lavTablesFitCp {lavaan} | R Documentation |
Pairwise maximum likelihood fit statistics
Description
Three measures of fit for the pairwise maximum likelihood estimation method that are based on likelihood ratios (LR) are defined:
,
, and
. Subscript
signifies a comparison of model-implied proportions of full response
patterns with observed sample proportions, subscript
signifies a comparison of model-implied proportions of full response
patterns with the proportions implied by the assumption of multivariate normality, and subscript
signifies
a comparison of model-implied proportions of pairs of item responses with the observed proportions of pairs of item responses.
Usage
lavTablesFitCf(object)
lavTablesFitCp(object, alpha = 0.05)
lavTablesFitCm(object)
Arguments
object |
An object of class |
alpha |
The nominal level of signifiance of global fit. |
Details
The statistic compares the log-likelihood of the model-implied proportions (
) with the observed proportions (
)
of the full multivariate responses patterns:
which asymptotically has a chi-square distribution with
where denotes the number of items with discrete response scales,
denotes the number of response options, and
denotes
the number of parameters to be estimated. Notice that
results may be biased because of large numbers of empty cells in the multivariate
contingency table.
The statistic is based on the
statistic, and compares the proportions implied by the model of interest (Model 1)
with proportions implied by the assumption of an underlying multivariate normal distribution (Model 0):
where is
for Model 0 and
is
for Model 1. Statistic
has a chi-square distribution with
degrees of freedom
where denotes the number of items with discrete response scales,
denotes the number of response options, and
denotes the number of polychoric correlations,
denotes the number of thresholds, and
is the number of parameters of the
model of interest. Notice that
results may be biased because of large numbers of empty cells in the multivariate contingency table. However,
bias may cancels out as both Model 1 and Model 0 contain the same pattern of empty responses.
With the statistic we only consider pairs of responses, and compare observed sample proportions (
) with model-implied proportions
of pairs of responses(
). For items
and
we obtain a pairwise likelihood ratio test statistic
where denotes the number of response options and
denotes sample size. The
statistic has an asymptotic chi-square distribution
with degrees of freedom equal to the information
minus the number of parameters (2(m-1) thresholds and 1 correlation),
As denotes the number of items, there are
possible pairs of items. The
statistic should therefore be applied with
a Bonferroni adjusted level of significance
, with
to keep the family-wise error rate at . The hypothesis of overall goodness-of-fit is tested at
and rejected as
soon as
is significant at
for at least one pair of items. Notice that with dichotomous items,
,
and
, so that hypothesis can not be tested.
References
Barendse, M. T., Ligtvoet, R., Timmerman, M. E., & Oort, F. J. (2016). Structural Equation Modeling of Discrete data: Model Fit after Pairwise Maximum Likelihood. Frontiers in psychology, 7, 1-8.
Joreskog, K. G., & Moustaki, I. (2001). Factor analysis of ordinal variables: A comparison of three approaches. Multivariate Behavioral Research, 36, 347-387.
See Also
Examples
# Data
HS9 <- HolzingerSwineford1939[,c("x1","x2","x3","x4","x5",
"x6","x7","x8","x9")]
HSbinary <- as.data.frame( lapply(HS9, cut, 2, labels=FALSE) )
# Single group example with one latent factor
HS.model <- ' trait =~ x1 + x2 + x3 + x4 '
fit <- cfa(HS.model, data=HSbinary[,1:4], ordered=names(HSbinary[,1:4]),
estimator="PML")
lavTablesFitCm(fit)
lavTablesFitCp(fit)
lavTablesFitCf(fit)