Conditional probabilistic analyses with WarpPLS

If an analysis suggests that two variables are causally linked, yielding a path coefficient of 0.25 for example, this essentially means in probabilistic terms that an increase in the predictor variable leads to an increase in the conditional probability that the criterion variable will be above a certain value. Yet, conditional probabilities cannot be directly estimated based on path coefficients; and those probabilities may be of interest to both researchers and practitioners.

By using the “Explore conditional probabilistic queries” menu option, WarpPLS users can estimate conditional probabilities via queries including combinations of latent variables, unstandardized indicators, standardized indicators, relational operators (e.g., > and <=), and logical operators (e.g., & and |). The article below provides an illustration of the use of this software feature in the context of a study of technology, innovation, and SMEs' export intensity.

Haddoud, M. Y., Kock, N., Onjewu, A. K. E., Jafari-Sadeghi, V., & Jones, P. (forthcoming). Technology, innovation and SMEs' export intensity: Evidence from Morocco. Technological Forecasting and Social Change. (Online availability details: Volume 191, June 2023.)

Link to full-text access from the publisher:

https://www.sciencedirect.com/science/article/abs/pii/S0040162523001609

Abstract:

This study seeks to understand the scarcely examined relationships between SMEs' foreign technology licensing, R&D expenditure, innovation and export intensity. Espousing an integrated open innovation and self-selection paradigm, observations of 446 Moroccan SMEs are analysed through structural equation modelling. The definitive path analysis showed that foreign technology licensing and R&D expenditure distinctively affect innovation and, in turn, innovation increases export intensity. In further insights, to illustrate how the distribution of these inputs enhances internationalisation, a probabilistic analysis shows that foreign technology licensing, R&D expenditure and innovation will incrementally stimulate export intensity by >71 %. The permutations of these variables in the fresh setting of Morocco summon scholars' empirical attention at the same time as policymakers' consideration.

(Many thanks to my co-authors, particularly Dr. Mohamed Haddoud, and to the review panel, for all of their work!)

Best regards to all!

Model with endogenous dichotomous variable

There are two main ways in which a model with an endogenous dichotomous variable can be analyzed in PLS-SEM - via the logistic regression variables technique, and via the conditional probabilistic queries technique.

Logistic regression variables technique

Starting in version 8.0 of WarpPLS, the menu option “Explore logistic regression” allows you to create a logistic regression variable as a new indicator that has both unstandardized and standardized values. Logistic regression is normally used to convert an endogenous variable on a non-ratio scale (e.g., dichotomous) into a variable reflecting probabilities. You need to choose the variable to be converted, which should be an endogenous variable, and its predictors. 

The new logistic regression variable is meant to be used as a replacement for the endogenous variable on which it is based. Two algorithms are available: probit and logit. The former is recommended for dichotomous variables; the latter for non-ratio variables where the number of different values (a.k.a. “distinct observations”) is greater than 2 but still significantly smaller than the sample size; e.g., 10 different values over a sample size of 100. The unstandardized values of a logistic regression variable are probabilities; going from 0 to 1. 

Since a logistic regression variable can be severely collinear with its predictors, you can set a local full collinearity VIF cap for the logistic regression variable. Predictor-criterion collinearity, or lateral collinearity, is rarely assessed or controlled in classic logistic regression algorithms. 

For more on this topic, see the links below.

Explore Logistic Regression in WarpPLS

https://youtu.be/PO65ySuhPW0

Using Logistic Regression in PLS-SEM with Composites and Factors

https://youtu.be/c9ySXN4NVus

Conditional probabilistic queries technique

How do we interpret the results of a model with an endogenous dichotomous variable, using the conditional probabilistic queries technique? Let us use the model below to illustrate the answer to this question. In this model we have one endogenous dichotomous variable “Success” that is significantly caused in a direct way by two predictors: “Projmgt” and “JSat”. The direct effect of a third predictor, namely "ECollab", is relatively small and borderline significant.

Let us assume that the unit of analysis is a team of people. The variable “Success” is coded as 0 or 1, meaning that a team is either successful or not. After standardization, the 0 and 1 will be converted into a negative and a positive number. The standardized version of the variable “Success” will have a mean of zero and a standard deviation of 1.

One way to interpret the results is the following. The probability that a team will be successful (i.e., that “Success” > 0) is significantly affected by increases in the variables “Projmgt” and “JSat”.

WarpPLS users are able, starting in version 6.0, to calculate conditional probabilities as shown below, without having to resort to transformations based on assumed underlying functions, such as those performed by logistic regression. In this screen shot, only latent variables are used, and they are all assumed to be standardized.

In the screen shot above, we can see that the probability that a team will be successful (i.e., that “Success” > 0), if “Projmgt” > 1 and “JSat” > 1, is 52.2 percent. Stated differently, if “Projmgt” and “JSat” are high (greater than 1 standard deviation above the mean), then the probability of success is slightly greater than chance.

A probability of 52.2 percent is not that high. The reason why it is not higher, in the context of the conditional probabilistic query above, is that we are not including the variable "ECollab" in the mix. Still, it does not seem like “Projmgt” and “JSat” being high are sufficient conditions for success, although they may be necessary conditions.

Consider a different set of conditional probabilities. If a team is successful (i.e., if “Success” > 0), what is the probability that “Projmgt” and “JSat” are low for that team. The answer, shown in the screen below, is 1.3 percent. That is a very low probability, suggesting that “Projmgt” and “JSat” matter as necessary but not sufficient elements for success.

These are among the conditional probabilistic queries that users are able to make starting in version 6.0 of WarpPLS. Bayes’ theorem is used to produce the answers to the queries.

Conditional probabilistic queries

If an analysis suggests that two variables are causally linked, yielding a path coefficient of 0.25 for example, this essentially means in probabilistic terms that an increase in the predictor variable leads to an increase in the conditional probability that the criterion variable will be above a certain value. Yet, conditional probabilities cannot be directly estimated based on path coefficients; and those probabilities may be of interest to both researchers and practitioners. By using the “Explore conditional probabilistic queries” menu option, users of WarpPLS can, starting in version 6.0, estimate conditional probabilities via queries including combinations of latent variables, unstandardized indicators, standardized indicators, relational operators (e.g., > and <=), and logical operators (e.g., & and |).

Related YouTube video:

Explore Conditional Probabilistic Queries in WarpPLS