lavTablesFit: Pairwise maximum likelihood fit statistics

Description Usage Arguments Details References See Also Examples

Description

Three measures of fit for the pairwise maximum likelihood estimation method that are based on likelihood ratios (LR) are defined: C_F, C_M, and C_P. Subscript F signifies a comparison of model-implied proportions of full response patterns with observed sample proportions, subscript M signifies a comparison of model-implied proportions of full response patterns with the proportions implied by the assumption of multivariate normality, and subscript P signifies a comparison of model-implied proportions of pairs of item responses with the observed proportions of pairs of item responses.

Usage

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lavTablesFitCf(object)
lavTablesFitCp(object, alpha = 0.05)
lavTablesFitCm(object)

Arguments

object

An object of class lavaan.

alpha

The nominal level of signifiance of global fit.

Details

C_F

The C_F statistic compares the log-likelihood of the model-implied proportions (π_r) with the observed proportions (p_r) of the full multivariate responses patterns:

C_F = 2N∑_{r}p_{r}\ln[p_{r}/\hat{π}_{r}],

which asymptotically has a chi-square distribution with

df_F = m^k - n - 1,

where k denotes the number of items with discrete response scales, m denotes the number of response options, and n denotes the number of parameters to be estimated. Notice that C_F results may be biased because of large numbers of empty cells in the multivariate contingency table.

C_M

The C_M statistic is based on the C_F 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):

C_M = C_{F1} - C_{F0},

where C_{F0} is C_F for Model 0 and C_{F1} is C_F for Model 1. Statistic C_M has a chi-square distribution with degrees of freedom

df_M = k(k-1)/2 + k(m-1) - n_{1},

where k denotes the number of items with discrete response scales, m denotes the number of response options, and k(k-1)/2 denotes the number of polychoric correlations, k(m-1) denotes the number of thresholds, and n_1 is the number of parameters of the model of interest. Notice that C_M 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.

C_P

With the C_P statistic we only consider pairs of responses, and compare observed sample proportions (p) with model-implied proportions of pairs of responses(π). For items i and j we obtain a pairwise likelihood ratio test statistic C_{P_{ij}}

C_{P_{ij}}=2N∑_{c_i=1}^m ∑_{c_j=1}^m p_{c_i,c_j}\ln[p_{c_i,c_j}/\hat{π}_{c_i,c_j}],

where m denotes the number of response options and N denotes sample size. The C_P statistic has an asymptotic chi-square distribution with degrees of freedom equal to the information (m^2 -1) minus the number of parameters (2(m-1) thresholds and 1 correlation),

df_P = m^{2} - 2(m - 1) - 2.

As k denotes the number of items, there are k(k-1)/2 possible pairs of items. The C_P statistic should therefore be applied with a Bonferroni adjusted level of significance α^*, with

α^*= α /(k(k-1)/2)),

to keep the family-wise error rate at α. The hypothesis of overall goodness-of-fit is tested at α and rejected as soon as C_P is significant at α^* for at least one pair of items. Notice that with dichotomous items, m = 2, and df_P = 0, 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

lavTables, lavaan

Examples

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# 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),
           estimator="PML")
lavTablesFitCm(fit)
lavTablesFitCp(fit)
lavTablesFitCf(fit)

yrosseel/lavaan documentation built on Dec. 7, 2018, 12:02 p.m.