Multi-Group Analysis (PLS-MGA) with MICOM

What multi-group analysis tests

Multi-group analysis (MGA) asks whether the relationships in a structural equation model are the same in different groups of respondents. A typical research question is: is the effect of trust on satisfaction stronger for online shoppers than for store shoppers? or does perceived usefulness drive adoption intention more for younger than for older users? The grouping variable, such as gender, country, channel, customer type, industry or experience level, acts as a categorical moderator. Rather than adding an interaction term, MGA estimates the full model separately in each group and compares the path coefficients.

In partial least squares structural equation modeling (PLS-SEM), MGA is the standard way to analyse categorical moderation and to check whether findings hold across subgroups, a point reviewers frequently raise under the heading of “unobserved” or “observed heterogeneity”. This tool implements the full workflow recommended in the PLS-SEM literature: measurement invariance testing with MICOM, followed by four tests of path coefficient differences.

Step 1: estimate the model in each group

The tool estimates your PLS path model with the path weighting scheme and mode A (reflective) outer weights, exactly as in the IPMA and cIPMA tools, three times: once for group 1, once for group 2 and once for the pooled sample. Indicators are standardised within each estimation, and composite signs are oriented so that loadings are mostly positive, which avoids spurious sign flips between groups and resamples. For each group, the tool then draws bootstrap samples (500 by default) to obtain standard errors and p-values for every path, and reports R² values and outer loadings and weights side by side.

Step 2: check measurement invariance with MICOM

Comparing path coefficients only makes sense if the constructs mean the same thing in both groups. If, say, the trust composite is formed mainly from one indicator in one group and from another indicator in the other group, a difference in the trust → satisfaction path may simply reflect different measurement. Henseler, Ringle and Sarstedt (2016) proposed the measurement invariance of composite models (MICOM) procedure for PLS-SEM, which has three steps:

1. Configural invariance. The same indicators, the same data treatment and the same algorithm settings are used in every group. This tool guarantees step 1 by design, because both groups are estimated from the same model syntax and settings.
2. Compositional invariance. For each construct, composite scores are computed for all respondents twice, once with group 1’s outer weights and once with group 2’s. Their correlation c should be close to 1. A permutation test, which randomly reassigns respondents to groups many times, gives the distribution of c under invariance; compositional invariance holds when the observed c is not below the 5% quantile of that distribution.
3. Equal composite means and variances. Using the pooled-sample weights, the tool compares the mean and the log-variance of each composite between groups. If both differences fall within their 95% permutation intervals, the construct shows full invariance.

When steps 1 and 2 hold, the model has partial measurement invariance, which is sufficient to compare path coefficients between groups with MGA. If step 3 also holds for all constructs, there is full measurement invariance, and the groups may also be pooled into a single analysis. If step 2 fails for a construct, MGA results involving that construct should not be interpreted; the tool flags this clearly.

Step 3: compare the path coefficients

For every structural path, the tool reports the coefficient in each group, their difference (group 1 minus group 2) and four p-values for the null hypothesis that the paths are equal:

• Permutation test (Chin and Dibbern, 2010). Group labels are randomly reshuffled (1,000 times by default), the model is re-estimated in both permuted groups, and the p-value is the share of permuted differences at least as large in absolute value as the observed one. It makes no distributional assumptions, keeps the type I error rate under control and is the test most often recommended (Hair et al., 2022). It works best when group sizes are not too different.
• Henseler’s PLS-MGA (Henseler, Ringle and Sinkovics, 2009). A non-parametric test that compares every bootstrap estimate of group 1 with every bootstrap estimate of group 2, after centring the bootstrap distributions on the original estimates. It gives the probability that the path in group 1 is smaller than in group 2; the tool reports the two-tailed version, as SmartPLS 4 does.
• Parametric test (Keil et al., 2000). A t-test that uses the bootstrap standard errors of both groups and assumes equal variances, with n₁ + n₂ − 2 degrees of freedom.
• Welch-Satterthwaite test. The same idea without the equal-variance assumption, with Satterthwaite’s approximation for the degrees of freedom. It is preferable to the parametric test when group sizes or standard errors differ.

When the tests agree, the conclusion is robust. When they disagree, which mostly happens with p-values close to .05, report the permutation test as the primary result and mention the others.

Worked example with the built-in data

The example contains 420 simulated customers, half buying online and half in store, who answered four trust items, four satisfaction items and three loyalty items on five-point scales. The model is TR → SAT, SAT → LOY and TR → LOY. MICOM finds compositional invariance for all three constructs (c ≥ .998, above the 5% permutation quantiles) and equal composite means and variances, so full measurement invariance is established. The trust → satisfaction path is much stronger online (β = 0.57) than in store (β = 0.22), a difference of 0.34 that all four tests find significant (p < .001). Satisfaction → loyalty is somewhat weaker online (0.35 vs 0.49), but this difference is not significant (permutation p = .089), and trust → loyalty does not differ (p = .45). The managerial reading is that building trust pays off in satisfaction mainly in the online channel. R² for satisfaction is .32 online but only .05 in store, which tells the same story.

Preparing your data

• Use one row per respondent with all indicator columns in the model and one categorical grouping column. Text or numeric codes both work (for example Male/Female or 1/2).
• Select the two groups to compare. Respondents in other groups, or with a missing group, are excluded. To compare three or more groups, run the tool for each pair and adjust the p-values for multiple comparisons, for example with Holm’s method.
• Write the model in the same lavaan-style syntax as the other SEM tools: TR =~ TR1 + TR2 + TR3 for constructs and SAT ~ TR for paths. With your own data, the “Generate from my variable names” button builds a starting model from item names such as TR1, TR2.
• Rows with missing values on any indicator are excluded (listwise deletion); clean or impute missing data first with the data cleaning tool if needed.

Sample size and group balance

Each group must be large enough to estimate the model on its own. A common minimum is ten times the largest number of arrows pointing at any construct, but power analyses suggest 100 or more per group for detecting moderate differences of about .2 in path coefficients. Very unequal group sizes reduce power and make the parametric test unreliable; the permutation test is also affected when one group is more than about twice the size of the other, so interpret results with care in that case, or draw a random subsample of the larger group as a sensitivity check.

MGA, interaction terms and CB-SEM

An alternative to MGA for a binary moderator is to add the moderator and its interaction with the predictor to a single model, as in the moderation analysis tool. MGA is more flexible because it allows every path, and the measurement model, to differ between groups at once, and it is the conventional choice in PLS-SEM. In covariance-based SEM, the equivalent is multi-group CFA and SEM with successive equality constraints (configural, metric and scalar invariance) and chi-square difference tests, as available in AMOS and lavaan. If your study is theory-testing with reflective common factors, report CB-SEM invariance tests; if it uses PLS-SEM, MICOM and PLS-MGA are the matching procedures.

Common mistakes

• Skipping measurement invariance. Path differences without at least partial invariance are not interpretable.
• Comparing p-values instead of testing the difference. “The path is significant in group 1 but not in group 2” does not show that the paths differ; only a test of the difference does.
• Running many pairwise comparisons without correcting for multiple testing.
• Interpreting tiny but significant differences in very large samples as important; look at the size of the difference as well.
• Forming groups by splitting a continuous variable at the median, which loses information; use a continuous moderator in the moderation tool instead.

Glossary of multi-group analysis terms

• Observed heterogeneity: differences between groups that are defined by a measured variable such as gender or country.
• Unobserved heterogeneity: hidden segments with different relationships, uncovered with methods such as latent class analysis or FIMIX-PLS.
• Configural invariance: the same model specification and settings in all groups.
• Compositional invariance: composites are formed in the same way in all groups, so they measure the same thing.
• Full invariance: compositional invariance plus equal composite means and variances across groups.
• Permutation distribution: the distribution of a statistic when group labels are randomly reassigned, used as a reference for “no group difference”.
• Bootstrap standard error: the standard deviation of a coefficient across samples drawn with replacement from a group.

Reporting in APA style

Report the groups and their sizes, the MICOM results, each group’s path coefficients with their significance, the differences and the p-values of the permutation test (and ideally one other test). Example: “Measurement invariance was assessed with MICOM; compositional invariance held for all constructs (c ≥ .998), and composite means and variances did not differ, indicating full invariance. A permutation-based multi-group analysis (1,000 permutations) showed that the effect of trust on satisfaction was stronger for online (β = 0.57) than for store customers (β = 0.22), Δβ = 0.34, p < .001; the other paths did not differ.”

Key references: Henseler, Ringle and Sarstedt (2016), “Testing measurement invariance of composites using partial least squares”, International Marketing Review; Henseler, Ringle and Sinkovics (2009), Advances in International Marketing; Chin and Dibbern (2010), in Handbook of Partial Least Squares; Keil et al. (2000), MIS Quarterly; Sarstedt, Henseler and Ringle (2011), Advances in International Marketing; Hair, Hult, Ringle and Sarstedt (2022), A Primer on PLS-SEM (3rd ed.).