The model fit evaluation approach using modification indices and expected parameter changes.

```
miPowerFit(lavaanObj, stdLoad = 0.4, cor = 0.1, stdBeta = 0.1,
intcept = 0.2, stdDelta = NULL, delta = NULL, cilevel = 0.9)
```

lavaanObj

The lavaan model object used to evaluate model fit

stdLoad

The amount of standardized factor loading that one would like to be detected (rejected). The default value is 0.4, which is suggested by Saris and colleagues (2009, p. 571).

cor

The amount of factor or error correlations that one would like to be detected (rejected). The default value is 0.1, which is suggested by Saris and colleagues (2009, p. 571).

stdBeta

The amount of standardized regression coefficients that one would like to be detected (rejected). The default value is 0.1, which is suggested by Saris and colleagues (2009, p. 571).

intcept

The amount of standardized intercept (similar to Cohen's
*d* that one would like to be detected (rejected). The default value is
0.2, which is equivalent to a low effect size proposed by Cohen (1988,
1992).

stdDelta

The vector of the standardized parameters that one would
like to be detected (rejected). If this argument is specified, the value
here will overwrite the other arguments above. The order of the vector must
be the same as the row order from modification indices from the
`lavaan`

object. If a single value is specified, the value will be
applied to all parameters.

delta

The vector of the unstandardized parameters that one would like
to be detected (rejected). If this argument is specified, the value here
will overwrite the other arguments above. The order of the vector must be
the same as the row order from modification indices from the `lavaan`

object. If a single value is specified, the value will be applied to all
parameters.

cilevel

The confidence level of the confidence interval of expected parameter changes. The confidence intervals are used in the equivalence testing.

A data frame with these variables:

lhs: The left-hand side variable, with respect to the operator in in the lavaan

`model.syntax`

op: The lavaan syntax operator: "~~" represents covariance, "=~" represents factor loading, "~" represents regression, and "~1" represents intercept.

rhs: The right-hand side variable

group: The level of the group variable for the parameter in question

mi: The modification index of the fixed parameter

epc: The expected parameter change if the parameter is freely estimated

target.epc: The target expected parameter change that represents the minimum size of misspecification that one would like to be detected by the test with a high power

std.epc: The standardized expected parameter change if the parameter is freely estimated

std.target.epc: The standardized target expected parameter change

significant.mi: Represents whether the modification index value is significant

high.power: Represents whether the power is enough to detect the target expected parameter change

decision.pow: The decision whether the parameter is misspecified or not based on Saris et al's method:

`"M"`

represents the parameter is misspecified,`"NM"`

represents the parameter is not misspecified,`"EPC:M"`

represents the parameter is misspecified decided by checking the expected parameter change value,`"EPC:NM"`

represents the parameter is not misspecified decided by checking the expected parameter change value, and`"I"`

represents the decision is inconclusive.se.epc: The standard errors of the expected parameter changes.

lower.epc: The lower bound of the confidence interval of expected parameter changes.

upper.epc: The upper bound of the confidence interval of expected parameter changes.

lower.std.epc: The lower bound of the confidence interval of standardized expected parameter changes.

upper.std.epc: The upper bound of the confidence interval of standardized expected parameter changes.

decision.ci: The decision whether the parameter is misspecified or not based on the confidence interval method:

`"M"`

represents the parameter is misspecified,`"NM"`

represents the parameter is not misspecified, and`"I"`

represents the decision is inconclusive.

The row numbers matches with the results obtained from the
`inspect(object, "mi")`

function.

In the lavaan object, one can inspect the modification indices and expected parameter changes. Those values can be used to evaluate model fit by two methods.

First, Saris, Satorra, and van der Veld (2009, pp. 570-573) used the power to detect modification indices and expected parameter changes to evaluate model fit. First, one should evaluate whether the modification index of each parameter is significant. Second, one should evaluate whether the power to detect a target expected parameter change is high enough. If the modification index is not significant and the power is high, there is no misspecification. If the modification index is significant and the power is low, the fixed parameter is misspecified. If the modification index is significant and the power is high, the expected parameter change is investigated. If the expected parameter change is large (greater than the the target expected parameter change), the parameter is misspecified. If the expected parameter change is low (lower than the target expected parameter change), the parameter is not misspecificied. If the modification index is not significant and the power is low, the decision is inconclusive.

Second, the confidence intervals of the expected parameter changes are
formed. These confidence intervals are compared with the range of trivial
misspecification, which could be (-`delta`

, `delta`

) or (0,
`delta`

) for nonnegative parameters. If the confidence intervals are
outside of the range of trivial misspecification, the fixed parameters are
severely misspecified. If the confidence intervals are inside the range of
trivial misspecification, the fixed parameters are trivially misspecified.
If confidence intervals are overlapped the range of trivial
misspecification, the decision is inconclusive.

Cohen, J. (1988). *Statistical power analysis for the
behavioral sciences* (2nd ed.). Hillsdale, NJ: Erlbaum.

Cohen, J. (1992). A power primer. *Psychological Bulletin, 112*(1),
155--159. doi:10.1037/0033-2909.112.1.155

Saris, W. E., Satorra, A., & van der Veld, W. M. (2009). Testing structural
equation models or detection of misspecifications? *Structural Equation
Modeling, 16*(4), 561--582. doi:10.1080/10705510903203433

`moreFitIndices`

For the additional fit indices
information

# NOT RUN { library(lavaan) HS.model <- ' visual =~ x1 + x2 + x3 textual =~ x4 + x5 + x6 speed =~ x7 + x8 + x9 ' fit <- cfa(HS.model, data = HolzingerSwineford1939, group = "sex", meanstructure = TRUE) miPowerFit(fit) model <- ' # latent variable definitions ind60 =~ x1 + x2 + x3 dem60 =~ y1 + a*y2 + b*y3 + c*y4 dem65 =~ y5 + a*y6 + b*y7 + c*y8 # regressions dem60 ~ ind60 dem65 ~ ind60 + dem60 # residual correlations y1 ~~ y5 y2 ~~ y4 + y6 y3 ~~ y7 y4 ~~ y8 y6 ~~ y8 ' fit2 <- sem(model, data = PoliticalDemocracy, meanstructure = TRUE) miPowerFit(fit2, stdLoad = 0.3, cor = 0.2, stdBeta = 0.2, intcept = 0.5) # }