LRT test — lavTestLRT • lavaan

LRT test for comparing (nested) lavaan models.

Usage

lavTestLRT(object, ..., method = "default", test = "default",
           A.method = "delta", scaled.shifted = TRUE,
           type = "Chisq", model.names = NULL)
anova(object, ...)

Arguments

object: An object of class lavaan.
...: additional objects of class lavaan.
method: Character string. The possible options are "satorra.bentler.2001", "satorra.bentler.2010", "satorra.2000", and "standard". See details.
test: Character string specifying which scaled test statistics to use, in case multiple scaled test= options were requested when fitting the model(s). See details.
A.method: Character string. The possible options are "exact" and "delta". This is only used when method = "satorra.2000". It determines how the Jacobian of the constraint function (the matrix A) will be computed. Note that if A.method = "exact", the models must be nested in the parameter sense, while if A.method = "delta", they only need to be nested in the covariance matrix sense.
scaled.shifted: Logical. Only used when method = "satorra.2000". If TRUE, we use a scaled and shifted test statistic; if FALSE, we use a mean and variance adjusted (Satterthwaite style) test statistic.
type: Character. If "Chisq", the test statistic for each model is the (scaled or unscaled) model fit test statistic. If "Cf", the test statistic for each model is computed by the lavTablesFitCf function. If "browne.residual.adf" (alias "browne") or "browne.residual.nt", the standard chi-squared difference is calculated from each model's residual-based statistic.
model.names: Character vector. If provided, use these model names in the first column of the anova table.

Value

An object of class anova. When given a single argument, it simply returns the test statistic of this model. When given a sequence of objects, this function tests the models against one another, after reordering the models according to their degrees of freedom.

Details

The anova function for lavaan objects simply calls the lavTestLRT function, which has a few additional arguments.

The only test= options that currently have actual consequences are "satorra.bentler", "yuan.bentler", or "yuan.bentler.mplus" because "mean.var.adjusted" and "scaled.shifted" are currently distinguished by the scaled.shifted argument. See lavOptions for details about test= options implied by robust estimator= options). The "default" is to select the first available scaled statistic, if any. To check which test(s) were calculated when fitting your model(s), use lavInspect(fit, "options")$test.

If type = "Chisq" and the test statistics are scaled, a special scaled difference test statistic is computed. If method is "satorra.bentler.2001", a simple approximation is used described in Satorra & Bentler (2001). In some settings, this can lead to a negative test statistic. To ensure a positive test statistic, we can use the method proposed by Satorra & Bentler (2010). Alternatively, when method="satorra.2000", the original formulas of Satorra (2000) are used. The latter is used for model comparison, when ... contain additional (nested) models. Even when test statistics are scaled in object or ..., users may request the method="standard" test statistic, without a robust adjustment.

Note

If there is a lavaan model stored in object@external$h1.model, it will be added to ...

References

Satorra, A. (2000). Scaled and adjusted restricted tests in multi-sample analysis of moment structures. In Heijmans, R.D.H., Pollock, D.S.G. & Satorra, A. (eds.), Innovations in multivariate statistical analysis: A Festschrift for Heinz Neudecker (pp.233-247). London, UK: Kluwer Academic Publishers.

Satorra, A., & Bentler, P. M. (2001). A scaled difference chi-square test statistic for moment structure analysis. Psychometrika, 66(4), 507-514. doi:10.1007/BF02296192

Satorra, A., & Bentler, P. M. (2010). Ensuring postiveness of the scaled difference chi-square test statistic. Psychometrika, 75(2), 243-248. doi:10.1007/s11336-009-9135-y

Examples

HS.model <- '
    visual  =~ x1 + b1*x2 + x3
    textual =~ x4 + b2*x5 + x6
    speed   =~ x7 + b3*x8 + x9
'
fit1 <- cfa(HS.model, data = HolzingerSwineford1939)
fit0 <- cfa(HS.model, data = HolzingerSwineford1939, 
            orthogonal = TRUE)
lavTestLRT(fit1, fit0)
#> 
#> Chi-Squared Difference Test
#> 
#>      Df    AIC    BIC   Chisq Chisq diff   RMSEA Df diff Pr(>Chisq)    
#> fit1 24 7517.5 7595.3  85.305                                          
#> fit0 27 7579.7 7646.4 153.527     68.222 0.26875       3  1.026e-14 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


## When multiple test statistics are selected when the model is fitted,
## use the type= and test= arguments to select a test for comparison.

## refit models, requesting 6 test statistics (in addition to "standard")
t6.1 <- cfa(HS.model, data = HolzingerSwineford1939,
            test = c("browne.residual.adf","scaled.shifted","mean.var.adjusted",
                     "satorra.bentler", "yuan.bentler", "yuan.bentler.mplus"))
#> Warning: lavaan->lav_options_set():  
#>    observed.information for ALL test statistics is set to h1.
t6.0 <- cfa(HS.model, data = HolzingerSwineford1939, orthogonal = TRUE,
            test = c("browne.residual.adf","scaled.shifted","mean.var.adjusted",
                     "satorra.bentler", "yuan.bentler", "yuan.bentler.mplus"))
#> Warning: lavaan->lav_options_set():  
#>    observed.information for ALL test statistics is set to h1.

## By default (test="default", type="Chisq"), the first scaled statistic
## requested will be used. Here, that is "scaled.shifted"
lavTestLRT(t6.1, t6.0)
#> 
#> Scaled Chi-Squared Difference Test (method = “satorra.2000”)
#> 
#> lavaan->lavTestLRT():  
#>    lavaan NOTE: The “Chisq” column contains standard test statistics, not the 
#>    robust test that should be reported per model. A robust difference test is 
#>    a function of two standard (not robust) statistics.
#> 
#>      Df    AIC    BIC   Chisq Chisq diff   RMSEA Df diff Pr(>Chisq)    
#> t6.1 24 7517.5 7595.3  85.305                                          
#> t6.0 27 7579.7 7646.4 153.527     62.699 0.26875       3  1.558e-13 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## But even if "satorra.bentler" were requested first, method="satorra.2000"
## provides the scaled-shifted chi-squared difference test:
lavTestLRT(t6.1, t6.0, method = "satorra.2000")
#> 
#> Scaled Chi-Squared Difference Test (method = “satorra.2000”)
#> 
#> lavaan->lavTestLRT():  
#>    lavaan NOTE: The “Chisq” column contains standard test statistics, not the 
#>    robust test that should be reported per model. A robust difference test is 
#>    a function of two standard (not robust) statistics.
#> 
#>      Df    AIC    BIC   Chisq Chisq diff   RMSEA Df diff Pr(>Chisq)    
#> t6.1 24 7517.5 7595.3  85.305                                          
#> t6.0 27 7579.7 7646.4 153.527     62.699 0.26875       3  1.558e-13 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## == lavTestLRT(update(t6.1, test = "scaled.shifted"), update(t6.0, test = "scaled.shifted"))

## The mean- and variance-adjusted (Satterthwaite) statistic implies
## scaled.shifted = FALSE
lavTestLRT(t6.1, t6.0, method = "satorra.2000", scaled.shifted = FALSE)
#> 
#> Scaled Chi-Squared Difference Test (method = “satorra.2000”)
#> 
#> lavaan->lavTestLRT():  
#>    lavaan NOTE: The “Chisq” column contains standard test statistics, not the 
#>    robust test that should be reported per model. A robust difference test is 
#>    a function of two standard (not robust) statistics.
#> 
#>      Df    AIC    BIC   Chisq Chisq diff   RMSEA Df diff Pr(>Chisq)    
#> t6.1 24 7517.5 7595.3  85.305                                          
#> t6.0 27 7579.7 7646.4 153.527     61.316 0.26805   2.875  2.479e-13 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

## Because "satorra.bentler" is not the first scaled test in the list,
## we MUST request it explicitly:
lavTestLRT(t6.1, t6.0, test = "satorra.bentler") # method="satorra.bentler.2001"
#> 
#> Scaled Chi-Squared Difference Test (method = “satorra.bentler.2001”)
#> 
#> lavaan->lavTestLRT():  
#>    lavaan NOTE: The “Chisq” column contains standard test statistics, not the 
#>    robust test that should be reported per model. A robust difference test is 
#>    a function of two standard (not robust) statistics.
#> 
#>      Df    AIC    BIC   Chisq Chisq diff   RMSEA Df diff Pr(>Chisq)    
#> t6.1 24 7517.5 7595.3  85.305                                          
#> t6.0 27 7579.7 7646.4 153.527     55.899 0.26739       3  4.414e-12 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## == lavTestLRT(update(t6.1, test = "satorra.bentler"),
##               update(t6.0, test = "satorra.bentler"))
## The "strictly-positive test" is necessary when the above test is < 0:
lavTestLRT(t6.1, t6.0, test = "satorra.bentler", method = "satorra.bentler.2010")
#> 
#> Scaled Chi-Squared Difference Test (method = “satorra.bentler.2010”)
#> 
#> lavaan->lavTestLRT():  
#>    lavaan NOTE: The “Chisq” column contains standard test statistics, not the 
#>    robust test that should be reported per model. A robust difference test is 
#>    a function of two standard (not robust) statistics.
#> 
#>      Df    AIC    BIC   Chisq Chisq diff   RMSEA Df diff Pr(>Chisq)    
#> t6.1 24 7517.5 7595.3  85.305                                          
#> t6.0 27 7579.7 7646.4 153.527     55.326 0.26731       3  5.851e-12 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

## Likewise, other scaled statistics can be selected:
lavTestLRT(t6.1, t6.0, test = "yuan.bentler")
#> 
#> Scaled Chi-Squared Difference Test (method = “satorra.bentler.2001”)
#> 
#> lavaan->lavTestLRT():  
#>    lavaan NOTE: The “Chisq” column contains standard test statistics, not the 
#>    robust test that should be reported per model. A robust difference test is 
#>    a function of two standard (not robust) statistics.
#> 
#>      Df    AIC    BIC   Chisq Chisq diff   RMSEA Df diff Pr(>Chisq)    
#> t6.1 24 7517.5 7595.3  85.305                                          
#> t6.0 27 7579.7 7646.4 153.527     55.899 0.26739       3  4.414e-12 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## == lavTestLRT(update(t6.1, test = "yuan.bentler"),
##               update(t6.0, test = "yuan.bentler"))
lavTestLRT(t6.1, t6.0, test = "yuan.bentler.mplus")
#> 
#> Scaled Chi-Squared Difference Test (method = “satorra.bentler.2001”)
#> 
#> lavaan->lavTestLRT():  
#>    lavaan NOTE: The “Chisq” column contains standard test statistics, not the 
#>    robust test that should be reported per model. A robust difference test is 
#>    a function of two standard (not robust) statistics.
#> 
#>      Df    AIC    BIC   Chisq Chisq diff   RMSEA Df diff Pr(>Chisq)    
#> t6.1 24 7517.5 7595.3  85.305                                          
#> t6.0 27 7579.7 7646.4 153.527     66.195 0.26856       3  2.785e-14 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## == lavTestLRT(update(t6.1, test = "yuan.bentler.mplus"),
##               update(t6.0, test = "yuan.bentler.mplus"))

## To request the difference between Browne's (1984) residual-based statistics,
## rather than statistics based on the fitted model's discrepancy function,
## use the type= argument:
lavTestLRT(t6.1, t6.0, type = "browne.residual.adf")
#> 
#> Chi-Squared Difference Test based on Browne's residual (ADF) Test
#> 
#>      Df    AIC    BIC   Chisq Chisq diff   RMSEA Df diff Pr(>Chisq)    
#> t6.1 24 7517.5 7595.3  82.408                                          
#> t6.0 27 7579.7 7646.4 106.942     24.534 0.15443       3  1.932e-05 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

## Despite requesting multiple robust tests, it is still possible to obtain
## the standard chi-squared difference test (i.e., without a robust correction)
lavTestLRT(t6.1, t6.0, method = "standard")
#> 
#> Chi-Squared Difference Test
#> 
#>      Df    AIC    BIC   Chisq Chisq diff   RMSEA Df diff Pr(>Chisq)    
#> t6.1 24 7517.5 7595.3  85.305                                          
#> t6.0 27 7579.7 7646.4 153.527     68.222 0.26875       3  1.026e-14 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
## == lavTestLRT(update(t6.1, test = "standard"), update(t6.0, test = "standard"))