influence.SEM (version 2.3)

Likedist: Likelihood Distance.

Description

A general model-based measure of case influence on model fit is likelihood distance (Cook, 1977, 1986; Cook & Weisberg, 1982) defined as

$$LD_i=2[L(\hat{\mathbf{\theta}})-L(\hat{\mathbf{\theta}}_{(i)})]$$

where \(\hat{\mathbf{\theta}}\) and \(\hat{\mathbf{\theta}}_{(i)}\) are the \(k \times 1\) vectors of estimated model parameters on the original and deleted \(i\) samples, respectively, where \(i = 1, \ldots, N\). The subscript (\(i\)) indicates that the estimate was computed on the sample excluding case \(i\). \(L(\hat{\mathbf{\theta}})\) and \(L(\hat{\mathbf{\theta}}_{(i)})\) are the log-likelihoods based on the original and the deleted \(i\) samples, respectively.

Usage

Likedist(model, data, ...)

Arguments

model

A description of the user-specified model using the lavaan model syntax. See lavaan for more information.

data

A data frame containing the observed variables used in the model. If any variables are declared as ordered factors, this function will treat them as ordinal variables.

…

Additional parameters for sem function.

Value

Returns a vector of \(LD_i\).

Details

The log-likelihoods \(L(\hat{\mathbf{\theta}})\) and \(L(\hat{\mathbf{\theta}}_{(i)})\) are computed by the function bollen.loglik using the formula 4B2 described by Bollen (1989, pag. 135).

The likelihood distance gives the amount by which the log-likelihood of the full data changes if one were to evaluate it at the reduced-data estimates. The important point is that \(L(\hat{\mathbf{\theta}}_{(i)})\) is not the log-likelihood obtained by fitting the model to the reduced data set. It is obtained by evaluating the likelihood function based on the full data set (containing all \(n\) observations) at the reduced-data estimates (Schabenberger, 2005).

References

Bollen, K.A. (1989). Structural Equations with latent Variables. New York, NY: Wiley.

Cook, R.D. (1977). Detection of influential observations in linear regression. Technometrics, 19, 15-18.

Cook, R.D. (1986). Assessment of local influence. Journal of the Royal Statistical Society B, 48, 133-169.

Cook, R.D., Weisberg, S. (1986). Residuals and influence in regressions. New York, NY: Chapman & Hall.

Pek, J., MacCallum, R.C. (2011). Sensitivity Analysis in Structural Equation Models: Cases and Their Influence. Multivariate Behavioral Research, 46, 202-228.

Schabenberger, O. (2005). Mixed model influence diagnostics. In SUGI, 29, 189-29. SAS institute Inc, Cary, NC.

See Also

bollen.loglik

Examples

# NOT RUN {
## not run: this example take several minutes
data("PDII")
model <- "
  F1 =~ y1+y2+y3+y4
"
# fit0 <- sem(model, data=PDII)
# LD <-Likedist(model,data=PDII)
# plot(LD,pch=19,xlab="observations",ylab="Likelihood distances")

## not run: this example take several minutes
## an example in which the deletion of a case yelds a solution 
## with negative estimated variances
model <- "
  F1 =~ x1+x2+x3
  F2 =~ y1+y2+y3+y4
  F3 =~ y5+y6+y7+y8
"

# fit0 <- sem(model, data=PDII)
# LD <-Likedist(model,data=PDII)
# plot(LD,pch=19,xlab="observations",ylab="Likelihood distances")
# }