Yves, I know that the statements are often made: “The CB-SEM model estimation requires a set of assumptions to be fulfilled, including the multivariate normality of data”. “If the data are to some extent nonnormal, use PLS-SEM”. But these statements are wrong! Comparisons between PLS and CB-SEM were generally done using the maximum likelihood estimator for CB-SEM, and ML of course is based on the assumption of multivariate normality.

As Jay pointed out, there are many estimation methods available for SEM with nonnormal data, such as weighted least-squares (WLS), unweighted least squares (ULS), and the robust versions weighted least-squares mean and variance adjusted (WLSMV) as well as ULSMV and robust maximum likelihood (MLR), and many more.

SEM is also not only based on covariances but also on means, therefore using the term CB-SEM is misleading.

PLS is a least-squares method, as the name implies. Therefore, in order to perform multi-sample analyses, many adequate estimation methods for SEM are available.

Kind regards,

Karin

There is much difference between multivariate normality and the univariate. When there is only one attribute, we use univariate normality test. When there are more than one attribute, for instance using width and length of Sepal to predict the species, we must use multivariate normality test to check if the data set follows a multivariate normal distribution. So, they can not be used instead of each other. and as you know, univariate normality does not lead to multivariate normality. I hope I have understood what you mean.

You should use multivariate normality test to check if the data set follows a multivariate normal distribution.

Hi,

I personally find those rather uninformative as data usually departs from multivariate normality. In most cases you cannot even find univariate normality. Consequently, I apply in SEM (there is only SEM versus PLS :) the Satorra-Bentler correction or the Yuan-Bentler correction of the chisquare test and standard errors.

Best,

Holger

In general this is a very challenging problem, particularly in high dimension.

The normality tests tend to be kind of uninformative and often over-powered, and thus likely to find trivial and uninteresting violations in moderate to large samples. Vice versa, they miss important violations in small samples. You should absolutely look through your data graphically as well as with hypothesis tests and, most importantly, an understanding of the variables substantively and theoretically. For instance if you anticipate that a variable might have an active ceiling or floor, that will help guide you, even if the sample size is small and the normality test says everything's fine.

How important deviations from normality are really depends on the analysis. It's not required for exogenous variables in SEM, which still seems to work out OK with some second-order corrections (e.g., Satorra-Bentler, etc.) with mild deviations. PLS can likely get away with some deviations from normality as well but its least squares criterion will be affected, much as the MVN likelihood is for SEM.

In Stata, they provide a pretty comprehensives suite although I think there are some newer ones. I'm sure that R has those---there's one based on information theoretic concepts I can't recall the name of at the moment but most of the tests out there are based on the third and fourth moments, which approximate the KL divergence anyway.

You may want to look at:

Doornik, J. A., and H. Hansen. 2008. An omnibus test for univariate and multivariate normality. Oxford Bulletin of Economics and Statistics 70: 927–939.

Dear Fatemeh Mahmoudi , Claire Su-Yeon Park , Holger Steinmetz and Jay Verkuilen

Thanks for the contributions. This issue arose because researchers in areas like marketing and management information systems defend that one of the criterions for choosing between PLS or CB-SEM is the data characteristics like a normal distribution (then we should opt for PLS). For example, according to Hair, Ringle, and Sarstedt (2011): “The CB-SEM model estimation requires a set of assumptions to be fulfilled, including the multivariate normality of data”. In another excerpt, they say “If the data are to some extent nonnormal, use PLS-SEM”.

Considering the second citation, I conclude that checking for univariate normality should be enough for opting to PLS. But it gets more complicated IF we are working with PLS-SEM AND need to execute a multi-group analysis. According to Chin et al. (2012), “The Partial Least Squares (PLS) approach has been popularized among researchers in part because the sample size requirements are much smaller for complex models than required for covariance based techniques, and there are fewer assumptions on data properties such as normality and heterogeneity. It is also considered a more appropriate choice when the emphasis is on prediction. However, to date the procedures used for multi-group comparisons have been relatively naïve being focused on discussions of the magnitude of differences between estimates or on t-test statistics when assessing differences. However, such techniques can be problematic if sample sizes are dissimilar or the data is not normally distributed.”

Considering Chin citation, it is ok to opt for PLS-SEM if data distribution is normal and there’s heterogeneity. However, if we are interested also in executing a multigroup analysis (under PLS approach), non normally distributed data becomes a problem.

Best,

Yves

Not really knowing anything about your variables, it's very hard to speak in more than generalities. I think you'd probably want to seek out local expertise.

A lot of SEM methods, broadly speaking, are designed explicitly for non-Gaussian variables and SEM tolerates some degree of non-Gaussian data, particularly if GLS, DWLS, or ULS is used instead of maximum likelihood. There are many adaptations of SEM to consider binary, ordinal, nominal, or count data or even mixed types of variables. The latter would be very difficult in a PLS framework. There are also multigroup versions of SEM. Ultimately the methods aren't really that different, though, and instrumental variables (IV) approaches to SEM (see Ken Bollen's recent work) are very much a bridge between the two worlds.

Jay Verkuilen

Thank you! Within our area, the normality assumption is used as a rule of thumb for choosing between structural equations modeling analysis methods and even for multivariate outliers detection (e.g., Mahalanobis distance). Those who have to opt for one or another statistics analysis methods under an applied social sciences context (like me) need to rely on those rules of thumbs (which in several occasions demonstrates a lot of inconsistencies) or follow the guidelines of experienced statisticians.

We realize that when experienced professionals like you and Holger Steinmetz show us new possible directions to apply in our expertise area. For example, Holger suggested to " apply the Satorra-Bentler correction or the Yuan-Bentler correction of the chi-square test and standard errors".

When I was looking for these tests I found a researcher with a similar question. Quoting him " Consequently, I checked several references and found (at least for me) divergent answers regarding the "requirements" for Satorra-Bentler test-statistics. One paper reported the following: "The variables in the data were multivariate nonnormally distributed, with normalized Mardia’s multivariate kurtosis of 6.03 (p < 001). We employed a maximum likelihood estimation method with robust standard errors together with the Satorra-Bentler rescaled chi-square statistic (Satorra and Bentler 1994), which compensates for nonnormality of variables." (Stern, Katz-Navon, Naveh, 2008, p. 1558)".

I quoted him to demonstrate my point. If this normality adjustment could always be adequate to our research, it would allow us to abandon this "normality rule of thumb" and focus more on the analysis objectives (predictive or confirmatory).

As you can see we are always trying to clarify those peculiarities, and so, we shouldn't have to rely on - blindly -, on the well-established rules of thumb in our area.

Thank you all for your support!

Best.

Yves, I know that the statements are often made: “The CB-SEM model estimation requires a set of assumptions to be fulfilled, including the multivariate normality of data”. “If the data are to some extent nonnormal, use PLS-SEM”. But these statements are wrong! Comparisons between PLS and CB-SEM were generally done using the maximum likelihood estimator for CB-SEM, and ML of course is based on the assumption of multivariate normality.

As Jay pointed out, there are many estimation methods available for SEM with nonnormal data, such as weighted least-squares (WLS), unweighted least squares (ULS), and the robust versions weighted least-squares mean and variance adjusted (WLSMV) as well as ULSMV and robust maximum likelihood (MLR), and many more.

SEM is also not only based on covariances but also on means, therefore using the term CB-SEM is misleading.

PLS is a least-squares method, as the name implies. Therefore, in order to perform multi-sample analyses, many adequate estimation methods for SEM are available.

Kind regards,

Karin

When I use AMOS, I observed normality by skewness and kurtosis. Since, may sample is large (> 200), the results from Kolmogorov-Smirnov are likely to show non-normality.

When I use PLS, I have not concern about non-normality because the technique is relax for this assumption. In top journal marketing, the researchers use PLS because their data is non-normal.

For example,

Ashraf, A. R., Thongpapanl, N., & Auh, S. (2014). The application of the technology acceptance model under different cultural contexts: The case of online shopping adoption. Journal of International Marketing, 22(3), 68-93.

Dear Karin Schermelleh-Engel

Thank you for your insights on the subject. You are precise when you say that " these statements are wrong." What is disturbing is that a whole field of research (applied social sciences) might be assuming wrong premises to analyze data and thus presenting false results.

Best regards

Dear Yves,

here is some more to swallow :)

McIntosh, C. N., Edwards, J. R., & Antonakis, J. (2014). Reflections on partial least squares path modeling. Organizational Research Methods, 17(2), 210-251. doi:10.1177/1094428114529165

Rönkkö, M., McIntosh, C. N., & Antonakis, J. (2015). On the adoption of partial least squares in psychological research: Caveat emptor. Personality and Individual Differences, 87, 76-84.

Rönkkö, M., McIntosh, C. N., Antonakis, J., & Edwards, J. R. (2016). Partial least squares path modeling: Time for some serious second thoughts. Journal of Operations Management, 47, 9-27. doi:10.1016/j.jom.2016.05.002

Rouse, A., & Corbitt, B. (2008). There's sem and" sem": A critique of the use of pls regression in information systems research. ACIS 2008 Proceedings, 81.

My personal impression is that the PLS folks tried to "surfe the SEM wave" and sell it as the better, more appropriate, "soft modeling" version of SEM. As these papers argue this is based on shaky ground and as Karin noted, true shortcomings in SEM wrt to deviations from multinormality or the low sample size problems of fit testing can nowadays avoided by using appropriate estimators and/or corrections. PLS is nothing more than regression with composites. I cannot say by myself if it does that appropriately but regression is not SEM.

Given that state of affairs I cannot really understand why the proponents of PLS still keep to this approach. Escalation of commitment?

Best,

Holger

Dear Holger Steinmetz

Thanks for the material! Surely I will enjoy the reading. 😊

In fact, PLS is being presented since a long time as a “silver bullet” within applied social sciences research (especially in marketing area). For example, this article below has 5857 citations since 2011.

Hair, J. F., Ringle, C. M., & Sarstedt, M. (2011). PLS-SEM: Indeed a silver bullet. Journal of Marketing theory and Practice, 19(2), 139-152.

In most of these articles, the explanation about why nonnormal data distribution is more suitable to PLS than Covariance-based SEM is not clear. The same is valid for the minimum sample estimation. If you go deeper, it’s possible to grasp the inconsistencies (even to nonstatistics professionals like me).

About your last phrase “Escalation of commitment?” Maybe. And it can be harmful to a whole research area. That’s why these rules of thumb should be even more discussed between statisticians and non-statisticians.

Wonderful debate!

Best,

Yves

Hi Yves Costa Netto

I indeed have the impression that the discussion on PLS has been led either by statisticians or by researchers focusing on statistical issues such as efficiency or (statistical) bias. My focus as a causally-focused researcher is how to enhance causal inference, and this most of all refers to concepts on a population/identification level. Statistical considerations like errors, estimation, bias, and efficiency are vital but come second. I have forgotten from whom the quote is "if you can't get it right on the infinite level you shouldn't be in the business" :)

With regard to the interest or routines of statisticians to be concerned with causal questions, you should take a look in the work of Judea Pearl. This is polemic, I agree, but has a kernel of truth. The fact that PLS always was sold with its focus on prediction (rather than explanation/causation) shows that this is the philosophy of PLS-folks themselves. The problem, however, occurs when the majority of PLS-users apply PLS to test their hypotheses (hence, focus on an explanation context). I surely have a limited "sample size" but almost all former colleagues and researchers who I know and who switched to PLS did that because their SEM told them unconfortable truths. The fact that PLS is so wide-spread in marketing should be a warning to everyone being confronted with marketing ;) If this sentence is too hard, relax; it is even worse in psychology :)

Below, I post some books paper if you are interested in Pearl's criticsm on an overly statistical focus and lack of causal concepts. Of special interest is the recent direct discussion between Pearl and Andrew Gelman, the great statistician on Pearl's blog:

http://causality.cs.ucla.edu/blog/index.php/2019/01/

Hope that helps.

Holger

Bollen, K. A., & Pearl, J. (2013). Eight myths about causality and structural equation modeling. In S. L. Morgan (Ed.), Handbook of Causal Analysis for Social Research (pp. 301-328). Dordrecht: Springer.

Pearl, J., Glymour, C., & Jewell, N. P. (2016). Causal inference in statistics: A primer: Wiley.

Pearl, J., & MacKenzie, D. (2018). The book of why. New York: Basic books.

Pearl, J. (2009). Causal inference in statistics: An overview. Statistics Surveys, 3(0), 96-146. doi:10.1214/09-SS057