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, ...)The lavaan model object used to evaluate model fit
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).
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).
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).
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).
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.
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.
The confidence level of the confidence interval of expected parameter changes. The confidence intervals are used in the equivalence testing.
arguments passed to modificationIndices,
except for delta, which is already an argument (which can be
substituted for stdDelta or specific sets of parameters using
stdLoad, cor, stdBeta, and intcept).
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 EPC if the parameter is freely estimated
target.epc: The target EPC 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 EPC 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: Lower confidence limit of standardized EPCs
upper.std.epc: Upper confidence limit of standardized EPCs
decision.ci: Decision whether the parameter is misspecified
based on the CI 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.
To decide whether a parameter should be freed, one can inspect its modification index (MI) and expected parameter change (EPC). Those values can be used to evaluate model fit by 2 methods.
Method 1: Saris, Satorra, and van der Veld (2009, pp. 570--573) used power (probability of detecting a significant MI) and EPC to decide whether to free a parametr. First, one should evaluate whether a parameter's MI is significant. Second, one should evaluate whether the power to detect a target EPC is high enough. The combination of criteria leads to the so-called "JRule" first implemented with LISREL (van der Veld et al., 2008):
If the MI is not significant and the power is low, the test is inconclusive.
If the MI is not significant and the power is high, there is no misspecification.
If the MI is significant and the power is low, the fixed parameter is misspecified.
If the MI is significant and the power is high, the EPC is investigated. If the EPC is large (greater than the the target EPC), the parameter is misspecified. If the EPC is low (lower than the target EPC), the parameter is not misspecificied.
Method 2: The confidence interval (CI) of an EPC is calculated.
These CIs are compared with the range of trivial
misspecification, which could be (-delta, delta) or (0,
delta) for nonnegative parameters.
If a CI overlaps with the range of trivial misspecification, the test is inconclusive.
If a CI completely exceeds the range of trivial misspecification, the fixed parameters are severely misspecified.
If a CI is completely within the range of trivial misspecification, the fixed parameters are trivially misspecified.
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. 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. 10.1080/10705510903203433
van der Veld, W. M., Saris, W. E., & Satorra, A. (2008). JRule 3.0 Users Guide. 10.13140/RG.2.2.13609.90729
moreFitIndices For the additional fit indices
information
# NOT RUN {
library(lavaan)
HS.model <- ' visual =~ x1 + x2 + x3 '
fit <- cfa(HS.model, data = HolzingerSwineford1939,
group = "sex", group.equal = c("loadings","intercepts"))
miPowerFit(fit, free.remove = FALSE, op = "=~") # loadings
miPowerFit(fit, free.remove = FALSE, op = "~1") # intercepts
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)
# }
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