Description Usage Arguments Details References See Also Examples

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.

1 2 3 | ```
lavTablesFitCf(object)
lavTablesFitCp(object, alpha = 0.05)
lavTablesFitCm(object)
``` |

`object` |
An object of class |

`alpha` |
The nominal level of signifiance of global fit. |

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.

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.

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.

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.

1 2 3 4 5 6 7 8 9 10 11 12 | ```
# 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)
``` |

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