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Chi-square difference tests are frequently used to test differences between nested models in confirmatory factor analysis, path analysis and structural equation modeling. Nested models are two models (or more if one is fitting a series of models) that are identical except that oneof the models constrains some of the parameters (the null model) and one does not have those constraints(the alternative model). Examples of this include the introduction of a set ofdichotomous predictors representing a single nominal (categorical) variable to the model, or a test for differences across groups in a multiplegroup model. Typically a chi-square difference test involves calculating the difference between the chi-squarestatistic for the null and alternative models, the resulting statistic is distributed chi-squarewith degrees of freedom equal to the difference in the degrees of freedom between the two models. However, when a model is run in Mplus using the MLM or MLR estimators, the following warning message is displayed, warning the user that the standard chi-square difference test is not valid:

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This page shows how to calculate the correct chi-square difference test statistic formodels estimated with the MLM or MLR estimator as described on the Mplus website (see http://www.statmodel.com/chidiff.shtmlfor additional information). For both MLM and MLR the Satorra-Bentler scaled chi-square difference test can be used. An additional test is available to test for differences in nested models that use the MLR estimator; this test is based on the log-likelihoods from nested models. Information on how to calculate the Satorra-Benter scaled chi-square test appears first, followed by information on using the log-likelihood to calculate a difference test. Each section includes both a discussion of the formulae and an example.

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Cautions

While the above message appears for a number of estimators, the procedure described below is for use with the MLM and MLR estimators only. If the model uses the MLMV or WLSMV estimator, the difftest command can be used to test for differences across nested models. A Wald test can also be used to test nested models using the model test command. See the Mplus manual for more information on these commands. If you are unsure of what estimator is being used in your model, you can find the estimator listed towards the top of the output file.

Also keep in mind that this type of test is only valid if the models are nested; that is, the modelsmust be the same except that one of the models includes additional constraintson the parameters.

The Satorra-Bentler scaled chi-square difference test

In order to calculate the Satorra-Bentler scaled chi-square difference test, we willneed a number of pieces of information. Below is a list of the information needed, along with the symbol (i.e., letter and number) used to represent each value.

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In order to calculate the test statistic, T, we first need to calculatethe value cd:

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Once we have calculated cd, we can compute:

T is distributed chi-square with degrees of freedom:

Example

Below are two Mplus input files. On the left is the null model, which constrainsthe paths from x2 to f1 and from x3 to f1 to be zero (using x2@0 x3@0 respectively). On the right is the alternative model which estimates the regression path from x2 to f1 and from x3 to f1.

Once we have run the model, we can find all the information we need to compute the test statistic in the section of the output labeled 'Chi-Square Test of Model Fit.' This section of the output appears below (all other output is omitted). Output for the null model appears on the left, and output for the alternative model appears on the right. The information we need appears in bold. Below the output, each of the values highlighted appears on the appropriate line. If your output does not include the scaling correction factors (c0 and c1), instructions for calculating them from other output appears below.

Once we have identified and noted all the necessary values in our output, the test statistic (T) is computed in two steps.

The test statistic T is distributed chi-square with df = d0-d1. We can look up the p-value for a chi-square statistic of 123.25, with two degrees of freedom using a table or some other method (chi2(2) = 123.25, p < 0.01).

Calculating the scaling correction factor

If your output does not include the scaling correction factor, you can calculate thisvalue from the unadjusted chi-square and the Satorra-Bentler scaled chi-square statistic.So, we are going to pretend that we have output that contains only the unadjusted chi-square,the Satorra-Bentler scaled chi-square statistic and the degrees of freedom.

Given those values, the scaling correction factors (c0 and c1) can be calculatedas shown below. c0 is the scaling correction factor for the null model, and c1is the scaling correction factor for the alternative model.

For example, taking the appropriate values from the output:

The scaling correction factors can be calculated:

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A test using the log-likelihood

For the MLR estimator there is an additional test for nested models.This test compares the log-likelihoods for the null and alternative models ratherthan the chi-square values. Below is a list of the information needed, along with the symbol (i.e., letter and number) used to represent each value.

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From this information the tests statistic TRd can be calculated (note that cd is calculated first then used to calculate TRd), along with the degrees of freedom:

Example

Below is an example of output from two nested models. The informationnecessary to calculate this test statistic appears in the sections of outputlabeled “Loglikelihood” and the section immediately after it labeled “Information Criteria” (all other output has been omitted). The informationneeded for the calculations appears in bold. Note that the log likelihoods andscaling correction factors identified as H0 should be used, not the values labeled H1. Below the output, each of the values highlighted appears on the appropriate line.