Model Fit
Model fit describes how well the model-implied covariance matrix matches the sample covariance matrix. In PLS-SEM, model fit measures play a more limited role than in CB-SEM because PLS-SEM does not optimize a single global scalar function; still, SmartPLS reports several fit criteria that researchers can consult alongside the model's predictive and explanatory results.
Fit Measures in SmartPLS
SmartPLS offers the following fit measures:
| Fit measure | What it captures |
|---|---|
| SRMR | Average discrepancy between observed and model-implied correlations |
| Exact fit criteria d_ULS and d_G | Bootstrap-based test of the exact overall model fit |
| NFI | Comparison of the model's Chi² against a null-model benchmark |
| Chi² | Chi-square value and degrees of freedom of the PLS path model |
For the approximate fit indices, such as SRMR and NFI, you can directly look at the outcomes of a PLS-SEM or PLSc-SEM model estimation (i.e., the results report) and compare these criteria's values against a threshold (e.g., SRMR < 0.08 and NFI > 0.90).
For the bootstrap-based test for the exact overall model fit measures d_ULS and d_G, you can consider the inference statistics for an assessment. To do so, run the bootstrap procedure using the "complete bootstrap" option in SmartPLS. When running the bootstrap procedure, note that it counts up to the specified number of bootstrapping samples twice:
- In the first round, SmartPLS uses the standard bootstrapping procedure to get the inference statistics for the model parameters (e.g., path coefficients, weights, etc.).
- In the second round, SmartPLS uses an adapted Bollen-Stine bootstrapping procedure, as described in Dijkstra and Henseler (2015; also see Bollen and Stine, 1992; Yuan and Hayashi, 2003), to create confidence intervals for the d_ULS, d_G, and SRMR criteria (note that SmartPLS has two computation runs in the second round: one for the saturated model and one for the estimated model).
The study by Cho et al. (2020) investigates structural equation modeling (SEM) based on components or composites using the generalized structured component analysis (GSCA) approach. While GSCA employs fit indexes such as the goodness-of-fit index (GFI) and the standardized root mean square residual (SRMR), their performance in this specific context had not been systematically examined. Through a simulation study, Cho et al. (2020) demonstrated that both GFI and SRMR effectively distinguish between correctly specified and misspecified models. Based on their findings, the authors propose practical cutoff criteria for these fit indexes across various sample sizes, providing guidance for model evaluation in applied research. These insights are also relevant for assessing model fit in composite-based approaches such as PLS-SEM (see also Schuberth et al., 2023).
Note: Also see the information on goodness of fit (GoF), which researchers should not use.
Standardized Root Mean Square Residual (SRMR)
While the root mean square residual (RMSR) is a measure of the mean absolute value of the covariance residuals, the standardized root mean square residual (SRMR) is based on transforming both the sample covariance matrix and the predicted covariance matrix into correlation matrices. Note: literature on PLS-SEM needs to better explain where and how the covariance matrix is derived in PLS-SEM (since it is different from CB-SEM, which is a full information method, whereas PLS-SEM is not). Most important, it remains unclear whether the researcher should use the estimated model (the more reasonable choice) or the saturated model to obtain the covariance matrix.
The SRMR is defined as the difference between the observed correlation and the model-implied correlation matrix. Thus, it allows assessing the average magnitude of the discrepancies between observed and expected correlations as an absolute measure of (model) fit.
| Threshold | Interpretation |
|---|---|
| SRMR < 0.10 | Good fit |
| SRMR < 0.08 (more conservative version; Hu and Bentler, 1999) | Good fit |
Henseler et al. (2014) introduce the SRMR as a goodness of fit measure for PLS-SEM that can be used to avoid model misspecification.
SmartPLS also provides bootstrap-based inference statistics of the SRMR criterion. For the interpretation of SRMR bootstrap confidence interval results, see the bootstrap-based test for the exact overall model fit below.
Bootstrap-based test for the exact overall model fit
The bootstrap-based test for the exact overall model fit tests the statistical (bootstrap-based) inference of the discrepancy between the empirical covariance matrix and the covariance matrix implied by the composite factor model. As defined by Dijkstra and Henseler (2015; also see Schuberth et al., 2022), d_ULS (i.e., the squared Euclidean distance) and d_G (i.e., the geodesic distance) represent two different ways to compute this discrepancy. The Bollen-Stine (1992) bootstrap routine provides the confidence intervals of these discrepancy values. The d_G criterion builds on PLS-SEM eigenvalue computations. However, the question remains how these eigenvalues differ from CB-SEM.
Note: The value of the d_ULS and d_G in itself do not pertain any value. Only the Bollen-Stine bootstrapping results of the bootstrap-based test for the exact overall model fit, as provided my SmartPLS, allow an interpretation of results. More specifically, since the d_ULS and d_G (and SRMR) confidence intervals are not obtained by running the regular bootstrapping procedure, but the adapted Bollen-Stine bootstrapping procedure, their results interpretation somewhat differs from the regular bootstrap outcomes.
For the bootstrap-based test for the exact overall model fit (i.e., d_ULS and d_G), you compare their original value against the confidence interval created from the sampling distribution. The confidence interval should include the original value. Hence, the upper bound of the confidence interval should be larger than the original value of the d_ULS and d_G fit criteria to indicate that the model has a “good fit”. Choose the confidence interval in a way that the upper bound is at the 95% or 99% point.
In other words, a model fits well if the difference between the correlation matrix implied by your model and the empirical correlation matrix is so small that it can be purely attributed to sampling error. Hence, the difference between the correlation matrix implied by your model and the empirical correlation matrix should be non-significant (p > 0.05). Otherwise, if the discrepancy is significant (p < 0.05), model fit has not been established.
Normed Fit Index (NFI) or Bentler and Bonett Index
One of the first fit measures proposed in the SEM literature is the normed fit index by Bentler and Bonett (1980). It computes the Chi² value of the proposed model and compares it against a meaningful benchmark. Since the Chi² value of the proposed model in itself does not provide sufficient information to judge model fit, the NFI uses the Chi² value from the null model, as a yardstick. Literature, however, does not explain how the PLS-SEM Chi² value differs from the CB-SEM one.
The NFI is then defined as 1 minus the Chi² value of the proposed model divided by the Chi² values of the null model. Consequently, the NFI results in values between 0 and 1. The closer the NFI to 1, the better the fit. NFI values above 0.9 usually represent acceptable fit. Lohmöller (1989) provides detailed information on the NFI computation of PLS path models. However, for the applied user, these explications are quite difficult to comprehend.
The NFI represents an incremental fit measure. As such, a major disadvantage is that it does not penalize for model complexity. The more parameters in the model, the larger (i.e., better) the NFI result. It is for this reason that this measure is not recommended, but alternatives such as the non-normed fit index (NNFI) or Tucker-Lewis index, which penalizes the Chi² values by the degrees of freedom (df). Lohmöller (1989) suggests computing the NNFI of PLS path models. However, the NNFI has not been implemented in SmartPLS, yet.
Chi² and Degrees of Freedom
Assuming a multinormal distribution, the Chi² value of a PLS path model with df degrees of freedom approximately is (N-1)*L, whereby N is the number of observations and L the maximum likelihood function as defined by Lohmöller (1989). The degrees of freedom (df) are defined as (K² + K) /2 – t, whereby is the number of manifest variables in the PLS path model and t the number of independent variables to estimate the model implied covariance matrix. However, future research must clearly define how to determine the degrees of freedom of composite models, common factor models, and mixed models when using PLS-SEM.
Estimated vs. Saturated Model
The distinction of estimated and saturated models in PLS-SEM is in its very early stages. Future research must provide detailed explanations and recommendations on the computation, usage and interpretation of these outcomes.
The saturated model assesses correlation between all constructs. The estimated model is a model which is based on a total effect scheme and takes the model structure into account. It is hence a more restricted version of the fit measure.
Researchers often struggle to choose between the estimated and saturated model when trying to report the fir of a PLS path model. At this stage, PLS-SEM literature is very vague on the use of fit criteria in general and, in specific, the choice between the estimated and saturated model. However, the estimated model seems to be a reasonable choice, if a researcher makes the questionable decision to report the fit results of the PLS path model.
Fit Measures by Model Type
Composite Model Fit Measures
If you want to obtain the composite model fit measures, use formative measurement models for all constructs in the PLS path model. After model estimation, refer to the estimated (or saturated?) model outcomes.
Common Factor Model Fit Measures
If you want to obtain the common factor model fit measures, use reflective measurement models for all constructs in the PLS path model. After model estimation, refer to the estimated (or saturated?) model SRMR outcomes. However, when assuming common factor models for all constructs in the PLS path model, the question remains why the researcher would not use CB-SEM in the first place to estimate and evaluate such a model.
Mixed Model Fit Measures
If you use both reflective and formative measurement models, SmartPLS 3.2.4 (and subsequent versions) provides the mixed model fit measures, considering common factor models for reflective measurement models and composite models for formative measurement models. However, at this stage, PLS-SEM literature does not provide much support for why and how a researcher would theoretically distinguish between constructs represented by common factors and composites in the same model, the need for their mixed use, and the necessity to report model fit in that case.
Note on Model Fit in PLS-SEM
Lohmöller (1989) already offers a set of fit measures. But he states that they have been introduced to provide a comparison to LISREL results rather than to represent an appropriate PLS-SEM index. More specifically, Lohmöller (1989) finds that some fit measures imply restrictive assumptions on the residual covariances, which PLS-SEM does not imply when estimating the model. For example, certain fit measures assume a common factor model, which requires uncorrelated outer residuals. In contrast, the outer residuals of composite models are not required to be uncorrelated. Hence, they are inappropriate for PLS-SEM.
However, when mimicking CB-SEM models with the consistent PLS (PLSc-SEM) approach, one also mimics common factor models with the PLS-SEM approach. Hence, when using PLSc-SEM for a path model that only includes reflectively measured constructs (i.e., common factor models), one may be interested in the model fit. Thereby, it is more comprehensively possible to mimic CB-SEM via the PLSc-SEM approach or to compare the results from the two approaches. Against this background, Sarstedt et al. (2017, 2021) conclude that "validation using goodness-of-fit measures is also relevant in a PLS-SEM context but less so compared to factor-based SEM".
Frequently Asked Questions
What does SRMR below 0.08 mean in PLS-SEM?
An SRMR value below 0.08 (or below 0.10 in the less conservative version) indicates a good fit between the observed correlation matrix and the model-implied correlation matrix. However, the literature on PLS-SEM has not fully clarified how the underlying covariance matrix should be derived, so SRMR results should be interpreted with some caution.
Should I report the exact fit test (d_ULS, d_G) or just SRMR?
Both can be reported. SRMR is an approximate fit index available directly from a standard PLS-SEM or PLSc-SEM run. The exact fit test for d_ULS and d_G additionally requires the bootstrap-based (Bollen-Stine) procedure and provides confidence intervals that let you test whether the discrepancy between the empirical and model-implied correlation matrix is statistically significant.
What is the difference between the estimated and saturated model?
The saturated model assesses correlations between all constructs without restrictions, while the estimated model is based on a total effect scheme that takes the specified model structure into account, making it a more restricted version of the fit measure. PLS-SEM literature is still vague on when to use which, but the estimated model is generally the more reasonable choice if you decide to report fit results at all.
Can I use model fit indices for formative (composite) models?
Yes. The composite model fit measures in SmartPLS are obtained when all constructs in the PLS path model are specified formatively. Reflective constructs instead yield the common factor model fit measures, and models that mix both measurement types yield the mixed model fit measures (available since SmartPLS 3.2.4).
Should I use model fit at all in PLS-SEM?
Use it cautiously. Fit criteria for PLS-SEM are still an early-stage research area, several thresholds are not fully settled, and validation using goodness-of-fit measures is less central in PLS-SEM than in factor-based SEM (Sarstedt et al., 2017, 2021). Researchers should rely primarily on predictive and explanatory assessment and treat model fit as a supplementary check.
Related SmartPLS Methods
- Goodness of fit (GoF)
- Discriminant validity assessment (HTMT)
- Consistent PLS-SEM (PLSc)
- CB-SEM
- Bootstrapping
References
- Bollen, K. A., & Stine, R. A. (1992). Bootstrapping goodness-of-fit measures in structural equation models. Sociological Methods & Research, 21(2), 205–229.
- Bentler, P. M., & Bonett, D. G. (1980). Significance tests and goodness-of-fit in the analysis of covariance structures. Psychological Bulletin, 88, 588–600.
- Cho, G., Hwang, H., Sarstedt, M., & Ringle, C. M. (2020). Cutoff criteria for overall model fit indexes in generalized structured component analysis. Journal of Marketing Analytics, 8, 189–202.
- Dijkstra, T. K., & Henseler, J. (2015). Consistent and asymptotically normal PLS estimators for linear structural equations. Computational Statistics & Data Analysis, 81(1), 10–23.
- Hair, J. F., Hollingsworth, C. L., Randolph, A. B., & Chong, A. Y. L. (2017). An updated and expanded assessment of PLS-SEM in information systems research. Industrial Management & Data Systems, 117(3), 442–458.
- Hair, J. F., Hult, G. T. M., Ringle, C. M., & Sarstedt, M. (2027). A primer on partial least squares structural equation modeling (PLS-SEM) (4th ed.). Sage.
- Henseler, J., Dijkstra, T. K., Sarstedt, M., Ringle, C. M., Diamantopoulos, A., Straub, D. W., Ketchen, D. J., Hair, J. F., Hult, G. T. M., & Calantone, R. J. (2014). Common beliefs and reality about partial least squares: Comments on Rönkkö & Evermann (2013). Organizational Research Methods, 17(2), 182–209.
- Hu, L.-t., & Bentler, P. M. (1998). Fit indices in covariance structure modeling: Sensitivity to underparameterized model misspecification. Psychological Methods, 3(4), 424–453.
- Lohmöller, J.-B. (1989). Latent variable path modeling with partial least squares. Physica.
- Sarstedt, M., Ringle, C. M., & Hair, J. F. (2025). Partial least squares structural equation modeling. In C. Homburg, M. Klarmann, & A. Vomberg (Eds.), Handbook of Market Research (pp. 1–56). Springer Nature Switzerland.
- Schuberth, F., Rademaker, M. E., & Henseler, J. (2023). Assessing the overall fit of composite models estimated by partial least squares path modeling. European Journal of Marketing, 57(6), 1678–1702.
- More literature ...
Cite correctly
Please always cite the use of SmartPLS!
Ringle, Christian M., Wende, Sven, & Becker, Jan-Michael. (2024). SmartPLS 4. Bönningstedt: SmartPLS. Retrieved from https://www.smartpls.com
