Invariance Syntax Builder

1. Say whether the items are continuous or ordered categorical, in the first box under The model. That choice decides which of the two sequences is written, which estimators are offered, which levels can be ticked, and whether the number of response categories and the parameterization are asked for at all.
2. Write the factor definitions, one factor to a line, exactly as they would be written in Mplus: F1 BY x1 x2 x3; The hyphen range notation is read, so F1 BY x1-x4; is the same definition. Each item may belong to one factor only.
3. Give the number of groups, the name of the grouping variable, and the code and the name of each group, written as 1 = male, 2 = female. The code is the value the grouping variable takes in the data file; the name is what Mplus calls that group, so it obeys the naming rules of a variable.
4. Choose the estimator. Which difference test the sequence can use follows from it: with MLMV, WLSMV and ULSMV the page pairs a SAVEDATA: DIFFTEST = file to an ANALYSIS: DIFFTEST = file between consecutive levels, and with the others it does not.
5. With ordered categorical items, give the number of response categories: one number for every item, and, for items that differ from it, a number of their own in the table under Give particular items a different number of categories. An item with fewer than four categories contributes no degrees of freedom to the threshold-invariance test, which is why the number matters here. A model that mixes binary items with items having three or more categories is not written by this version of the generator: its minimal identification constraints must be item-specific, and silently applying one all-item rule would test a different model.
6. Tick the levels of the sequence to write, choose how the factors are identified, and, with ordered categorical items, choose the parameterization. These are the choices that are not part of the description of the model.
7. Press Write the inputs. Each input appears with the checks it passed under it. Copy one with the button beside its heading, or press Download every input as a separate .inp file to save them all, then run them in Mplus in the order they are written.

The generator reports descriptions for which it cannot write an identified sequence: malformed factor definitions, items assigned to more than one factor, invalid or reserved Mplus names, duplicate group names, fewer than two groups, category counts outside 2–12, and mixed binary/non-binary ordered indicators requiring item-specific identification constraints that this version does not implement.

Write the sequence for these indicators

Continuous indicators: configural, metric, scalar, strict Ordered categorical indicators: the sequence of Wu and Estabrook (2016)

Write these levels of the sequence

The configural model is always written, because every later level is compared with it or with a level built on it.

Name the data file and the columns, so that the inputs run as they stand

Everything in here is optional. Left empty, the inputs are still complete Mplus syntax, but the FILE line names data.dat and the NAMES statement lists the grouping variable and the items only, so both have to be corrected before the inputs are run.

Title the run The name of the level is added to it in each input.

Read the data from this file

List every column of the data file, in order This becomes the NAMES statement, so it must name every column of the file in the order they are in it, whether the model uses them or not. The hyphen range notation is read here too. USEVARIABLES is written from the items of the model in either case.

Missing values are coded The value that stands for a missing answer in the data file, such as -999. Written as MISSING = ALL (-999);.

Begin every file name with The inputs are named prefix_configural.inp and so on.

With continuous indicators the observed variable has a metric of its own. Its mean and its variance are estimated in every group, so an intercept and a residual variance can be estimated in every group as well, and the sequence configural, metric, scalar, strict (Millsap, 2011) adds one type of constraint at a time to a model that is already identified.

With ordered categories nothing is observed but the proportion of respondents in each category and the association between items. The model puts a normally distributed latent response variable behind each item and cuts it at thresholds, and that latent response has no metric of its own: its location and its scale are conventions. Wu and Estabrook (2016) write the transformation that changes those conventions without changing the probability of any response pattern. It has four parts in each group: a shift and a rescaling for every latent response, and a shift and a rescaling for every factor, so $2(p+m)$ constraints per group are needed to identify a model with p items and m factors.

The consequence is that the identification condition of a baseline model is not neutral with respect to the invariance tests that follow it. Once loadings are held equal across groups, a condition such as fixing the variance of every latent response to one in every group stops being a free choice of metric and becomes a restriction, and the same invariance hypothesis tested from two differently identified baselines gives two different chi-squares. Wu and Estabrook show a two-group example in which loading invariance is rejected under two identification conditions and retained under a third, with the same data and the same true parameters. Their answer is to identify each level of invariance separately, which is what this page writes.

Three consequences follow for the sequence itself.

• Invariance of loadings alone, of intercepts alone, and of residual variances alone cannot be tested. A model with only one of these constrained is statistically equivalent to the configural model, so the difference test has no degrees of freedom to spend. Threshold invariance has to be established first (Wu & Estabrook, 2016, Proposition 5 and Section 6).
• How much of threshold invariance can be tested depends on the number of categories. A shift and a rescaling of one latent response can always be chosen to make one threshold match, and can be chosen to make two thresholds match, but not three. An item with C categories therefore contributes $\max (C-3,0)(G-1)$ degrees of freedom to the test of threshold invariance across G groups: none for a binary or three-category item, one per extra group for a four-category item.
• Binary items are a separate case. With one threshold per item, threshold invariance and threshold-and-loading invariance are both statistically equivalent to the configural model (Wu & Estabrook, 2016, Propositions 3 and 6), and the model that holds thresholds invariant while freeing the intercepts and the scales of the other groups is not even identified. When all items are binary this page drops those two levels and goes from the configural model to the model with invariant thresholds, loadings and intercepts. Mixed binary/polytomous models require item-specific identification constraints and are refused rather than approximated. The intercepts are not the only class that can carry that step: holding the residual variances of the latent responses equal instead, with the intercepts left free, imposes the same number of restrictions on the configural model, and applied accounts of the binary sequence often take that route. The two models are each a restriction of the configural one and neither is a restriction of the other, so a difference test between them is not defined; this page writes the first of the two and names it in the heading of the input.

The two articles differ in how far they take the sequence and in what they call the levels. Wu and Estabrook (2016) name each model by the parameters it holds invariant and set out the whole hierarchy of sixteen models, of which ten are statistically distinct for polytomous items and five for binary items. Svetina, Rutkowski and Rutkowski (2020) follow the same identification conditions but write out three models only, which they name after the propositions they come from: the baseline model, Proposition 4 for threshold invariance and Proposition 7 for threshold and loading invariance. They also point out that the resulting order — thresholds first, then loadings — differs from the order in which applied work usually proceeds, where loadings are tested before thresholds and intercepts. One detail in their tutorial is a slip: the text and the annotation of the Proposition 7 input say that the item intercept is fixed at one in the first group and give [y1-y4@1];, while Condition 19 of Wu and Estabrook, which that step implements, sets those intercepts to zero. This page writes @0.

• Meredith, W. (1993). Measurement invariance, factor analysis and factorial invariance. Psychometrika, 58(4), 525–543. https://doi.org/10.1007/BF02294825
• Millsap, R. E. (2011). Statistical approaches to measurement invariance. Routledge.
• Millsap, R. E., & Yun-Tein, J. (2004). Assessing factorial invariance in ordered categorical measures. Multivariate Behavioral Research, 39(3), 479–515. https://doi.org/10.1207/S15327906MBR3903_4
• Muthén, B., & Asparouhov, T. (2002). Latent variable analysis with categorical outcomes: Multiple-group and growth modeling in Mplus (Mplus Web Notes No. 4).
• Svetina, D., Rutkowski, L., & Rutkowski, D. (2020). Multiple-group invariance with categorical outcomes using updated guidelines: An illustration using Mplus and the lavaan/semTools packages. Structural Equation Modeling, 27(1), 111–130. https://doi.org/10.1080/10705511.2019.1602776
• Vandenberg, R. J., & Lance, C. E. (2000). A review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational research. Organizational Research Methods, 3(1), 4–70. https://doi.org/10.1177/109442810031002
• Wu, H., & Estabrook, R. (2016). Identification of confirmatory factor analysis models of different levels of invariance for ordered categorical outcomes. Psychometrika, 81(4), 1014–1045. https://doi.org/10.1007/s11336-016-9506-0