Conduct Additional Modeling and Analysis for 'seminr'.

SEMinRExtras adds functionality to the SEMinR package.

SEMinR (Ray, Danks, & Calero Valdez, 2026) is a domain specific language for modeling and estimating structural equation models. This is a supplementary package for SEMinR and not a standalone package. This package serves to provide additional extra methods and functions that can be used to analyze PLS-SEM models.

SEMinRExtras provides advanced SEM tools which are compatible with SEMinR. It implements several methods for evaluating and validating PLS-SEM models:

• Cross-Validated Predictive Ability Test (CVPAT) — Compare model predictive performance against benchmarks or alternative models (Liengaard et al., 2021; Sharma et al., 2022).
• Predictive Contribution of the Mediator (PCM) — Evaluate whether a mediating construct improves out-of-sample predictive accuracy by comparing Direct Antecedent (DA) and Earliest Antecedent (EA) prediction approaches (Danks, 2021).
• Combined Importance-Performance Map Analysis (cIPMA) — IPMA with NCA integration to identify constructs that are both important and necessary (Ringle & Sarstedt, 2016; Sarstedt et al., 2024; Hauff et al., 2024).
• Composite Overfit Analysis (COA) — Detect observation-level overfitting in PLS composite models via predictive deviance trees and parameter instability analysis.
• Necessary Condition Analysis (NCA) — Test whether predictors are necessary conditions for an outcome, complementing PLS-SEM's sufficiency logic (Dul, 2016; Richter et al., 2020).
• NCA-ESSE — Effect Size Sensitivity Extension that assesses how robust NCA results are to extreme response patterns (Becker et al., 2026).
• Confirmatory Tetrad Analysis (CTA-PLS) — Empirically test whether a construct's measurement model is reflective or formative, with indicator borrowing for constructs with fewer than 4 indicators (Gudergan et al., 2008).
• Unobserved Heterogeneity — Two complementary segmentation approaches for detecting latent classes in PLS-SEM:
  • FIMIX-PLS — EM-based probabilistic segmentation assuming normally distributed residuals (Hahn et al., 2002; Sarstedt et al., 2011).
  • PLS-POS — Deterministic hill-climbing segmentation that maximizes the sum of R-squared across segments, making no distributional assumptions (Becker et al., 2013).
• Congruence Testing — Bootstrapped congruence coefficient testing for construct validity (Franke, Sarstedt, & Danks, 2021).

SEMinRExtras also serves to host the example models used in the PLS-SEM in R workbook (Hair et al., 2026).

Functions

The demo files for Hair et al. (2026)

In order to access the demo files for the textbook, you can run the demo() function after loading the SEMinRExtras package.

• seminr-help-debugging
• seminr-pls-cvpat
• seminr-pls-pcm
• seminr-pls-cipma
• seminr-pls-coa
• seminr-pls-nca
• seminr-pls-fimix
• seminr-pls-cta
• seminr-pls-pos
• seminr-pls-congruence
• seminr-primer-v2-chap2
• seminr-primer-v2-chap3
• seminr-primer-v2-chap4
• seminr-primer-v2-chap5
• seminr-primer-v2-chap6
• seminr-primer-v2-chap7
• seminr-primer-v2-chap8

E.g. demo("seminr-help-debugging", package = "seminrExtras")

The Example model: Corporate Reputation

We are applying the CVPAT process to the corporate reputation example bundled with SEMinR. Since we will be comparing two models, we will first estimate and plot both models. Below you will find the competing and established models which will be compared.

Established Model

Competing Model

Example

# Create measurement model ----
corp_rep_mm_ext <- constructs(
  composite("QUAL", multi_items("qual_", 1:8), weights = mode_B),
  composite("PERF", multi_items("perf_", 1:5), weights = mode_B),
  composite("CSOR", multi_items("csor_", 1:5), weights = mode_B),
  composite("ATTR", multi_items("attr_", 1:3), weights = mode_B),
  composite("COMP", multi_items("comp_", 1:3)),
  composite("LIKE", multi_items("like_", 1:3)),
  composite("CUSA", single_item("cusa")),
  composite("CUSL", multi_items("cusl_", 1:3))
)

alt_mm <- constructs(
  composite("QUAL", multi_items("qual_", 1:8), weights = mode_B),
  composite("PERF", multi_items("perf_", 1:5), weights = mode_B),
  composite("CSOR", multi_items("csor_", 1:5), weights = mode_B),
  composite("ATTR", multi_items("attr_", 1:3), weights = mode_B),
  composite("COMP", multi_items("comp_", 1:3)),
  composite("LIKE", multi_items("like_", 1:3)),
  composite("CUSA", single_item("cusa")),
  composite("CUSL", multi_items("cusl_", 1:3))
)

# Create structural model ----
corp_rep_sm_ext <- relationships(
  paths(from = c("QUAL", "PERF", "CSOR", "ATTR"), to = c("COMP", "LIKE")),
  paths(from = c("COMP", "LIKE"), to = c("CUSA", "CUSL")),
  paths(from = c("CUSA"),         to = c("CUSL"))
)
alt_sm <- relationships(
  paths(from = c("QUAL", "PERF", "CSOR", "ATTR"), to = c("COMP", "LIKE")),
  paths(from = c("COMP", "LIKE"), to = c("CUSA")),
  paths(from = c("CUSA"),         to = c("CUSL"))
)


# Estimate the models ----
established_model <- estimate_pls(
  data = corp_rep_data,
  measurement_model = corp_rep_mm_ext,
  structural_model  = corp_rep_sm_ext,
  missing = mean_replacement,
  missing_value = "-99")

competing_model <- estimate_pls(
  data = corp_rep_data,
  measurement_model = alt_mm,
  structural_model  = alt_sm,
  missing = mean_replacement,
  missing_value = "-99")

# Function to compare the Loss of two models
compare_results <- assess_cvpat_compare(established_model = established_model,
                                        alternative_model = competing_model,
                                        testtype = "two.sided",
                                        nboot = 2000,
                                        technique = predict_DA,
                                        seed = 123,
                                        noFolds = 10,
                                        reps = 10,
                                        cores = 1)


print(compare_results,
      digits = 3)

# Assess the base model ----
assess_results <- assess_cvpat(established_model,
                               seed = 123,
                               cores = 1)
print(assess_results$CVPAT_compare_LM,
      digits = 3)
print(assess_results$CVPAT_compare_IA,
      digits = 3)

First, conduct CVPAT analysis of the established model.

# Assess the base model ----
assess_results <- assess_cvpat(established_model,
                               seed = 123,
                               cores = 1)
print(assess_results$CVPAT_compare_LM,
      digits = 3)
#>         PLS Loss LM Loss   Diff Boot T value Boot P Value
#> COMP       1.196   1.211 -0.015        0.542        0.588
#> LIKE       1.915   2.063 -0.148        3.761        0.000
#> CUSA       0.994   0.983  0.011       -0.489        0.625
#> CUSL       1.560   1.600 -0.040        2.914        0.004
#> Overall    1.416   1.464 -0.048        3.482        0.001
#>
#> CVPAT as per Sharma et al. (2023).
print(assess_results$CVPAT_compare_IA,
      digits = 3)
#>         PLS Loss IA Loss   Diff Boot T value Boot P Value
#> COMP       1.196   2.023 -0.827        8.580        0.000
#> LIKE       1.915   3.103 -1.187        8.293        0.000
#> CUSA       0.994   1.374 -0.379        5.004        0.000
#> CUSL       1.560   2.663 -1.102        7.572        0.000
#> Overall    1.416   2.290 -0.874       10.301        0.000
#>
#> CVPAT as per Sharma et al. (2023).

The established model has significantly lower predictive loss compared to both the naive benchmark IA and the LM model. Thus, we can say that the established model has predictive relevance.

Now we compare the results:

# Function to compare the Loss of two models
compare_results <- assess_cvpat_compare(established_model = established_model,
                                        alternative_model = competing_model,
                                        testtype = "two.sided",
                                        nboot = 2000,
                                        technique = predict_DA,
                                        seed = 123,
                                        noFolds = 10,
                                        reps = 10,
                                        cores = 1)

print(compare_results,
      digits = 3)
#>         Base Model Loss Alt Model Loss   Diff Boot T value Boot P Value
#> COMP              1.198          1.195  0.003       -0.460        0.645
#> LIKE              1.923          1.933 -0.010        0.883        0.378
#> CUSA              0.988          0.992 -0.004        0.809        0.419
#> CUSL              1.562          1.715 -0.152        3.286        0.001
#> Overall           1.418          1.459 -0.041        3.293        0.001
#>
#> CVPAT as per Sharma, Liengaard, Hair, Sarstedt, & Ringle, (2023).
#>   Both models under comparison have identical endogenous constructs with identical measurement models.
#>   Purely exogenous constructs can differ in regards to their relationships with both nomological
#>   partners and measurement indicators.

The established model has significantly lower predictive loss compared to the competing model. Thus, we can say that the established model has superior predictive performance compared to the competing model.

Confirmatory Tetrad Analysis (CTA-PLS)

CTA-PLS (Gudergan et al., 2008) empirically tests whether a construct's measurement model is consistent with a reflective (common factor) specification. Under a reflective model, all model-implied vanishing tetrads equal zero. If any tetrad is significantly non-zero, the reflective specification is rejected in favour of a formative one.

CTA-PLS requires at least 4 indicators per construct. When borrow = TRUE (the default), constructs with only 2 or 3 indicators can still be tested by borrowing indicators from structurally connected constructs. The borrowing rules follow Gudergan et al. (2008, Table 1): a reflective construct with 3 own indicators borrows 1 from an adjacent reflective construct (all tetrads vanish under H0), while a construct with 2 own indicators borrows 2 from any adjacent construct (only the cross-tetrad tau_1342 vanishes).

library(seminr)
library(seminrExtras)

mobi_mm <- constructs(
  composite("Image",        multi_items("IMAG", 1:5)),
  composite("Expectation",  multi_items("CUEX", 1:3)),
  composite("Value",        multi_items("PERV", 1:2)),
  composite("Satisfaction", multi_items("CUSA", 1:3)),
  composite("Loyalty",      multi_items("CUSL", 1:3))
)

mobi_sm <- relationships(
  paths(from = "Image",       to = c("Expectation", "Satisfaction", "Loyalty")),
  paths(from = "Expectation", to = c("Value", "Satisfaction")),
  paths(from = "Value",       to = "Satisfaction"),
  paths(from = "Satisfaction", to = "Loyalty")
)

mobi_pls <- estimate_pls(data = mobi,
                          measurement_model = mobi_mm,
                          structural_model  = mobi_sm)

# CTA-PLS with borrowing (default) — constructs with 2-3 indicators
# borrow from adjacent constructs to form testable 4-tuples
cta_result <- assess_cta(mobi_pls, nboot = 5000, seed = 123)
print(cta_result)
summary(cta_result)

# Without borrowing — only constructs with >= 4 indicators are tested
cta_no_borrow <- assess_cta(mobi_pls, nboot = 5000, borrow = FALSE)
print(cta_no_borrow)

Unobserved Heterogeneity (FIMIX-PLS and PLS-POS)

FIMIX-PLS and PLS-POS are complementary approaches for detecting unobserved heterogeneity in PLS-SEM. FIMIX-PLS uses probabilistic (EM-based) assignment and assumes normally distributed residuals, while PLS-POS uses deterministic (hill-climbing) assignment with no distributional assumptions. Both methods partition the sample into K segments with segment-specific path coefficients.

FIMIX-PLS (Finite Mixture PLS)

FIMIX-PLS (Hahn et al., 2002; Sarstedt et al., 2011) probabilistically assigns observations to K segments, each with segment-specific structural path coefficients.

assess_fimix() estimates a single K-segment solution, while assess_fimix_compare() compares solutions across K values using information criteria (AIC, BIC, CAIC, etc.) to help select the optimal number of segments.

library(seminr)
library(seminrExtras)

# Estimate a PLS model
corp_pls <- estimate_pls(
  data = corp_rep_data,
  measurement_model = corp_rep_mm_ext,
  structural_model  = corp_rep_sm_ext,
  missing = mean_replacement,
  missing_value = "-99")

# FIMIX with K=2 segments
fimix_k2 <- assess_fimix(corp_pls, K = 2, nstart = 10, seed = 123)
print(fimix_k2)
summary(fimix_k2)
plot(fimix_k2)

# Compare across K=2..4 using information criteria
fimix_compare <- assess_fimix_compare(corp_pls,
                                       K_range = 2:4,
                                       nstart = 10,
                                       seed = 123)
print(fimix_compare)
plot(fimix_compare)

PLS-POS (Prediction-Oriented Segmentation)

PLS-POS (Becker et al., 2013) maximizes the sum of R-squared across all endogenous constructs across K segments using deterministic hill-climbing. Unlike FIMIX-PLS, it makes no distributional assumptions and can detect heterogeneity in both structural and formative measurement models.

assess_pos() estimates a single K-segment solution, while assess_pos_compare() compares solutions across multiple K values using the objective criterion (sum of R-squared).

# PLS-POS with K=2 segments (using corp_pls from above)
pos_k2 <- assess_pos(corp_pls, K = 2, nstart = 10, seed = 123)
print(pos_k2)
summary(pos_k2)
plot(pos_k2, type = "rsquared")

# Compare across K=2..4
pos_compare <- assess_pos_compare(corp_pls,
                                   K_range = 2:4,
                                   nstart = 10,
                                   seed = 123)
print(pos_compare)
plot(pos_compare)

Congruence Testing

Congruence testing (Franke, Sarstedt, & Danks, 2021) evaluates whether PLS composite weights are stable across bootstrap samples by computing congruence coefficients. A congruence coefficient close to 1 indicates that the composite weight pattern is robust.

cong_result <- congruence_test(mobi_pls,
                                nboot = 2000,
                                seed = 123)
print(cong_result)
summary(cong_result)

References

• Becker, J.-M., Rai, A., Ringle, C. M., & Voelckner, F. (2013). Discovering Unobserved Heterogeneity in Structural Equation Models to Avert Validity Threats. MIS Quarterly, 37(3), 665-694.
• Becker, J.-M., Richter, N. F., Ringle, C. M., & Sarstedt, M. (2026). Must-have, or maybe not? A sensitivity-based extension to necessary condition analysis. Journal of Business Research, 206, 115920.
• Dul, J. (2016). Necessary Condition Analysis (NCA): Logic and methodology of "necessary but not sufficient" causality. Organizational Research Methods, 19(1), 10-52.
• Franke, G. R., Sarstedt, M., & Danks, N. P. (2021). An empirical comparison of factor score estimation methods. Journal of Business Research, 130, 318-334.
• Hahn, C., Johnson, M. D., Herrmann, A., & Huber, F. (2002). Capturing Customer Heterogeneity using a Finite Mixture PLS Approach. Schmalenbach Business Review, 54, 243-269.
• Gudergan, S. P., Ringle, C. M., Wende, S. & Will, A. (2008). Confirmatory Tetrad Analysis in PLS Path Modeling. Journal of Business Research, 61(12), 1238-1249.
• Hair, J.F. (Jr), Hult, T.M., Ringle, C.M., Sarstedt, M., Danks, N.P., and Adler, S. (2026). Partial Least Squares Structural Equation Modeling (PLS-SEM) Using R (Second Edition) - A Workbook. Springer.
• Hauff, S., Richter, N. F., Sarstedt, M., & Ringle, C. M. (2024). Importance and Performance in PLS-SEM and NCA: Introducing the Combined Importance-Performance Map Analysis (cIPMA). Journal of Retailing and Consumer Services, 78, 103723.
• Liengaard, B. D., Sharma, P. N., Hult, G. T. M., Jensen, M. B., Sarstedt, M., Hair, J. F., & Ringle, C. M. (2021). Prediction: coveted, yet forsaken? Introducing a cross-validated predictive ability test in partial least squares path modeling. Decision Sciences, 52(2), 362-392.
• Ray, S., Danks, N.P., Calero Valdez, A. (2026) SEMinR: Domain-specific language for building, estimating, and visualizing structural equation models in R.
• Richter, N. F., Schubring, S., Hauff, S., Ringle, C. M., & Sarstedt, M. (2020). When predictors of outcomes are necessary: guidelines for the combined use of PLS-SEM and NCA. Industrial Management & Data Systems, 120(12), 2243-2267.
• Sarstedt, M., Becker, J.-M., Ringle, C. M. & Schwaiger, M. (2011). Uncovering and Treating Unobserved Heterogeneity with FIMIX-PLS. Schmalenbach Business Review, 63(1), 34-62.
• Ringle, C. M. & Sarstedt, M. (2016). Gain More Insight from Your PLS-SEM Results: The Importance-Performance Map Analysis. Industrial Management & Data Systems, 119(9), 1865-1886.
• Sarstedt, M., Richter, N. F., Hauff, S. & Ringle, C. M. (2024). Combined Importance-Performance Map Analysis (cIPMA): A SmartPLS 4 Tutorial. Journal of Marketing Analytics, 12, 746-760.
• Sharma, P. N., Liengaard, B. D., Hair, J. F., Sarstedt, M., & Ringle, C. M. (2022). Predictive model assessment and selection in composite-based modeling using PLS-SEM: extensions and guidelines for using CVPAT. European Journal of Marketing, 57(6), 1662-1677.