PLS-SEM Post Hoc Power Analysis in SmartPLS
SmartPLS reports a post hoc minimum sample size after estimating a partial least squares structural equation modeling (PLS-SEM) model. The result is labeled Technically required sample size and helps researchers assess whether the number of observations used in the analysis is technically sufficient to detect structural relationships of the estimated magnitude (Hair, Hult, Ringle, & Sarstedt, 2027; Kock & Hadaya, 2018).
This result is a retrospective statistical-power diagnostic. It should be interpreted together with the structural model results, the study's research design, and the substantive relevance of the estimated path coefficients.
Important: The technically required sample size is available for the PLS-SEM algorithm. It is not a general result provided for every algorithm in SmartPLS.
A technically sufficient sample size does not mean that the sample is representative of the research population. Statistical power and population representativeness are separate aspects of research quality.
What is the technically required sample size?
The technically required sample size is the estimated minimum number of observations needed to detect a structural path coefficient of a given magnitude at a specified significance level and statistical power.
In SmartPLS, this assessment is conducted after the PLS-SEM model has been estimated. It therefore answers the following retrospective question:
Given the magnitude of the estimated structural path coefficient that drives the sample size requirement, how many observations are technically required to detect an effect of this size under the assumptions of the calculation?
The required sample size depends strongly on the magnitude of the structural path coefficient. Smaller absolute path coefficients require larger samples because weak effects are more difficult to distinguish from sampling variation. Larger absolute path coefficients generally require fewer observations.
Where can I find the result in SmartPLS?
After running the PLS-SEM algorithm, open the results report and navigate to:
Algorithm → Posthoc minimum sample size
SmartPLS displays the Technically required sample size in this section. Compare this value with the actual number of valid observations used to estimate the PLS-SEM model.
How does SmartPLS determine the technically required sample size?
The calculation draws on the inverse square root method developed by Kock and Hadaya (2018). The method estimates the minimum sample size from the absolute magnitude of the structural path coefficient that determines the technical requirement.
Under the commonly used assumptions of a 5% significance level, 80% statistical power, and a directional test, the inverse square root method is expressed as:
[ N_{\text{technical}}
\left\lceil
\left(
\frac{2.486}{|\beta|_{\min}}
\right)^2
\right\rceil
]
where:
- (N_{\text{technical}}) is the technically required sample size;
- (|\beta|_{\min}) is the absolute magnitude of the structural path coefficient that drives the sample size requirement; and
- (\lceil \cdot \rceil) means that the result is rounded up to the next whole observation.
The constant 2.486 combines the critical value for a 5% significance level with the critical value for 80% statistical power. The original inverse square root method uses the statistically significant path coefficient with the smallest absolute magnitude in the model (Kock & Hadaya, 2018).
The method is preferable to the traditional 10-times rule, which does not account for effect magnitude and can substantially underestimate the required sample size. The inverse square root method is straightforward to apply and generally provides a conservative estimate of the technical minimum sample size (Hair et al., 2027; Kock & Hadaya, 2018).
Example calculations
The following table illustrates how the technically required sample size increases as the absolute path coefficient becomes smaller.
| Absolute path coefficient (|\beta|) | Technically required sample size |
| --------------------------------------: | ---------------------------------: |
| 0.10 | 619 |
| 0.15 | 275 |
| 0.20 | 155 |
| 0.25 | 99 |
| 0.30 | 69 |
| 0.40 | 39 |
For example, an absolute path coefficient of 0.20 yields:
[ N_{\text{technical}}
\left\lceil \left( \frac{2.486}{0.20} \right)^2 \right\rceil
155
]
Accordingly, at least 155 valid observations are technically required to detect a path coefficient of this magnitude under the stated assumptions.
How should I interpret the SmartPLS result?
Compare the technically required sample size with the actual analytical sample size, that is, the number of valid observations included in the PLS-SEM estimation.
| Comparison | Interpretation |
|---|---|
| Actual sample size is equal to or larger than the technically required sample size | The sample meets the technical statistical-power requirement under the assumptions of the calculation. |
| Actual sample size is smaller than the technically required sample size | The analysis may have insufficient statistical power to detect a structural relationship of the relevant magnitude reliably. |
Meeting the technical requirement does not guarantee that a path coefficient is statistically significant, unbiased, substantively important, or correctly specified. It only indicates that the sample size is technically adequate in relation to the path coefficient magnitude, significance level, and statistical power assumed by the calculation.
When the actual sample is smaller than the technical requirement, researchers should interpret nonsignificant or unstable structural model results cautiously. An insufficient sample size increases the risk of failing to detect an effect that exists in the population, which is known as a Type II error.
Why does the smallest path coefficient matter?
The weakest relevant structural relationship generally determines the overall technical sample size requirement. This is because detecting a small path coefficient requires more statistical information than detecting a large path coefficient.
Researchers should nevertheless interpret this principle in light of the study's theory and objectives. A very small coefficient for an exploratory path, a control variable, or a theoretically unimportant relationship can produce a very large sample size requirement. The path that determines the study-wide requirement should therefore be a focal and substantively meaningful relationship that the study is intended to detect.
A very large technically required sample size for a near-zero path coefficient is a mathematical consequence of the inverse square root method. It is not a SmartPLS calculation error.
Post hoc assessment versus a priori sample size planning
The SmartPLS result is a post hoc assessment because it uses the PLS-SEM results obtained from the analyzed sample. It describes the technical sample size associated with an effect magnitude observed after data collection.
For planning a new study, an a priori sample size analysis is preferable. Before collecting data, researchers should identify the smallest path coefficient that is theoretically relevant and use prior studies, meta-analyses, pilot data, or substantive judgment to specify the expected minimum effect.
Prospective planning helps researchers collect enough observations from the outset. The post hoc SmartPLS result remains useful as a diagnostic after model estimation, but it does not replace a well-justified sample size analysis conducted before data collection.
Technical sample size adequacy is not population representativeness
The technically required sample size addresses whether the number of observations is sufficient to detect a structural relationship of a particular magnitude. It does not assess whether the observations adequately represent the target population.
A sample can be larger than the technically required sample size and still be unrepresentative. For example, a large convenience sample may systematically exclude relevant groups from the population. Conversely, a carefully selected probability sample may represent the population well but still be too small to detect weak structural relationships with adequate statistical power.
Population representativeness depends on the sampling and data collection process, including:
- the definition of the target population;
- the adequacy of the sampling frame;
- probability versus nonprobability sampling;
- coverage of relevant population groups;
- selection and self-selection processes;
- nonresponse and attrition;
- recruitment channels;
- weighting and calibration procedures; and
- the heterogeneity of the research population.
Therefore, researchers should not use the technically required sample size as evidence that their sample is representative.
Correct interpretation: The analytical sample meets the technical statistical-power requirement for the focal PLS-SEM relationships under the assumptions of the calculation.
Incorrect interpretation: The sample is sufficiently large and is therefore representative of the target population.
What does the technically required sample size not establish?
The SmartPLS post hoc minimum sample size does not evaluate:
- population representativeness;
- sampling bias or nonresponse bias;
- measurement model reliability and validity;
- structural model specification;
- omitted variables or endogeneity;
- causal identification;
- effect size relevance;
- out-of-sample predictive performance; or
- the quality of the underlying data and theory.
These aspects require separate research design considerations and additional PLS-SEM assessment procedures.
Recommended workflow
When assessing sample size in a PLS-SEM study, we recommend the following workflow:
- Define the target population and sampling strategy before collecting data.
- Identify the smallest focal effect that the study must be able to detect.
- Plan the sample size prospectively using prior evidence and appropriate assumptions about significance level and statistical power.
- Allow for missing data and attrition when setting the recruitment target.
- Estimate the model with the PLS-SEM algorithm using the valid analytical sample.
- Inspect the Posthoc minimum sample size result and compare the technically required sample size with the actual sample size.
- Use bootstrapping for statistical inference on path coefficients, loadings, and weights.
- Report technical sample size adequacy and population representativeness separately.
When the study includes subgroup analyses, multigroup analysis, complex sampling, or substantial missing data, the final recruitment target may need to exceed the technically required sample size reported for the overall PLS-SEM model.
Reporting the technically required sample size
Researchers should report the analytical sample size, the SmartPLS result, and the limitation regarding population representativeness. A concise report may state:
The PLS-SEM model was estimated in SmartPLS 4 using an analytical sample of (N = \text{actual sample size}). The Posthoc minimum sample size result indicated a technically required sample size of (N = \text{required sample size}), based on the inverse square root method (Kock & Hadaya, 2018). The analytical sample therefore met/did not meet the technical statistical-power requirement. This result concerns the ability to detect the focal structural relationship and does not establish that the sample is representative of the target population.
Researchers should separately describe how the sample was recruited, how the target population was defined, and whether potential coverage, selection, or nonresponse biases were assessed.
Frequently asked questions
What is the Posthoc minimum sample size in SmartPLS?
It is a retrospective sample size diagnostic available after running the PLS-SEM algorithm. SmartPLS reports a Technically required sample size, which indicates the estimated minimum number of observations needed to detect the structural relationship that drives the sample size requirement under the calculation's assumptions.
Is the technically required sample size available for every SmartPLS algorithm?
No. The result is available for the PLS-SEM algorithm. It should not be described as a general output of all SmartPLS algorithms.
Is the technically required sample size the same as the required sample size for the entire study?
Not necessarily. The final study sample may need to be larger because of the sampling design, population heterogeneity, missing data, attrition, subgroup analyses, multigroup analysis, or precision requirements. The SmartPLS result addresses a specific technical statistical-power requirement.
Does meeting the technically required sample size prove that my PLS-SEM results are valid?
No. It addresses only sample size adequacy in relation to the relevant path coefficient magnitude and the assumptions of the calculation. Researchers must still assess the measurement models, structural model, coefficient uncertainty, predictive performance, data quality, and research design.
Does a large enough sample mean that it is representative?
No. Sample size and representativeness are different. Representativeness depends on how observations were selected from the target population. A large convenience sample can remain systematically biased.
Why is the technically required sample size very large?
The required sample size increases rapidly as the absolute path coefficient approaches zero. Check whether the path that drives the requirement is a focal and substantively meaningful relationship. A near-zero exploratory or control path should not automatically determine the study-wide sample size requirement.
Does the post hoc minimum sample size replace bootstrapping?
No. The procedures answer different questions. The post hoc minimum sample size assesses technical sample size adequacy. Bootstrapping provides standard errors, confidence intervals, and significance tests for estimated model parameters.
Can I use the SmartPLS result to plan a future study?
The result can provide an empirical reference when the prior and planned studies are closely comparable. However, prospective planning should be based on the smallest theoretically relevant expected effect and should be completed before data collection.
Further reading
For a comprehensive introduction to sample size assessment and the evaluation of PLS-SEM results, see Hair et al. (2027). The inverse square root method is introduced and evaluated by Kock and Hadaya (2018).
References
- 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.
- Kock, N., & Hadaya, P. (2018). Minimum sample size estimation in PLS-SEM: The inverse square root and gamma-exponential methods. Information Systems Journal, 28(1), 227–261.
- Ringle, C. M., Wende, S., & Becker, J.-M. (2024). SmartPLS 4. SmartPLS.
- 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.
- 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
