miPowerFit {semTools}

Modification indices and their power approach for model fit evaluation

Description

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

Usage

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

Arguments

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.

...

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).

Details

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.

Value

A data frame with these variables:

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

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

3. rhs: The right-hand side variable

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

5. mi: The modification index of the fixed parameter

6. epc: The EPC if the parameter is freely estimated

7. 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

8. std.epc: The standardized EPC if the parameter is freely estimated

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

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

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

12. 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.

13. se.epc: The standard errors of the expected parameter changes.

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

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

16. lower.std.epc: Lower confidence limit of standardized EPCs

17. upper.std.epc: Upper confidence limit of standardized EPCs

18. 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.

Author(s)

Sunthud Pornprasertmanit ([e-mail])

References

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

van der Veld, W. M., Saris, W. E., & Satorra, A. (2008). JRule 3.0 Users Guide. doi:10.13140/RG.2.2.13609.90729

See Also

moreFitIndices For the additional fit indices information

Examples

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)