Assessing Inner Model

STEP 1: COLLINEARITY ASSESSMENT

To assess collinearity, we apply the same measures as in the evaluation of formative measurement models (i.e., tolerance and VIF values).

For instance, Y1 and Y2 jointly explain Y3. Likewise, Y2 and Y3 act as predictors of Y4. Therefore, one needs to check whether there are critical levels of collinearity between each set of predictor variables, that is, Y1 and Y2 as well as Y2 and Y3.

Analogous to the assessment of formative measurement models, we consider tolerance values below 0.20 (VIF value above 5) in the predictor constructs as critical levels of collinearity. If a critical level of collinearity is indicated by the tolerance or VIF guidelines, one should consider eliminating constructs, merging predictors into a single construct, or creating higher-order constructs to treat collinearity problems.

Step 2: Structural Model Path Coefficients

After running the PLS-SEM algorithm, estimates are obtained for the structural model relationships (i.e., the path coefficients), which represent the hypothesized relationships among the constructs. The path coefficients have standardized values approximately between –1 and +1 (values can be smaller/larger but usually fall in between these bounds).

Estimated path coefficients close to +1 represent strong positive relationships (and vice versa for negative values) that are usually statistically significant (i.e., different from zero in the population). The closer the estimated coefficients are to 0, the weaker are the relationships. Very low values close to 0 are usually not significantly different from zero.

Whether a coefficient is significant ultimately depends on its standard error that is obtained by means of bootstrapping.

Critical t-values

If t-value is greater than or equal to 1.645, then the relationship is significant at alpha < 0.05.

Bootstrap Confidence Interval

It provides additional information on the stability of a coefficient estimate.

It shows the range into which the true population parameter will fall assuming a certain level of confidence (e.g., 95%).

If a confidence interval for an estimated coefficient does not include zero, then the null hypothesis is rejected, and we assume a significant effect.

The range of the confidence interval provides the researcher with an indication of how stable the estimate is. If the confidence interval of a coefficient is wider, then its stability is lower.

Step 3: Coefficients of determination (R²)

PLS-SEM aims at maximizing R²of endogenous variable in a path model.

Researchers should evaluate the model's predictive accuracy via the R². It can be viewed as the combined effect of exogenous variables on an endogenous variable. In other words, it represents the amount of variance in the endogenous constructs explained by all exogenous variable linked to it.

The R² value ranges from 0 to 1, with higher levels indicating higher levels of predictive accuracy.

Rule of Thumb

Issues in using R²

R² increases when additional predictor constructs are included into the model. In other words, the more predictor included in the model, the R² will apparently increase without any stability. Therefore, it is advisable to use adjusted R², which controls for model complexity when comparing different models (Wherry, 1931) .

R-squared measures the proportion of the variation in dependent variable (Y) explained by independent variables (X) for a linear regression model. Adjusted R-squared adjusts the statistic based on the number of independent variables in the model.

The reason, why this is important, is because you can “game” R-squared by adding more and more independent variables, irrespective of how well they are correlated to your dependent variable. Obviously, this isn’t a desirable property of a goodness-of-fit statistic. Conversely, adjusted R-squared provides an adjustment to the R-squared statistic such that an independent variable that has a correlation to Y increases adjusted R-squared and any variable without a strong correlation will make adjusted R-squared decrease. That is the desired property of a goodness-of-fit statistic.

Which one to be used?

In the case of a linear regression with more than one variable, use adjusted R-squared. For a single independent variable model, both statistics are interchangeable.

Step 4: Effect Size (f²)

In addition to evaluating the R² values of all endogenous constructs, the change in the R² value when a specified exogenous construct is omitted from the model can be used to evaluate whether the omitted construct has a substantive impact on the endogenous constructs.

Rule of Thumb (Cohen et al., 1988)

Step 5: Predictive Relevance

Using PLS for prediction purposes requires a measure of predictive capability.

This measure is an indicator of the model’s out-of-sample predictive power or predictive relevance. When a PLS path model exhibits predictive relevance, it accurately predicts data not used in the model estimation. In the structural model, Q² values larger than zero for a specific reflective endogenous latent variable indicate the path model’s predictive relevance for a particular dependent construct.

Suggested approach – Blindfolding.

Wold (1982, p. 30), “The cross-validation test of Stone (1974) and Geisser (1975) fits soft modeling like hand in glove”

Blindfolding in SmartPLS 3.0

Guideline:

Make sure the number of observations used in the model estimation divided by the omission distance D is not integer (round number).
Chose D values between 5 and 12.
If Q²> 0 indicate that exogenous variable have predictive relevance for endogenous construct under consideration.
The Q² values of 0.02, 0.15, and 0.35 signify small, medium, and large predictive relevance of certain latent variable, thus explaining the endogenous latent variable under evaluation.

Summary of Indices for Structural Model Assessment