lavTablesFitCp | R Documentation |

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.

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

`object` |
An object of class |

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

`C_F`

The `C_F`

statistic compares the log-likelihood of the model-implied proportions (`\pi_r`

) with the observed proportions (`p_r`

)
of the full multivariate responses patterns:

```
C_F = 2N\sum_{r}p_{r}\ln[p_{r}/\hat{\pi}_{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(`\pi`

). For items `i`

and `j`

we obtain a pairwise likelihood ratio test statistic `C_{P_{ij}}`

```
C_{P_{ij}}=2N\sum_{c_i=1}^m \sum_{c_j=1}^m
p_{c_i,c_j}\ln[p_{c_i,c_j}/\hat{\pi}_{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 `\alpha^*`

, with

```
\alpha^*= \alpha /(k(k-1)/2)),
```

to keep the family-wise error rate at `\alpha`

. The hypothesis of overall goodness-of-fit is tested at `\alpha`

and rejected as
soon as `C_P`

is significant at `\alpha^*`

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.

`lavTables, lavaan`

```
# 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)
```

Embedding an R snippet on your website

Add the following code to your website.

For more information on customizing the embed code, read Embedding Snippets.