Chi-square difference.

Description

Quantifies case influence on overall model fit by change in the test statistic

\Delta_{\chi^2_i}=\chi^2-\chi^2_{(i)}

where \chi^2 and \chi^2_{(i)} are the test statistics obtained from original and deleted i samples.

Usage

Deltachi(model, data, ..., scaled = FALSE)

Arguments

Value

Returns a vector of \Delta_{\chi^2_i}.

Note

If for observation i model does not converge or yelds a solution with negative estimated variances, the associated value of \Delta_{\chi^2_i} is set to NA.

This function is a particular case of fitinfluence, see example below.

Author(s)

Massimiliano Pastore

References

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

Rosseel, Y. (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48, 1-36.

Rosseel, Y. (2022). The lavaan tutorial. URL: https://lavaan.ugent.be/tutorial/.

Examples

## not run: this example take several minutes
data("PDII")
model <- "
  F1 =~ y1+y2+y3+y4
"

# fit0 <- sem(model, data=PDII)
# Dchi <- Deltachi(model,data=PDII)
# plot(Dchi,pch=19,xlab="observations",ylab="Delta chisquare")

## 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)
# Dchi <- Deltachi(model,data=PDII)
# plot(Dchi,pch=19,xlab="observations",ylab="Delta chisquare",main="Deltachi function")

## the case that produces negative estimated variances
# sem(model,data=PDII[-which(is.na(Dchi)),])

## same results 
# Dchi <- fitinfluence("chisq",model,data=PDII)$Dind$chisq
# plot(Dchi,pch=19,xlab="observations",ylab="Delta chisquare",main="fitinfluence function")

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

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).

Value

Returns a vector of LD_i.

Note

If for observation i model does not converge or yelds a solution with negative estimated variances, the associated value of LD_i is set to NA.

Author(s)

Massimiliano Pastore, Gianmarco Altoe'

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: 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")

Industrialization and Democracy indicators.

Description

Simulated data set from covariance matrix reported in Bollen (1989).

Usage

data(PDII)

Format

This data frame contains 75 obs. of 11 variables:

• x1: num, gross national product per capita.

• x2: num, consumption per capita.

• x3: num, percentage of the labor force in industrial occupations.

• y1: num, freedom of the press in 1960.

• y2: num, freedom of group opposition in 1960.

• y3: num, fairness of elections in 1960.

• y4: num, elective nature and effectiveness of the legislative body in 1960.

• y5: num, freedom of the press in 1965.

• y6: num, freedom of group opposition in 1965.

• y7: num, fairness of elections in 1965.

• y8: num, elective nature and effectiveness of the legislative body in 1965.

References

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

Examples

data(PDII)

Simulated data set.

Description

Simulated data set.

Usage

data(Q)

Format

This data frame contains 919 obs. of 10 ordinal discrete variables.

Examples

data(Q)

Log-Likelihood of a sem model (Internal function).

Description

Internal function, called by Likedist.

Usage

bollen.loglik(N, S, Sigma)

Arguments

Details

The log-likelihood is computed by the function bollen.loglik using the formula 4B2 described by Bollen (1989, pag. 135).

Value

Returns the Log-likelihood.

Author(s)

Massimiliano Pastore, Gianmarco Altoe'

References

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

See Also

Likedist

Examples

data("PDII")
model <- "
  F1 =~ y1+y2+y3+y4
"
fit0 <- sem(model, data=PDII)
N <- fit0@Data@nobs[[1]]
S <- fit0@SampleStats@cov[[1]]
Sigma <- fitted(fit0)$cov
bollen.loglik(N,S,Sigma)

Explores case influence.

Description

It explores case influence. Cases with extreme values of the considered measure of influence are reported. Extreme values are determined using the boxplot criterion (Tukey, 1977) or user-defined cut-offs. Cases for which deletion leads to a model that does not converge or yelds a solution with negative estimated variances are also reported. In addition, explore.influence provides a graphical representation of case influence.

Usage

explore.influence(x, cut.offs = 'default', 
                     plot = 'TRUE', cook = 'FALSE', ...)

Arguments

Value

A list with the following components:

Author(s)

Gianmarco Altoe'

References

Tukey, J. W. (1977). Exploratory data analysis. Reading, MA: Addison-Wesley.

Examples

data("PDII")
model <- "
F1 =~ y1+y2+y3+y4
"
fit0 <- sem(model, data=PDII,std.lv=TRUE)
## not run
# gCD <- genCookDist(model,data=PDII,std.lv=TRUE)
# explore.influence(gCD,cook=TRUE)

##
## not run: this example take several minutes
model <- "
F1 =~ x1+x2+x3
F2 =~ y1+y2+y3+y4
F3 =~ y5+y6+y7+y8
"

# fit0 <- sem(model, data=PDII)
# FI <- fitinfluence('rmsea',model,PDII)
# explore.influence(FI)

Case influence on model fit.

Description

This function evaluate the case's effect on a user-defined fit index.

Usage

fitinfluence(index, model, data, ...)

Arguments

Details

For each case evaluate the influence on one or more fit indices: the difference between the chosen fit index calculated for the SEM target model M and the same index computed for the SEM model M_{(i)} excluding case i.

Value

Returns a list:

Note

If for observation i model does not converge or yelds a solution with negative estimated variances, the associated value of influence is set to NA.

Author(s)

Massimiliano Pastore

References

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

Examples

## not run: this example take several minutes
data("PDII")
model <- "
  F1 =~ y1+y2+y3+y4
"

# fit0 <- sem(model, data=PDII)
# FI <- fitinfluence("cfi",model,data=PDII)
# plot(FI$Dind,pch=19)

## 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)
# FI <- fitinfluence(c("tli","rmsea"),model,PDII)
# explore.influence(FI$Dind$tli)
# explore.influence(FI$Dind$rmsea)

Generalized Cook Distance.

Description

Case influence on a vector of parameters may be quantified by generalized Cook's Distance (gCD; Cook 1977, 1986):

gCD_i=(\hat{\mathbf{\theta}}-\hat{\mathbf{\theta}}_{(i)})' _a\hat{\mathbf{\Sigma}}(\hat{\mathbf{\theta}}_{(i)})^{-1} (\hat{\mathbf{\theta}}-\hat{\mathbf{\theta}}_{(i)})

where \hat{\mathbf{\theta}} and \hat{\mathbf{\theta}}_{(i)} are l \times 1 vectors of parameter estimates obained from the original and delete i samples, and _a\hat{\mathbf{\Sigma}}(\hat{\mathbf{\theta}}_{(i)}) is the estimated asymptotic covariance matrix of the parameter estimates obtained from reduced sample.

Usage

genCookDist(model, data, ...)

Arguments

Value

Returns a vector of gCD_i.

Note

If for observation i model does not converge or yelds a solution with negative estimated variances, the associated value of gCD_i is set to NA.

Author(s)

Massimiliano Pastore

References

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.

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

Examples

## not run: this example take several minutes
data("PDII")
model <- "
  F1 =~ y1+y2+y3+y4
"
# fit0 <- sem(model, data=PDII)
# gCD <- genCookDist(model,data=PDII)
# plot(gCD,pch=19,xlab="observations",ylab="Cook distance")

## not run: this example take several minutes
## an example in which the deletion of a case produces 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)
# gCD <- genCookDist(model,data=PDII)
# plot(gCD,pch=19,xlab="observations",ylab="Cook distance")

Case influence on model parameters.

Description

Computes direction of change in parameter estimates with

\Delta \hat{\theta}_{ji}=\frac{\hat{\theta}_j-\hat{\theta}_{j(i)}}{[VAR(\hat{\theta}_{j(i)})]^{1/2}}

where \hat{\theta}_j and \hat{\theta}_{j(i)} are the parameter estimates obtained from original and deleted i samples.

Usage

parinfluence(parm, model, data, cook = FALSE, ...)

Arguments

Value

Returns a list:

Note

If for observation i model does not converge or yelds a solution with negative estimated variances or NA parameter values, the associated values of \Delta \hat{\theta}_{ji} are set to NA.

Author(s)

Massimiliano Pastore

References

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

Examples

## not run: this example take several minutes
data("PDII")
model <- "
  F1 =~ y1+y2+y3+y4
"
# fit0 <- sem(model, data=PDII)
# PAR <- c("F1=~y2","F1=~y3","F1=~y4")
# LY <- parinfluence(PAR,model,PDII)
# str(LY)
# explore.influence(LY$Dparm[,1])

## 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)
# PAR <- c("F2=~y2","F2=~y3","F2=~y4")
# LY <- parinfluence(PAR,model,PDII)

## not run: this example take several minutes
## dealing with ordinal data
data(Q)
model <- "
 F1 =~ it1+it2+it3+it4+it5+it6+it7+it8+it9+it10
"

# fit0 <- sem(model, data=Q, ordered=colnames(Q))
# LY <- parinfluence("F1=~it4",model,Q,ordered=colnames(Q))
# explore.influence(LY$Dparm[,1])

Fitted values and residuals

Description

It calculates the expected values and the residuals of a sem model.

Usage

sem.fitres(object)
obs.fitres(object)
lat.fitres(object)

Arguments

Details

The main function, sem.fitres(), calls one of the other two routines depending on the type of the model. If model does not contain latent variables, sem.fitres() calls the function obs.fitres(), otherwise calls the function lat.fitres().

The functions obs.fitres() and lat.fitres() are internal functions, do not use it directly.

Value

Returns a data frame containing: 1) The observed model variables; 2) The expected values on dependent variables (indicated with hat.); 3) The residuals on dependent variables (indicated with e.)

Note

In order to compute more interpretable fitted values and residuals, model is forced to have meanstrucure = TRUE and std.lv = TRUE.

Author(s)

Massimiliano Pastore

Examples

data("PDII")
model <- "
  F1 =~ y1+y2+y3+y4
"

fit0 <- sem(model, data=PDII)
out <- sem.fitres(fit0)
head(out)

par(mfrow=c(2,2))
plot(e.y1~hat.y1,data=out)
plot(e.y2~hat.y2,data=out)
plot(e.y3~hat.y3,data=out)
plot(e.y4~hat.y4,data=out)

qqnorm(out$e.y1); qqline(out$e.y1)
qqnorm(out$e.y2); qqline(out$e.y2)
qqnorm(out$e.y3); qqline(out$e.y3)
qqnorm(out$e.y4); qqline(out$e.y4)