1 Introduction
2 Field of application
3 Methodological framework for testing differences between parameters
3.1 The standard/Student’s t confidence interval
3.2 The percentile bootstrap confidence interval
3.3 The basic bootstrap confidence interval
4 Guideline on testing parameter differences in partial least squares path modeling
Step 1 | Use PLS or PLSca to obtain the model parameter estimates: \((\hat{\theta }_k;\hat{\theta }_l).\)
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Step 2 | Calculate the difference of the parameter estimates: \(\Delta \hat{\theta }=\hat{\theta }_k-\hat{\theta }_l.\)
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Step 3 | Create B bootstrap samples of the original data set and calculate the parameter estimates \(\hat{\theta }_{ki}^*\) and \(\hat{\theta }_{li}^*\), and their difference \(\Delta \hat{\theta }_{i}^*=\hat{\theta }_{ki}^*-\hat{\theta }_{li}^*\) for every bootstrap sample, with \(i=1,...,N.\)
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Step 4 | Estimate the variance of the estimated parameter difference as \({\widehat{{{\text{Var}}(\Delta {\hat{\theta}}^*)}}}=(B-1)^{-1}\sum \limits _{i=1}^{B}{(\Delta {\hat{\theta }}_{i}^*-\overline{\Delta{\hat{\theta}}^*})^2},\quad {\text {with}}\quad {\overline{\Delta{\hat{\theta}}^*}}=B^{-1}\sum \limits _{i=1}^{B}{\Delta{\hat{\theta}}_{i}^*}. \qquad \qquad\) (4) |
Step 5 | Estimate the \(\frac{\alpha }{2}\) and \(1-\frac{\alpha }{2}\) sample quantile of \(\Delta \hat{\theta }^*\) given by \(\hat{F}_{\Delta \theta ^*}^{-1}(\frac{\alpha }{2})\) and \(\hat{F}_{\Delta \theta ^*}^{-1}(1-\frac{\alpha }{2}).\)
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- Steps 1 and 2 are needed for all approaches except for the percentile bootstrap CI. | |
- To apply the standard/Student’s t CI (Eq. 1), additionally Step 3 and 4 are necessary. | |
5 Empirical example
Type of CI (α=5 %) | Lower bound | Upper bound |
---|---|---|
Standard | 0.046 | 0.450 |
Percentile | 0.044 | 0.496 |
Basic | 0.001 | 0.452 |
Type of CI (α=5 %) | Lower bound | Upper bound |
---|---|---|
Standard | −0.099 | 0.488 |
Percentile | −0.048 | 0.508 |
Basic | −0.120 | 0.437 |