Regression Bootstrapping
Regression bootstrapping is a nonparametric resampling procedure for testing whether the estimates of a linear regression model are statistically significant. Rather than relying on distributional assumptions, it repeatedly re-estimates the regression model on randomly drawn subsamples of the original data to determine standard errors, confidence intervals, and t-values for each estimate. Researchers use this method whenever they need statistically defensible significance tests for regression coefficients without assuming a specific sampling distribution. In SmartPLS, bootstrapping can be applied directly to linear regression models estimated in the software.
How Regression Bootstrapping Works
SmartPLS uses bootstrapping to determine the significance of estimates from a linear regression model. Bootstrapping involves creating subsamples of randomly drawn observations from the original dataset (with replacement). Each subsample is then used to estimate the linear regression model. This process is repeated until a large number of random subsamples have been created (e.g., 10,000).
The parameter estimates obtained from the subsamples are used to derive the 95% confidence intervals for significance testing. In addition, bootstrapping provides the standard errors for the estimates, which allow t-values to be calculated to assess the significance of each estimate.
Hair et al. (2027) explain bootstrapping in more detail.
Bootstrapping Settings in SmartPLS
Subsamples
Bootstrapping creates subsamples of observations randomly drawn from the original dataset (with replacement). The number of observations in each bootstrap subsample is identical to the number of observations in the original sample (SmartPLS also accounts for the smaller number of observations in the original sample if you use case-by-case deletion to handle missing values).
To ensure stability of results, the number of subsamples should be large. For an initial evaluation, it may be advisable to choose a smaller number of bootstrap subsamples (e.g., 1,000) to be randomly drawn and estimated with the PLS-SEM algorithm, as this requires less time. However, a large number of bootstrap subsamples (e.g., 10,000) should be used to produce the final results.
Note: Larger numbers of bootstrap subsamples increase the computing time.
Confidence Interval Method
Specifies the bootstrapping method used to estimate non-parametric confidence intervals. The following bootstrapping methods are available (for more details, see Hair et al., 2022):
- Percentile Bootstrap (default)
- Studentized Bootstrap
- Bias-Corrected and Accelerated (BCa) Bootstrap
By default, we recommend using the percentile bootstrap. If you have concerns about a non-normal bootstrap distribution, you can alternatively use bias-corrected and accelerated (BCa) bootstrapping.
Test Type
Specifies whether a one-tailed or two-tailed significance test is performed.
Significance Level
Specifies the significance level of the test statistic.
Parallel Processing
If selected, the bootstrapping algorithm will be run on multiple processors (if your computer has more than one core). As each subsample can be calculated individually, subsamples can be calculated in parallel. Using parallel computing will reduce the computation time.
Frequently Asked Questions
What does regression bootstrapping test?
It tests whether the estimates of a linear regression model are statistically significant. SmartPLS repeatedly draws subsamples from the original data (with replacement), re-estimates the regression model on each one, and uses the resulting parameter estimates to derive standard errors, confidence intervals, and t-values for the original estimates.
How many bootstrap subsamples should I use?
For an initial check, a smaller number (e.g., 1,000 subsamples) is sufficient and faster to compute. For final results reported in a study, use a large number of subsamples (e.g., 10,000) to ensure the stability of the estimated standard errors and confidence intervals. Keep in mind that more subsamples increase computing time.
Which confidence interval method should I choose?
The percentile bootstrap is the SmartPLS default and recommended in most cases. If you are concerned that the bootstrap distribution is non-normal, use the bias-corrected and accelerated (BCa) bootstrap instead. The studentized bootstrap is also available as a further alternative.
Does parallel processing change the bootstrapping results?
No. Parallel processing only affects computation speed by distributing the calculation of individual subsamples across multiple processor cores; it does not change how subsamples are drawn or how significance is assessed.
Should I use a one-tailed or two-tailed test?
This depends on your hypotheses. SmartPLS lets you specify whether a one-tailed or two-tailed significance test is performed, together with the significance level of the test statistic.
Related SmartPLS Methods
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.
- Davison, A. C., & Hinkley, D. V. (1997). Bootstrap methods and their application. Cambridge University Press.
- Efron, B., & Tibshirani, R. J. (1993). An introduction to the bootstrap. Chapman & Hall.
- 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
