From Blindfolding Q² to PLSpredict and CVPAT: The Evolution of Predictive Assessment in PLS-SEM

Part 1. The History of Stone–Geisser Q² and Blindfolding

Using collected data to assess a model’s predictive ability rather than fit alone originates in two independent statistical papers published in 1974: Stone (1974) and Geisser (1974). Both proposed “predictive sample reuse”: hide some data, estimate the model using the remainder, compare predictions with hidden observations, and repeat until each data point has been hidden exactly once. This principle later became collectively known as Stone–Geisser Q².

In PLS-SEM, the standard computational technique for Stone–Geisser Q² is Blindfolding: omit some endogenous-variable data using omission distance D, reconstruct them using a model estimated from the remainder, and calculate Q² from reconstruction errors. Present in several SmartPLS versions, it was once considered a standard predictive-assessment step in PLS-SEM reporting (Hair et al., 2022), before its complete removal from SmartPLS 4.

Part 2. Why Did SmartPLS 4 Remove Blindfolding?

Official SmartPLS documentation gives two concise reasons: “We have discontinued support for blindfolding in SmartPLS 4 and removed the algorithm” and “The blindfolding method does not provide an out-of-sample assessment of predictive power” (SmartPLS, Blindfolding documentation). The fundamental issue is that Blindfolding reconstructs hidden data using a model estimated from the remainder of the same sample. It therefore remains an in-sample assessment despite resembling cross-validation.

Part 3. In-Sample Explanation and Out-of-Sample Prediction

Shmueli and Koppius (2011) distinguish explanatory modeling—testing and explaining theory through fit and statistical relationships within the observed sample—from predictive modeling, which evaluates performance on new, previously unseen data. R², path coefficients, and p-values belong to the former; Q²predict, RMSE, MAE, and CVPAT to the latter.

The objectives are compatible, but meeting one does not establish the other. A complete PLS-SEM model should present both evidence sets rather than stop at R² and implicitly infer predictive ability, as the hypothetical example in part 9 illustrates.

Part 4. Formulas: How Do Q² and Q²predict Differ?

The algebra looks identical: 1 minus a ratio of sums of squares. This explains confusion between Q² and Q²predict. The decisive difference is how SSE is calculated: Q² uses data partially “known” through omission, whereas Q²predict uses observations entirely unseen during estimation.

Part 5. How K-Fold Cross-Validation Works

Part 6. Using PLSpredict in SmartPLS 4

Part 7. Interpreting IA, LM, RMSE, and MAE

These four abbreviations address two different questions. IA (Indicator Average) is the simplest benchmark, predicting with training-sample means and used to calculate Q²predict. LM (Linear Model) is actual OLS regression using direct antecedent indicators as predictors, used to rank predictive ability through RMSE or MAE. Squaring makes RMSE penalize large errors more than MAE; MAE weights errors in proportion to their magnitudes (Chai and Draxler, 2014; Willmott and Matsuura, 2005).

The counting rule of Shmueli, Ray, Estrada, and Chatla (2019), applied after confirming positive Q²predict. It is a descriptive heuristic rather than a statistical test; part 8 explains how CVPAT formalizes the question through a p-value.

See “IA, LM, RMSE, and MAE in PLSpredict: Four Abbreviations, Two Different Questions” in this series for a detailed discussion and an actual PLSpredict-table example.

Part 8. CVPAT: Theory, Hypotheses, and Interpretation

CVPAT uses the same k-fold cross-validation to calculate each observation’s loss: mean squared prediction error across endogenous indicators. It calculates each observation’s loss difference between PLS-SEM and a benchmark (IA or LM), then averages these to obtain D̄.

Liengaard et al. (2021) describe T = D̄ divided by the square root of (S²/N) as a paired t-test, related to Diebold and Mariano’s (1995) time-series forecasting test, compared with a t distribution with N − 1 degrees of freedom or estimated through bootstrapping. Sharma et al. (2022) extend this framework into a practical procedure using IA and LM as the default benchmarks, as implemented in SmartPLS 4.

See “CVPAT: Theory, Hypotheses, and Interpretation” in this series for detailed treatment of loss functions, test statistics, and an actual CVPAT-table example.

Part 9. Illustration: High R² but Weak Prediction

The two hypothetical models below show why high R² does not guarantee predictive ability and why CVPAT is needed rather than stopping at R² or PLSpredict’s counting rule.

A hypothetical illustration, not data from a particular empirical study, demonstrating that explanation and prediction can be distinct.

From R² alone, Model A looks clearly “better” than Model B. Under CVPAT, however, only Model B has significantly lower predictive loss than the benchmark. Reporting Model A’s R² and concluding “good predictive ability” would be unsupported, whereas Model B, despite its more modest R², has statistically tested predictive evidence.

Part 10. Common Errors

See “Common Errors When Reporting Q², PLSpredict, and CVPAT” in this series for examples and a complete presubmission checklist.

Part 11. Updated Evidence and Recommendations for 2026

As of 2026, all current SmartPLS 4 versions have completely removed Blindfolding, so guides or sample articles still instructing users to “run Blindfolding for Q²” rely on outdated documentation. PLSpredict, published in 2019, and CVPAT, introduced in 2021 with practical extensions in 2022, now form the standard predictive-assessment framework supported and recommended by the software.

Recent applications increasingly use both tools in the same article. Jumani’s (2026) extended TPB study is one example: it runs PLSpredict (indicator-level Q²predict, RMSE, MAE) and CVPAT (loss differences and p-values against both IA and LM) on the same model in the sequence recommended here.

Zhang, Adams, Cheah, Teeroovengadum, and Ramayah’s (2026) systematic review quantifies the gap in one field. Reviewing 62 PLS-SEM articles in leading educational leadership and management journals published through 8/2025, they found 17 (27,4%) still reporting Blindfolding Q², while only 3 (4,8%) reported CVPAT. PLSpredict was explicitly described as an out-of-sample assessment in only one article (Thien and Yeap, 2023). No reviewed articles used confirmatory tetrad analysis (CTA) to assess measurement models.

Part 12. Academic Results-Table and Reporting Templates

Report the measurement model first, followed by the structural model (R², paths, f²), then prediction, presenting PLSpredict before CVPAT.

A combined PLSpredict/CVPAT table for all principal endogenous variables; fill each cell with actual SmartPLS 4 results.