New Fit Indices

Based on the unweighted approximation error, new fit indices can be calculated by substituting the noncentrality parameter in their well-established counterparts. An unweighted Root-Mean-Square Error of Approximation RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} can be calculated analogous to equation (3), yielding

(10) $\begin{array}{c}{\text{RMSEA}}_u=\sqrt{\text{max}\left(\frac{{\hat{\lambda }}_{\text{u}}}{n·\text{df}},,,0\right)}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {RMSEA}_u = \sqrt{\text {max} \left( \frac{{\hat{\lambda }}_\text {u}}{n \cdot \text {df}}, 0 \right) }. \end{aligned}$$\end{document}

For correlation structure models, RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} yields the average absolute correlation residual due to approximation error per $\text{df}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {df}$$\end{document} . For instance, RMSEA ${}_u=0.05$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u=0.05$$\end{document} denotes that the average absolute correlation residual per $\text{df}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {df}$$\end{document} due to systematic error is 0.05. Clearly, this interpretation is rather simple and intuitive as compared to that of the original RMSEA, for which the scaling of the noncentrality parameter needs to be considered.

In order to obtain an asymptotically unbiased estimate for the population value of equation (10), minor adjustments are required. For approaches (1.1) and (1.2) a correction constant can be obtained as follows, which is derived using Taylor expansions of moments of functions of random variables, which is outlined in more detail for approach (2).

For approach (1.2) Maydeu-Olivares (Reference Maydeu-Olivares2017) has given the following adjustment. For a simplified notation, let

$\begin{array}{c}{F}_{\text{ULS}}=T_{\text{ULS}}/n\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F_\text {ULS} = T_\text {ULS}/n \end{aligned}$$\end{document}

Then, the estimate of Eq. (10) based on approach (1.2) is

$\begin{array}{c}{\text{RMSEA}}_{u\left(1.2\right)}=\frac{1}{k_{\left(1.2\right)}}\sqrt{\text{max}\left(\frac{F_{\text{ULS}}-\text{tr}\left({\hat{\Sigma }}_e\right)}{\text{df}},,,0\right)}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {RMSEA}_{u(1.2)} = \frac{1}{k_{(1.2)}} \sqrt{\text {max} \left( \frac{F_\text {ULS}-\text {tr} \left( \varvec{{\hat{\Sigma }}_e} \right) }{\text {df}}, 0 \right) } \end{aligned}$$\end{document}

in which k is the correction constant, that is

$\begin{array}{c}{k}_{\left(1.2\right)}=1-\frac{{\sigma }_F^2}{4·F_{\text{ULS}}^2},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} k_{(1.2)} = 1 - \frac{\sigma ^2_F}{4 \cdot F_\text {ULS}^2}, \end{aligned}$$\end{document}

with

$\begin{array}{c}{\sigma }_F^2=2·\text{tr}\left(n^{-1}{\hat{\Sigma }}_e\right)+4·{(s-\hat{\sigma })}^{\prime }\left(n^{-1}{\hat{\Sigma }}_e\right)(s-\hat{\sigma })\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma ^2_F = 2 \cdot \text {tr}(n^{-1} \varvec{{\hat{\Sigma }}_e}) + 4 \cdot ({\varvec{s}} - \varvec{{\hat{\sigma }}})' (n^{-1} \varvec{{\hat{\Sigma }}_e}) ({\varvec{s}} - \varvec{{\hat{\sigma }}}) \end{aligned}$$\end{document}

The same approach can be used for approach (1.1), replacing ${\hat{\Sigma }}_e$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{{\hat{\Sigma }}_e}$$\end{document} with $n^{-1}\text{tr}\left(U_{\left(1.1\right)},\Gamma \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{-1} \text {tr}\left[ {\varvec{U}}_{(1.1)} \varvec{\Gamma } \right] $$\end{document} .

For approach (2), a bias corrected RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} estimate can be obtained using

$\begin{array}{c}{\text{RMSEA}}_{u\left(2\right)}=\frac{1}{k_{\left(2\right)}}\sqrt{\text{max}\left(\frac{\text{ADR}·F_{\text{ULS}}}{\text{df}},,,0\right)},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {RMSEA}_{u(2)} = \frac{1}{k_{(2)}} \sqrt{\text {max} \left( \frac{\text {ADR} \cdot F_\text {ULS}}{\text {df}}, 0 \right) }, \end{aligned}$$\end{document}

with

$\begin{array}{c}{k}_{\left(2\right)}=1-\frac{{\sigma }_F^2}{8·F_{\text{ULS}}^2}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} k_{(2)} = 1 - \frac{\sigma ^2_F}{8 \cdot F_\text {ULS}^2}. \end{aligned}$$\end{document}

The correction factor as well as standard errors of $f\left(F_{\text{ULS}}\right)={\text{RMSEA}}_{u\left(2\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(F_\text {ULS}) = \text {RMSEA}_{u(2)}$$\end{document} can also be estimated using the Delta method (e.g., Wolter, Reference Wolter1985). Specifically, expected value and variance are

$\begin{array}{c}\begin{array}{cc}E\left(f,\left(,F_{\text{ULS}},\right)\right)& =f\left(F_{\text{ULS}}\right)+\frac{f^{\prime \prime }\left(F_{\text{ULS}}\right)}{2}{\sigma }_F^2\\ & =\sqrt{\frac{\text{ADR}·F_{\text{ULS}}}{d}}-\frac{1}{8}\sqrt{\frac{\text{ADR}}{F_{\text{ULS}}^3·d}}{\sigma }_F^2\\ & =\sqrt{\frac{\text{ADR}·F_{\text{ULS}}}{d}}\left(1,-,\frac{{\sigma }_F^2}{8·F_{\text{ULS}}^2}\right)\end{array}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} E\left[ f(F_\text {ULS})\right]&= f(F_\text {ULS}) + \frac{f''(F_\text {ULS})}{2} \sigma ^2_F \\&= \sqrt{\frac{\textrm{ADR} \cdot F_\text {ULS}}{d}} - \frac{1}{8} \sqrt{\frac{\textrm{ADR}}{F_\text {ULS}^3 \cdot d}} \sigma ^2_F \\&= \sqrt{\frac{\textrm{ADR} \cdot F_\text {ULS}}{d}} \left( 1 - \frac{\sigma ^2_F}{8 \cdot F_\text {ULS}^2}\right) \end{aligned} \end{aligned}$$\end{document}

and

$\begin{array}{c}\begin{array}{cc}\text{Var}\left(f,\left(,F_{\text{ULS}},\right)\right)& ={\left(f^{\prime },\left(F_{\text{ULS}}\right)\right)}^2{\sigma }_F^2\\ & =\frac{{\sigma }_F^2·\text{ADR}}{4·F_{\text{ULS}}·d},\end{array}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \textrm{Var}\left[ f(F_\text {ULS})\right]&= \left[ f'(F_\text {ULS})\right] ^2 \sigma ^2_F \\&= \frac{\sigma ^2_F \cdot \textrm{ADR}}{4 \cdot F_\text {ULS} \cdot d}, \end{aligned} \end{aligned}$$\end{document}

given that

$\begin{array}{c}{f}^{\prime }\left(F_{\text{ULS}}\right)=\sqrt{\frac{\text{ADR}}{4·F_{\text{ULS}}·d}}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f'(F_\text {ULS}) = \sqrt{\frac{\textrm{ADR}}{4 \cdot F_\text {ULS} \cdot d}} \end{aligned}$$\end{document}

and

$\begin{array}{c}{f}^{\prime \prime }\left(F_{\text{ULS}}\right)=-\sqrt{\frac{\text{ADR}}{16·F_{\text{ULS}}^3·d}}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f''(F_\text {ULS}) = -\sqrt{\frac{\textrm{ADR}}{16 \cdot F_\text {ULS}^3\cdot d}} \end{aligned}$$\end{document}

RMSEA values are commonly reported together with their $90\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90\%$$\end{document} confidence interval. Assuming that the sampling distribution of the indices approximately follows a normal distribution in large samples, the interval may be calculated as

$\begin{array}{c}{\text{RMSEA}}_u\pm z_{1-\alpha /2}·se,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{RMSEA}_u \pm z_{1-\alpha /2} \cdot se, \end{aligned}$$\end{document}

in which se is the asymptotic standard error of $RMSE{A}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$RMSEA_u$$\end{document} .

For approach (1.2), the asymptotic standard error proposed by Maydeu-Olivares (Reference Maydeu-Olivares2017) can be used with one minor modification, replacing the number of nonredundant covariances/correlation with $\text{df}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {df}$$\end{document} . This yields

$\begin{array}{c}s{e}_{\left(1\right)}=\sqrt{\frac{{\sigma }_F^2}{k_{\left(1\right)}^2·4·\text{df}·F_{\text{ULS}}}}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} se_{(1)} = \sqrt{\frac{\sigma ^2_F}{k^2_{(1)} \cdot 4 \cdot \text {df} \cdot F_\text {ULS}}}. \end{aligned}$$\end{document}

For approach (2), a similar approach can be used. Carrying the $1/k_{\left(2\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/k_{(2)}$$\end{document} scaling forward, the asymptotic standard error is otherwise identical to the square root of expression of $\text{Var}\left(f,\left(,F_{\text{ULS}},\right)\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Var}\left[ f(F_\text {ULS})\right] $$\end{document} , given above, yielding

$\begin{array}{c}s{e}_{\left(2\right)}=\sqrt{\frac{{\sigma }_F^2·\text{ADR}}{k_{\left(2\right)}^2·4·F_{\text{ULS}}·\text{df}}}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} se_{(2)} = \sqrt{\frac{\sigma ^2_F \cdot \textrm{ADR}}{k_{(2)}^2 \cdot 4 \cdot F_\text {ULS} \cdot \text {df}}}. \end{aligned}$$\end{document}

An unweighted version of the Comparative Fit Index (CFI) can be obtained in a similar manner. The CFI considers the relation between the noncentrality parameter of the fitted model to that of a less complex base model (usually an independence model). Again, the noncentrality parameter is replaced by ${\hat{\lambda }}_{\text{u}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\lambda }}_\text {u}$$\end{document} . Specifically, the CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} may be written as

(11) $\begin{array}{c}{\text{CFI}}_u=1-\frac{\text{max}\left({\hat{\lambda }}_{\text{u}},,,0\right)}{\text{max}\left({\hat{\lambda }}_{\text{u}},,,{\hat{\lambda }}_{B·\text{u}}\right)},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {CFI}_u = 1 - \frac{\text {max} \left( {\hat{\lambda }}_\text {u}, 0 \right) }{\text {max} \left( {\hat{\lambda }}_\text {u}, {\hat{\lambda }}_{B \cdot \text {u}} \right) }, \end{aligned}$$\end{document}

in which ${\hat{\lambda }}_{B·\text{u}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\lambda }}_{B \cdot \text {u}}$$\end{document} is the unweighted approximation error estimate for the base model. Because estimates of ${\lambda }_{\text{u}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _\text {u}$$\end{document} are not systematically biased, no additional adjustments are required.

The interpretation of CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} may also be considered as somewhat simpler than that pertaining to the original CFI. Specifically, CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} expresses the size of systematic error variance of the base model relative to that of the fitted model. For instance, CFI ${}_u=0.90$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u = 0.90$$\end{document} denotes that the squared correlation residuals of a specific model are $90\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90\%$$\end{document} smaller than those of the baseline model.

The present considerations are restricted to RMSEA and CFI, because they are prototypical indices which include a noncentrality parameter estimate in their original formulation. Other relative fit indices that use a similar expression as the CFI, such as the Tucker-Lewis Index (TLI) or Normed Fit Index (NFI), lack this property and are, therefore, not suitable for the current purpose.

While specific values of RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} have a simple interpretation that can be directly connected to the absolute or squared correlation residuals, it is not obvious which values might indicate a good, acceptable or poor fit. There are two option for identifying suitable cutoff values for the new indices. First, by expert consensus based on empirical examples, which is the approach with which cutoff values for most fit indices have been established (e.g., Hu & Bentler, Reference Hu and Bentler1998). Second, by comparing established and new indices. If they are in a relatively simple relation, this may provide sufficient information to establish suitable cutoffs. While the first approach is beyond the scope of this article, the second approach will be pursued in the next section using simulation studies.

To assist with the calculation of the new indices, the supplementary materials to this article contain a data example (’example_data.txt’) as well as the corresponding R-code (’R_example.pdf’) for calculating RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} based on approaches (1.1), (1.2), and (2). The data is metric and contains 4 variables and 1000 cases. The corresponding analysis assumes one latent variable. The R-code can be also used for evaluating other data sets and models. For more information, see the instructions in the materials.

Method

Data were generated based on population correlation matrices ${\Sigma }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma }_0$$\end{document} for a given q-factorial model structure. These matrices were calculated from predefined $p\times q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times q$$\end{document} matrices of item loadings $K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{K}}$$\end{document} , a $q\times q$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \times q$$\end{document} matrix $\Phi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Phi }$$\end{document} of factor inter-correlations, and a $p\times p$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \times p$$\end{document} diagonal matrix $\Psi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Psi }$$\end{document} of residuals, that is

$\begin{array}{c}{\Sigma }_0=K\Phi K^{\prime }+\Psi .\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varvec{\Sigma }_0 = {\varvec{K}} \varvec{\Phi } {\varvec{K}}' + \varvec{\Psi }. \end{aligned}$$\end{document}

Factor loadings were chosen in a way that population correlation matrix ${\Sigma }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma }_0$$\end{document} yielded a predefined ULS-discrepancy from the best approximating correlation matrix implied by the model. Specifically, loadings were chosen in a way that the true approximation discrepancies yielded predefined target values for each of the simulation settings. This was achieved by numerical optimization, iteratively optimizing the population model discrepancy with respect to the fit indices’ target values based on the model parameters using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm as implemented in the optim-procedure in R. Two target value settings were considered: (i) RMSEA ${}_u=0.05$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u=0.05$$\end{document} ; CFI ${}_u=0.95$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u=0.95$$\end{document} and (ii) RMSEA ${}_u=0.05$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u=0.05$$\end{document} ; CFI ${}_u=0.99$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u=0.99$$\end{document} .

The source of systematic error was the $q^{\mathrm{th}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{th}$$\end{document} latent variable, which, in addition to the first $q-1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q-1$$\end{document} latent variables, was considered for generating the data, whereas the analysis model only considered the structure of the first $q-1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q-1$$\end{document} latent variables. For introducing additional variability to the way in which systematic misfit was generated, two different types of loading patterns were used for the $q^{\mathrm{th}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{th}$$\end{document} latent variable.

For the first type, the $q^{\mathrm{th}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{th}$$\end{document} latent variable had loadings on the first and last indicator pertaining to each of the prior $q^{\ast }=q-1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^*=q-1$$\end{document} latent variables. For instance, for a three-factorial model structure with overall $p=8$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=8$$\end{document} variables, the loading matrix would be

$\begin{array}{c}K={\left(\begin{array}{cccccccc}{k}_{11}& k_{21}& k_{31}& k_{41}& 0& 0& 0& 0\\ 0& 0& 0& 0& k_{52}& k_{62}& k_{72}& k_{82}\\ k_{13}^{\ast }& 0& 0& k_{43}^{\ast }& k_{53}^{\ast }& 0& 0& k_{83}^{\ast }\end{array}\right)}^{\prime }.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\varvec{K}} = \begin{pmatrix} k_{11} &{}k_{21}&{}k_{31}&{}k_{41} &{}0 &{}0 &{}0 &{} 0 \\ 0 &{}0 &{}0 &{}0 &{}k_{52} &{}k_{62}&{}k_{72}&{}k_{82} \\ k^*_{13}&{}0 &{}0 &{}k^*_{43}&{}k^*_{53}&{}0 &{}0 &{}k^*_{83} \end{pmatrix}'. \end{aligned}$$\end{document}

For the second type, the $q^{\mathrm{th}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{th}$$\end{document} latent variable had positive loadings on the first half of the indicators pertaining to each of the prior $q^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^*$$\end{document} latent variables, and negative loadings on the second half. Consequently, the three-factorial structure with $p=8$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=8$$\end{document} variables would be

$\begin{array}{c}K={\left(\begin{array}{cccccccc}{k}_{11}& k_{21}& k_{31}& k_{41}& 0& 0& 0& 0\\ 0& 0& 0& 0& k_{52}& k_{62}& k_{27}& k_{82}\\ k_{13}^{\ast }& k_{23}^{\ast }& -k_{33}^{\ast }& -k_{43}^{\ast }& k_{53}^{\ast }& k_{63}^{\ast }& -k_{73}^{\ast }& -k_{83}^{\ast }\end{array}\right)}^{\prime }.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\varvec{K}} = \begin{pmatrix} k_{11} &{}k_{21} &{}k_{31} &{}k_{41} &{}0 &{}0 &{}0 &{} 0 \\ 0 &{}0 &{}0 &{}0 &{}k_{52} &{}k_{62} &{}k_{27} &{}k_{82} \\ k^*_{13}&{}k^*_{23}&{}-k^*_{33}&{}-k^*_{43}&{}k^*_{53}&{}k^*_{63}&{}-k^*_{73}&{}-k^*_{83} \end{pmatrix}'. \end{aligned}$$\end{document}

The two loading patterns are henceforth referred to as misfit type I and misfit type II.

For the first $q^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^*$$\end{document} latent variables, loadings adhered to a simple structure, that is, each variable had loadings on one latent variable only. For simplicity, loadings on the first $q^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^*$$\end{document} latent variables had identical loadings k. Similarly, loadings on the $q^{\mathrm{th}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{th}$$\end{document} latent variable had identical values $k^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k^*$$\end{document} . Correlations between the first $q^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^*$$\end{document} factors were set to 0.3 by default. The $q^{\mathrm{th}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{th}$$\end{document} latent variable was assumed independent from the remaining $q-1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q-1$$\end{document} latent variables.

Altogether, 56 model settings were considered. In addition to misfit type and target values for RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} , the number of variables p, the number of latent variables $q^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^*$$\end{document} , and sample size n were varied. Specifically, the combinations of $p=8$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=8$$\end{document} , 12, and 18 and $q^{\ast }=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^*=1$$\end{document} , 2 and 3 were considered, omitting the combinations of $q^{\ast }=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^*=1$$\end{document} and $p=18$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=18$$\end{document} as well as $q^{\ast }=3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^*=3$$\end{document} and $p=8$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=8$$\end{document} . Sample sizes were $n=250$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=250$$\end{document} and 1000. The specific values for k and $k^{\ast }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k^*$$\end{document} for the different model settings, rounded to three decimal places, are given in Table 1.

Table 1. True loadings for the population models for the different model settings.

For each of the 56 settings, 1000 data sets were sampled from a multivariate normal distribution based on corresponding correlation matrices ${\Sigma }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma }_0$$\end{document} . These correlation matrices were then analyzed using confirmatory factor analyses based on ML and ULSMV estimation. For comparing fit indices between metric and dichotomous variables, additional dichotomized data sets were generated from the existing data by splitting metric variables at their mean. The dichotomized data were analyzed based on their tetrachoric correlation matrices using ULSMV.Footnote 2

For the analyses based on ML estimation, all approximation error estimates were calculated using approach (1.1), (1.2), and (2). For the ULSMV analyses, only approaches (1.1) and (1.2) were included, because of the equivalence of (1.1) and (2) under this specific condition. Finally, RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} based on the different approaches, as well as their noncentrality-parameter-based counterparts were calculated for each model and averaged within simulation settings.

Results

Results pertaining to models using ML and ULSMV estimation are presented separately. First, the results of metric data analyses with ML estimation are considered. Tables 2 and 3 give RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} estimates as well as the coverage probabilities of the corresponding $90\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90\%$$\end{document} -confidence intervals based on the different approaches for misfit type I and II respectively. Clearly, the average estimates of RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} are very close to their target values of 0.05 irrespective of the estimation approach. This holds true across all simulation settings, implying that the unweighted approximation error estimate performs well if calculated based on multivariate normal indicators and the ML test statistic. For sample sizes of $n=1000$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=1000$$\end{document} deviations of the average estimate from the target value are seldom larger than $\pm 0.001$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm 0.001$$\end{document} .

While neither type of misfit, nor the complexity of the model appears to have a distinct impact on the accuracy of average values, approach (2) has a tendency to overestimate the true RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} value in conditions in which the true CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} value is 0.99. These are also the conditions, in which average loadings are distinctly higher than for CFI ${}_u=0.95$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u=0.95$$\end{document} (see also Table 1). In such conditions, in which reproduced correlations are large, absolute deviations between sample and model implied correlations are more strongly weighted by typical discrepancy functions. Accordingly, because ADR-estimates are obtained based on the ML discrepancy function, this might have caused this slight overestimation.

Comparing values of the original RMSEA, it becomes obvious that they do not directly compare to RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} . For more complex models, as well as higher loadings, typical RMSEA values clearly exceed RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} values. Particularly, in simulation settings with CFI ${}_u=0.99$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u=0.99$$\end{document} , this trend can be observed. The reason for this is obvious. Because the elements of ${\Sigma }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\Sigma }_0$$\end{document} are comparably more extreme than in these settings, standard errors are smaller, and the scaling of the corresponding (weighted) discrepancy is relatively increased. Thus, the noncentrality parameter as well as the RMSEA are increased accordingly.

Table 2. Average RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and RMSEA values as well as $90\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90\%$$\end{document} -CI coverages of ML confirmatory factor models based on 1000 replications per cell. (Misfit Type I).

Est.: RMSEA estimate and CIC: $90\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90\%$$\end{document} -confidence interval coverage pertaining to the respective approach; Orig. Est.: unadjusted RMSEA.

The coverage probabilities of the $90\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90\%$$\end{document} -confidence interval (CIC) are adequate. Particularly, if $n=1000$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=1000$$\end{document} , results are close to $90\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90\%$$\end{document} . For smaller sample sizes, CICs remain somewhat below their target. Also, confidence intervals coverages pertaining to approaches (1.1) and (1.2) are somewhat closer to $90\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90\%$$\end{document} than those of approach (2). For bias type II and CFI ${}_u=0.99$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u=0.99$$\end{document} , all confidence intervals are somewhat too wide for $n=1000$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=1000$$\end{document} , irrespective of the condition.

Table 3. Average RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and RMSEA values as well as $90\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90\%$$\end{document} -CI coverages of ML confirmatory factor models based on 1000 replications per cell. (Misfit Type II).

Est.: RMSEA estimate and CIC: $90\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90\%$$\end{document} -confidence interval coverage pertaining to the respective approach; Orig. Est.: unadjusted RMSEA.

Table 4 gives the CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} estimates based on the different approaches for the metric data analyses with ML estimation. Similar to the RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} estimates, values are consistently close to their target value.

Table 4. Average CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and CFI values of ML confirmatory factor models based on 1000 replications per cell.

Orig.: unadjusted CFI.

Comparing the original CFI with CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} , the former is continuously smaller. For settings in which CFI ${}_u=0.99$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u=0.99$$\end{document} , this difference in size increases as the number of latent variables is increased, whereby CFI-values become consistently smaller with increasing model complexity. This difference is also likely connected to the different average correlation sizes within the respective conditions.

Despite the difference in average value, the corresponding unweighted and noncentrality-parameter-based indices measure very similar aspects of misspecification. This can be verified by the substantial correlations between RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and RMSEA (mean $=0.918$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = 0.918$$\end{document} ; range $=0.598:0.991$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = 0.598: 0.991$$\end{document} ) as well as those between CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and CFI (mean $=0.913$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = 0.913$$\end{document} ; range $=0.553:0.989$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ = 0.553: 0.989$$\end{document} ) within the individual simulation conditions. Note that all indices within conditions have the same target values and, therefore, decreased estimate variances lead to smaller correlations if the sample is large and the model has many parameters.

Second, the simulation results pertaining to the ULSMV analyses for metric as well as dichotomous variables are considered. The underlying model structures were fully identical for the two variable types respectively. Adjustments were performed using approach (1.1) and (1.2).

Tables 5 and 6 give RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and RMSEA for the ULSMV analyses for each of the data settings separated for misfit type I and II respectively. Results pertaining to CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and CFI are contained in Table 7.

Table 5. Average RMSEAu and RMSEA values as well as 90%-CI coverages of ULSMV confirmatory factor models based on 1000 replications per cell. (Misfit Type I).

Est.: RMSEA estimate and CIC: $90\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90\%$$\end{document} -confidence interval coverage pertaining to the respective approach; Orig. Est.: unadjusted RMSEA.

Similar to the ML analyses, RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} consistently approach their respective target values of 0.05 and 0.95/0.99 across conditions. This holds equally true for metric as well as dichotomous data. Although there are descriptive differences between approaches (1.1) and (1.2.) within singular condition, there are no notable differences in average performance.

If the sample size is $n=1000$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=1000$$\end{document} , deviations from the true values only rarely exceed $\pm 0.001$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm 0.001$$\end{document} for metric data and $\pm 0.002$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm 0.002$$\end{document} for dichotomous data. However, if the sample size is small and if many factors are modeled based on comparably few variables, fit indices have a notable bias. In these cases, RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} values are underestimated and CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} values are overestimated. Specifically, RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} underestimated its target value by up to 0.010 points and CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} are increased by up to 0.013 points if the target value is 0.95 and 0.003 points if the target value is 0.99. Interestingly, these effects do not differ in strength between metric and dichotomous data analyses.

The deviations from the target value in small samples do most likely not result from a specific property of the new fit indices. This may be inferred from the strong differences of the original RMSEA and CFI between different sample sizes within otherwise identical conditions, which are similar in size to the target value deviations for $n=250$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=250$$\end{document} .

While there are no strong differences between metric and dichotomous data models with respect to average value of the new indices, differences between confidence interval coverages are more pronounced. Coverages pertaining to the metric data models are reasonably close to their target of $90\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90\%$$\end{document} if $n=1000$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=1000$$\end{document} . For $n=250$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=250$$\end{document} there are some individual conditions (misfit type II, p=18, CFI ${}_u=0.99$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u=0.99$$\end{document} ) in which CIC pertaining to approach (1.2) are very poor. This may be explained in part by the biased estimates in the respective conditions.

The CIC pertaining to the dichotomous data models perform reasonably well for approach (1.2) if $n=1000$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=1000$$\end{document} . However, confidence intervals pertaining to approach (1.1) and, even more notable, CICs in samples with $n=250$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=250$$\end{document} are too small across conditions. Again, part of this can be attributed the moderate estimation bias in small samples. Nevertheless, reliable RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} confidence intervals for dichotomous data models are only obtained using approach (1.2) in larger samples.

Table 6. Average RMSEAu and RMSEA values as well as 90%-CI coverages of ULSMV confirmatory factor models based on 1000 replications per cell. (Misfit Type II).

Est.: RMSEA estimate and CIC: $90\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90\%$$\end{document} -confidence interval coverage pertaining to the respective approach; Orig. Est.: unadjusted RMSEA.

Because dichotomous data were obtained directly from the metric data within single replications of the simulation study, correlations of fit indices based on the two types of data can be considered. Specifically, the average correlation between fit indices for metric and dichotomous data models was 0.999 for RMESA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} across conditions and adjustment approaches. Thus, not only do they approach the same target value, they are also similarly sensitive to the respective discrepancies.

As was expected, conventional noncentrality-parameter-based fit indices differ strongly between metric and dichotomous data. Specifically, RMSEA values for models based on dichotomous indicators are consistently smaller (by at least 0.020 points) than those obtained from metric variable models, which matches previous results of Monroe and Cai (Reference Monroe and Cai2015) and Xia and Yang (Reference Xia and Yang2018). Also, CFI values from dichotomous data models are continuously larger than those obtained from metric data models. Thus, despite the identical model structures, fit assessments based on different variable types would lead to different conclusions using conventional fit indices. Interestingly, while RMSEA and CFI differ between metric and dichotomous data, they are virtually identical for analyses based on ML and ULSMV within corresponding conditions.

Table 7. Average CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and CFI values of ULSMV confirmatory factor models based on 1000 replications per cell.

Orig.: unadjusted CFI.

Deriving cutoff values for RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} and CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} by comparing them to their traditional counterparts is only possible to a limited extent, because of their non-constant relation across simulation conditions. Particularly if the target value of CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} is 0.99, this non-constant relation is most pronounced. Nevertheless, because RMSEA ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} does not differ strongly from the RMSEA, although the former yields somewhat smaller values, similar cutoff conventions, as suggested by Hu and Bentler (Reference Hu and Bentler1999), may be viable. For CFI ${}_u$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_u$$\end{document} , which always exceeds the CFI, a value of 0.99 may relate sufficiently to an CFI value of 0.95. However, it may be more advisable to refine cutoff criteria based on empirical experience, similar to the initial determination of cutoffs pertaining to RMSEA and CFI, rather than to restrict to relying on an analogy that only partly applies.

Another perspective for identifying a suitable cut-off value may be to simulate metric data with predefined target values for the original RMSEA and CFI. In this way, cut-off values for the new indices can be specifically linked to their established counterparts. A table containing the expected values of the new fit indices based on target values on RMSEA $=0.05$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=0.05$$\end{document} and CFI $=0.05$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=0.05$$\end{document} (for present simulation conditions) is contained in the supplementary materials of this article.