan*.

### Noncentrality-Based Fit Indices

This group of indices compares the hypothesized model against the perfect saturated model. It specifically uses the noncentrality parameter $\hat{\lambda} = \chi^2 - df$, with the df being adjusted by different model/data characterictics. Specific indices include Root Mean Square Error of approximation (RMSEA), McDonald’s centrality index (Mc), gamma-hat ($\hat{\Gamma}$), and adjusted gamma-hat ($\hat{\Gamma}_{adj}$).

We will show an example with the Holzinger and Swineford (1939) example. You first estimate your SEM/CFA model as usual

```
HS.model <- ' visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9 '
fit <- bcfa(HS.model, data=HolzingerSwineford1939, std.lv=TRUE)
```

You then need to pass the model to the `blavFitIndices()`

function

`gl_fits <- blavFitIndices(fit)`

Finally, you can describe the posterior distribution for each of the
indices with their `summary()`

function. With this call, we
see the 3 central tendency measures (mean median, and mode), the
standard deviation, and the 90% Credible Interval

```
##
## Posterior summary statistics and highest posterior density (HPD)
## 90% credible intervals for devm-based fit indices:
##
## EAP Median MAP SD lower upper
## BRMSEA 0.099 0.098 0.097 0.005 0.090 0.107
## BGammaHat 0.956 0.957 0.958 0.005 0.949 0.963
## adjBGammaHat 0.907 0.908 0.910 0.010 0.891 0.922
## BMc 0.902 0.904 0.906 0.010 0.886 0.918
```

### Incremental Fit Indices

Another group of fit indices compares the hypothesized model with the
*worst* possible model, so they are called incremental indices.
Such indices compare your model’s
$\chi^2_H$
to the *null* model’s
$\chi^2_0$
in different ways. Indices include the Comparative Fit Index (CFI),
Tucker-Lewis Index (TLI), and Normed Fit Index (NFI).

To estimate these indices we need to define and estimate the
respective *null* model. The standard *null* model used by
default in frequentist SEM programs (like *lavaan*) includes only
the indicators variances and intercepts, and no covariances between
items.

You can specify your *null* model by including only the
respective indicator variances in your model syntax, such as

```
HS.model_null <- '
x1 ~~ x1
x2 ~~ x2
x3 ~~ x3
x4 ~~ x4
x5 ~~ x5
x6 ~~ x6
x7 ~~ x7
x8 ~~ x8
x9 ~~ x9 '
fit_null <- bcfa(HS.model_null, data=HolzingerSwineford1939)
```

Once you have your hypothesized and null models, you pass both to the
`blavFitIndices`

function, and now it will provide both types
of fit indices

```
gl_fits_all <- blavFitIndices(fit, baseline.model = fit_null)
summary(gl_fits_all, central.tendency = c("mean","median","mode"), prob = .90)
```

```
##
## Posterior summary statistics and highest posterior density (HPD)
## 90% credible intervals for devm-based fit indices:
##
## EAP Median MAP SD lower upper
## BRMSEA 0.099 0.098 0.097 0.005 0.090 0.107
## BGammaHat 0.956 0.957 0.958 0.005 0.949 0.963
## adjBGammaHat 0.907 0.908 0.910 0.010 0.891 0.922
## BMc 0.902 0.904 0.906 0.010 0.886 0.918
## BCFI 0.930 0.931 0.933 0.008 0.917 0.942
## BTLI 0.883 0.884 0.888 0.013 0.862 0.903
## BNFI 0.910 0.910 0.912 0.007 0.897 0.921
```

The `summary()`

method now presents the central tendency
measure you asked for, standard deviation, and credible interval for the
noncentrality and incremental fit indices.

### Access the indices posterior distributions

You can also extract the posterior distributions for the respective
indices, this way you can explore further details. For example,
diagnostic plots using the `bayesplot`

package.

```
dist_fits <- data.frame(gl_fits_all@indices)
head(dist_fits)
```

```
## BRMSEA BGammaHat adjBGammaHat BMc BCFI BTLI BNFI
## 1 0.10593079 0.9499973 0.8933976 0.8883160 0.9187946 0.8644212 0.8988141
## 2 0.10014752 0.9550694 0.9042111 0.8995598 0.9274858 0.8789319 0.9071878
## 3 0.09712886 0.9576242 0.9096576 0.9052313 0.9319535 0.8863909 0.9115299
## 4 0.09818988 0.9567336 0.9077589 0.9032536 0.9308408 0.8845333 0.9105678
## 5 0.09633310 0.9582868 0.9110703 0.9067033 0.9334671 0.8889181 0.9131040
## 6 0.09211234 0.9617252 0.9184007 0.9143465 0.9386225 0.8975255 0.9178845
```

Once we have saved the posterior distributions, we can explore the the histogram and scatterplots between indices.

```
mcmc_pairs(dist_fits, pars = c("BRMSEA","BGammaHat","BCFI","BTLI"),
diag_fun = "hist")
```

### Summary

You can estimate posterior distributions for
$\chi^2$
based global fit indices. Notice that here we only presented the fit
indices based on the recommended method *devM* and with the
recommended number of parameters metric *loo*. These can be
adjusted by the user if desired.

The general recommendation is to prefer $\hat{\Gamma}$ and CFI, as these have shown to be less sensitive to model and data characteristics.

These defaults and recommendations are made based on previous simulation research. For more details about the fit indices please see Garnier-Villarreal and Jorgensen (2020).

### References

*Psychological Methods*25 (1): 46–70. https://doi.org/dx.doi.org/10.1037/met0000224.

*A Study of Factor Analysis: The Stability of a Bi-Factor Solution*. Supplementary Educational Monograph 48. Chicago: University of Chicago Press.