Code and procedures for the ICIS 2023 article: "Should we use Interactions? It Depends! Predictive Validation of Interaction Terms in PLS-SEM"

Danks & Ray (2023)

Procedure for generating point predictions from PLS-SEM models including a non-linear term generated using the product indicator approach.

1. Identify training ($x_{is}$) and holdout ($x_{oos}$) sets of cases.

• Here holdout data can be new data or data that has been partitioned during cross validation.

2. Estimate parameters of PLS-SEM model using training data ($x_{is}$) only and applying the product indicators approach for generating the non-linear term.

• Retain both initial descriptive statistics of the training data mean ($m_{is}$) and standard deviation ($s_{is}$), and
• Estimated measurement weights ($w_{is}$) and loadings ($l_{is}$), and structural path coefficients ($B_{is}$).

3. Generate the holdout ($x_{oos}$) product indicator data for the non-linear term.

• Standardize holdout data using training standard deviation $s_{is}$ and mean $m_{is}$.
• Generate the preliminary holdout product indicators data by all possible pairwise products of the indicators of the exogenous variables ($x_i$) and of the moderator variable ($m_j$) using holdout data ($x_{oos}$).
• Append the holdout data from step 1 with the new manufactured indicators from step 3.

4. Standardize holdout data from step 3 using training standard deviation $s_{is}$ and mean $m_{is}$.
5. Predict exogenous construct scores from outer weights:

• Predict the construct scores of exogenous constructs using holdout data from step 5 and the training measurement weights ($w_{is}$): $$X = x.w_{is}$$

6. Predict the endogenous construct scores:

• Multiply the predicted construct scores ($X$) by structural paths ($B_{is}$): $$Y = X.B$$

7. Predict the indicator scores of endogenous constructs:

• Multiply the predicted construct scores ($Y$) with the measurement loadings ($l_{is}$): $$y=Y.l_{is}$$

8. Unstandardize predictions.

• Use the training data standard deviation $s_{is}$ and mean $m_{is}$ to bring the predictions back to the original scale. Multiply each predicted observation by its corresponding standard deviation and add its corresponding mean.

Procedure for generating point predictions from PLS-SEM models including a non-linear term generated using the orthogonal approach.

1. Identify training ($x_{is}$) and holdout $x_{oos}$) sets of cases.

• Here holdout data can be new data or data that has been partitioned during cross validation.

2. Estimate parameters of PLS-SEM model using training data ($x_{is}$) only and applying the product indicators approach for generating the non-linear term.

• Retain both initial descriptive statistics of the training data mean ($m_{is}$) and standard deviation ($s_{is}$), and
• Estimated measurement weights ($w_{is}$) and loadings ($l_{is}$), and structural path coefficients ($B_{is}$).
• In addition to the measurement and structural model estimates, the parameter estimates for the linear models used in the residual centering ($b_{is}$) need to be retained.

3. Generate the holdout product indicator data for the non-linear term.

• Standardize holdout data using training standard deviation ($s_{is}$) and mean ($m_{is}$).
• Generate the preliminary holdout product indicators data by all possible pairwise products of the indicators of the exogenous variables ($x_i$) and of the moderator variable ($m_j$) using holdout data ($x_{oos}$).

4. Residual center the holdout data for indicators of the non-linear term

• For each of the preliminary indicators of the non-linear term (from 3) use the parameter estimates of the relevant residual centering linear model ($b_{is}$) and the holdout data ($x_{oos}$) to generate a predicted score for the preliminary indicator.
• Deduct the predicted holdout preliminary indicator scores calculated in step 4. from the preliminary holdout product indicator scores from step 3.
• Append the holdout data from step 1 with the new residual centered indicators from step 4.

5. Standardize holdout data from step 4 using training standard deviation $s_{is}$ and mean $m_{is}$.
6. Predict exogenous construct scores from outer weights:

• Predict the construct scores of exogenous constructs using holdout data from step 5 and the training measurement weights ($w_{is}$): $$ X = x.w_{is}$$

7. Predict the endogenous construct scores:

• Multiply the predicted construct scores ($X$) by structural paths ($B_{is}$): $$Y = X.B$$

8. Predict the indicator scores of endogenous constructs:

• Multiply the predicted construct scores ($Y$) with the measurement loadings ($l_{is}$): $$y=Y.l_{is}$$

9. Unstandardize predictions.

• Use the training data standard deviation $s_{is}$ and mean $m_{is}$ to bring the predictions back to the original scale. Multiply each predicted observation by its corresponding standard deviation and add its corresponding mean.