Robustness Checks using #SmartPLS4 - Linearity - Endogeneity - Heterogeneity

Structural equation modeling in SmartPLS relies on assumptions that can quietly break results—so robustness checks matter before interpreting paths. One key assumption is linearity: relationships between constructs should behave linearly rather than curving. In SmartPLS4, linearity is tested by adding quadratic effects for each path and running bootstrapping (recommended 5,000–10,000, using a two-tailed bias-corrected approach). When the quadratic terms show no significant p-values—and the path coefficients confirm all quadratic effects are insignificant—the model’s linearity assumption is treated as satisfied.

The next robustness step targets endogeneity, a problem where predictors correlate with error terms, distorting causal interpretation. The workflow uses the GoFian copula approach (implemented as “goian copula” in the interface). The procedure starts with single relationships: each path is tested for endogeneity, then results are copied out (e.g., into Excel) for tracking. A crucial pattern emerges: some individual paths can be significant for endogeneity while others are not. In the example, one relationship pair (from “commitment to leadership” and “leadership to reliability” are mentioned as the kinds of paths being combined) produces significant endogeneity when tested alone, but when the model tests different combinations, some pairs remain problematic while others show no endogeneity. The method scales up by testing combinations of two, then three, then four relationships—essentially checking whether endogeneity appears only in isolated paths or persists across sets.

After endogeneity checks, the robustness process moves to heterogeneity—specifically unobserved heterogeneity, where distinct subgroups in the data imply meaningfully different parameter estimates. If subgroup effects cancel out when estimating on the full sample, overall results can become misleading. SmartPLS addresses this using the FIX (finite mixture segmentation) approach. The key practical question is how many segments to test. The transcript uses a rule-of-thumb sample size of at least 85 per segment, based on a medium effect size (.15) and 80% power. With a total sample around 341, that implies testing roughly four segments (1–4).

For each candidate segmentation (1, 2, 3, and 4 segments), model selection criteria are collected—AIC3, CAIC, AIC4, BIC, and MDL5—along with entropy. The example ends with an ambiguous picture: AIC3 and CAIC point to different segment counts (AIC3 favoring four segments while CAIC favors two). AIC4 and BIC also favor two segments, and MDL5 points inconsistently (in the example, MDL5 favors one segment). Because the criteria do not unambiguously converge on a single segmentation solution, the analysis avoids further segmentation and instead treats the full dataset as a single group. That decision is used to conclude there is no clear evidence of unobserved heterogeneity affecting the results.