Applications of SEM are usually based on the assumption that the analyzed data stem from a single population, so that a unique global model represents all the observations well. However, in many real-world applications, this assumption of homogeneity is unrealistic. 

1. To examine heterogeneity effects (e.g., cross-cultural or gender differences) in a business research.

2. To determine the existence of significant differences across group-specific parameter estimates (e.g., outer weights, outer loadings, and path coefficients). 

3. To test for variations between different groups in two identical models when the groups are known.

4. To identify meaningful differences in multiple relationships across group-specific results.

Step 1: Data Preparation

One of the important aspects when dealing with MGA is sufficient statistical power once the data’s sample is divided into subgroups. In other words, when performing MGA, it is imperative to ensure that the subgroups have sufficient power to detect the moderating effect (Becker et al., 2013; Hair et al., 2017). 

The easiest way to ensure statistical power is to have large sample sizes for both groups. A statistical power of 80% should be computed and reported using G*Power to establish whether the absence of any moderating effect is due to low statistical power rather than the lack of a true moderating effect (Aguinis et al., 2017; Cohen, 1988; Hair et al., 2017; Memon et al., 2020).

Even if total sample size is relatively large, unequal sample sizes across the moderator-based subgroups would decrease statistical power and lead to the underestimation of moderating effects (Hair et al., 2017). To address this issue, researchers should try to select similar sample sizes for each group so that sample variance is maximized (see Aguinis et al., 2017). 

Step 2: Generate Data Groups for MGA

After data preparation, the second step is to create groups by selecting the categorical variable of interest from the dataset. Theory and observation play an important role in generating the data groups. For instance, if empirical studies show that males and females produce different results in terms of the effects of a product’s price and quality on actual purchase behaviour, then the researcher needs to position gender as a moderator to examine the overall relationships. This process would be similar if the researcher is interested in comparing a path coefficient across more than two groups, as in cross-cultural studies (Indonesia vs Thailand, Indonesia vs Singapore, and Indonesia vs Malaysia).

Step 3: Test for Measurement Invariance

Measurement invariance or measurement equivalence confirms that the measurement models specify measures of the same attribute under different conditions (Henseler et al., 2016). Differences in the paths (or β values) between latent variables could stem from different meanings attributed by a group’s respondents to the phenomena being examined, rather than from true differences in the structural relationships. 

The reasons for these discrepancies include: 

1. Crossnational differences that emerge from culture-specific response styles (e.g., Johnson et al., 2005), such as acquiescence, i.e. the tendency to agree with questions regardless of their content (Sarstedt and Mooi, 2019); 

2. Individual characteristics (e.g., gender, ethnicity, etc.) that entail responding to instruments in systematically different ways; and 

3. The use of of available scale options differently, i.e. the tendency to choose or not to choose extremes. 

Hult et al. (2008) stressed that failure to establish invariance can easily lead to the low power of statistical tests, the poor precision of estimators, and misleading results. Therefore, it is a fundamental step prior to conducting MGA because it gives researchers the confidence that group differences in model estimates do not result from the distinctive content and/or meanings of the latent variable across groups. 

Three steps in the measurement invariance of composite models (MICOM) procedure, namely, 

1. Assessments configural invariance (Step I), 

2. Assessment of compositional invariance (Step II), and 

3. Assessment of the equality of a composite’s mean value and variance across groups (Step III) (see Hair et al., 2018 for further explanation of these invariance steps). 

If both configural invariance (Step 1) and compositional invariance (Step 2) are established, then partial measurement invariance is confirmed, and researchers can proceed to compare the path coefficients with the MGA. 

On the other hand, full measurement invariance is established if, in addition to fulfilling partial measurement invariance (Step 1 and Step 2), composites exhibit equal means and variances across the groups (Step 3). With full measurement invariance, pooling the data is a possible option (i.e. it will increase statistical power), rendering MGA unnecessary (see Henseler et al., 2016). 

However, if Step 1, Step 2, and either requirement of Step 3 (either equality of composite variance or equality of composite mean) are achieved, then the researcher can claim partial measurement invariance and proceed with MGA.

Step 4: Tests for Multigroup Comparisons 

Once measurement invariance, either partial or full, is established using MICOM, the researcher can begin assessing group differences using MGA in PLSPM. MGA is often used to compare parameters (e.g., path coefficients, outer weights, outer loadings, etc.) between two or more groups when they are known a priori (Hair et al., 2018). 

To make group comparisons, SmartPLS offers five different assessment approaches based on bootstrapping (Hair et al., 2018). The SmartPLS software has three approaches, i.e. Henseler's bootstrap-based MGA (Henseler et al., 2009), the Parametric Test (Keil et al., 2000), and the Welch-Satterthwait Test (Welch, 1947). The fourth approach to assess group comparisons is the permutation test (Chin and Dibbern, 2010), which can be estimated using the MICOM path coefficient option in SmartPLS. However, if researchers would like to compare more than two groups (e.g., Malaysia vs Australia, Malaysia vs Singapore, and Australia vs Singapore), they can opt for the fifth approach, which is the Omnibus Test of Group Differences (OTG) (Sarstedt et al., 2011). 

Each of these approaches has its advantages and disadvantages (see Hair et al., 2018). For instance, the parametric test approach is rather lenient and subject to type I errors. This approach also relies on the assumption of normal distribution, so it is not consistent with the nature of PLS that uses non-parametric assumptions (Hair et al., 2018; Sarstedt et al., 2011). The Welch-Satterthwaite test, meanwhile, is a variant of the parametric test that does not assume equal variances when comparing the means of two groups. In contrast, both Henseler’s PLS-MGA test (Henseler et al., 2009) and the permutation test use nonparametric assumptions 

Henseler’s PLS-MGA procedure (Henseler et al., 2009) is a result of the probability value of a one-tailed test by comparing each bootstrap estimate of one group to all the bootstrap estimates of the same parameter in the other group (Hair et al., 2011). Since the bootstrap distributions for this procedure are not necessarily symmetrical, the use of a two-sided hypothesis is not possible (Henseler et al., 2009) This approach appears to be appropriate; however, the interpretation of the results may be somewhat challenging to due to the condition of the one-tailed test. To keep the explanation simple, researchers must be aware that Henseler’s PLS-MGA result is significant at the 5% probability level, whereby the p-value for the difference in group-specific path coefficients should be smaller than 0.05 or larger than 0.95. Next, the permutation test is a separate option that is run concurrently with Step II of the test for measurement invariance. That is, the output of path coefficients from the measurement invariance test is a means of comparing the path coefficients of subgroups. Notably, Chin and Dibbern (2010) and Hair et al. (2018) recommend the permutation test due to its ability to control for type I errors and its relatively conservative nature compared to the parametric test. Nevertheless, researchers must ensure that there are no large differences in group-specific sample sizes to prevent adverse consequences on the permutation test’s performance. Hair et al. (2018) suggested that when one group’s sample is more than double the size of the other group, researchers are recommended to choose between two options, which are (i) to select Henseler’s PLS-MGA approach (in the case of testing a one-sided hypothesis) or (ii) to randomly draw another sample for the large group that is comparable in size to the smaller group, and subsequently compare the two samples using the permutation test. 

The fifth approach, the OTG, combines the bootstrapping procedure with the permutation test to mimic an overall F-test. Although it sounds like an ANOVA F-test to analyse more than two groups, this procedure maintains the Type I error level (familywise error rate) and delivers an acceptable level of statistical power while not relying on distribution assumptions. The key disadvantage of this procedure is that the OTG has not yet been included in SmartPLS. Researchers interested in the OTG approach can attempt it using the R code to generate results (available under the download section at https://www.pls-sem.net/downloads/advanced-issues-in-pls-sem1/). Moreover, a researcher cannot assume that use of the OTG is sufficient, given that this procedure always demonstrates significant parameter differences, especially when the number of bootstraps runs increases. On top of that, OTG results do not provide a clear answer on the presence of specific differences between path coefficients across groups. Consequently, researchers must conduct all possible pairwise group comparisons while controlling the familywise error rate (Type I error) to avoid alpha inflation. This is easily controlled using either the Bonferroni correction or the Šidák correction (Hair et al., 2018). The Bonferroni correction can be applied with the formula alpha/m. For example, if researchers compare five groups, there would be m=10 comparisons, yielding a significance level of 0.05/10 (i.e., 0.005), instead of 0.05. On the other hand, the Šidák correction uses a different formula, that is, 1-(1- alpha)1/m. For example, for five groups with 10 comparisons, one would use a significance level of 1-(1-0.05)1/10 (i.e., 0.005116) instead of 0.05. Both corrections are plausible, but researchers should note that the Bonferroni correction has weaker statistical power than the Šidák correction in detecting pairwise group comparisons when there is an exponential increase in the number of comparisons (see Hair et al., 2018). Readers can refer to some examples of using the OTG with the Šidák correction in papers published in tourism (see Ting et al. 2019) and retailing (Osakwe et al., 2020). 

Download the data here, and draw the following model. 

Steps 1: Data Preparation

Check the minimum sample size requirement by using G*Power software.  

Step 2: Generate Data Groups 

In the data, there are two groups, 1 represents Indonesia and 2 represents Malaysia. Create the groups. 

Step 3: Measurement Invariance Test using MICOM

1. Assess configural invariance. 

2. Assess compositional Invariance using permutation. 

3. Assess the equality of a composite’s mean value and variance across groups. 

Step 4: Test of MGA Comparisons 

Find the report of  PLS-MGA results, Parametric PLS Multigroup Test results, Welch-Satterthwaite Test results, and bootstrapping results for path coefficient comparison. 

Cheah, J. H., Thurasamy, R., Memon, M. A., Chuah, F., & Ting, H. (2020). Multigroup analysis using smartpls: step-by-step guidelines for business research. Asian Journal of Business Research, 10(3), I-XIX.

Matthews, L. (2017). Applying multigroup analysis in PLS-SEM: A step-by-step process. In Partial least squares path modeling (pp. 219-243). Springer, Cham.

Nawanir, G., Lim, K. T., Ramayah, T., Mahmud, F., Lee, K. L., & Maarof, M. G. (2020). Synergistic effect of lean practices on lead time reduction: mediating role of manufacturing flexibility. Benchmarking: An International Journal, 27(5), 1815-1842.