Both standardized and unstandardized partial regression and path coefficients can be > |1|. In the case of standardized coefficients, values > |1| may point to collinearity and/or suppression issues.

A standardized coefficient greater than 1 means that is explaining more than 100% of the variance, which is not possible. Can you post a path diagram of the model you are testing?

Christian Geiser and David L Morgan thank you both for your answers.

this path diagram of the model is attached.

TotalPosi is the path of "positive affect" on "SA" with mediating "efficacy". (L*O)

Best regard

A standardized coefficient value that exceeds 1 (or less than 0) indicates an issue with the model. It points to a Heywood case ensued from a mis-specified model, multicollinearity, or an outlier. A fix could be done by ensuring construct validity (i.e., deleting the problematic item), removing collinearity, or constraining the variance of the concerned variable to “1” and naming its paths with its indicators using the same term (Collier, 2020). You could go through pages 88–89 of Collier’s (2020) book for germane insights. Here is the full citation.

Collier, J. E. (2020). Applied structural equation modeling using AMOS: Basic to advanced techniques. Routledge. https://www.taylorfrancis.com/books/mono/10.4324/9781003018414/applied-structural-equation-modeling-using-amos-joel-collier

Good luck,

I do not see any problems with the structure of your model, based on the diagram you presented. As a next step, I would recommend rewriting your model to treat each of the relations among your unmeasured variables as a correlation rather than a directed coefficient -- in essence a Confirmatory Factor Analysis rather than a full-scale Structural Equation Model. This might help you localize the problem.

David L Morgan and Mohialdeen Alotumi and Christian Geiser thank you for your answers. These are all extremely helpful.

David L Morgan It is not true that a standardized partial regression or path coefficient > |1| necessarily means that more than 100% of the variance are explained or that this is an improper solution. Standardized path coefficients can be > |1| without there being > 100% explained variance since these are not generally equal to correlation coefficients. These are partial regression coefficients, and in situations with highly correlated predictors (suppression), the coefficients can exceed |1| without R^1 > 1.

Mohialdeen Alotumi A standardized partial regression or path coefficient > |1| does not necessarily indicate a Heywood case or improper solution (unlike a correlation coefficient, which is bound between -1 and 1). There is nothing in the computational formula that keeps partial standardized regression coefficients from exceeding |1|, and this can happen when independent variables for the same outcome are highly correlated. The same is true for standardized factor loadings (those can also exceed |1| without there being an improper solution) when variables load on more than one factor and the factors are correlated.

1. My understanding of suppression is that it occurs when a direct effect is larger than the original zero order correlation, but I do see how that allows for an effect larger than 1.

2. As for multi-collinearity, I understand how it can make it difficult to estimate the separate effects of either variable, but again I do see how that could result in an effect greater than 1.

3. As for cross-loadings, I did not see any in her path diagram.

David L Morgan please see:

https://scholar.archive.org/work/6vc2eahqlza75hee6qkm7fhwfa/access/wayback/http://www.ssicentral.com/lisrel/techdocs/HowLargeCanaStandardizedCoefficientbe.pdf

Article On the Occurrence of Standardized Regression Coefficients Gr...

https://www.researchgate.net/post/Can-standardized-regression-weights-be-greater-than-1-What-doaes-it-means

David L Morgan Mohialdeen Alotumi

Below is an example of a 2-variable regression model (Y regressed on X1 and X2; full Mplus output file in the attached *.out file) that yields standardized regression coefficients > |1| without yielding R^2 > 1. The standardized regression coefficients are -1.056 and 1.444; R^2 = .761. The solution is not improper (not a Heywood case). There is no negative variance estimate or correlation estimate > |1|.

Correlations

Y X1 X2

________ ________ ________

Y 1.000

X1 0.100 1.000

X2 0.600 0.800 1.000

STANDARDIZED MODEL RESULTS (STDYX Standardization)

Two-Tailed

Estimate S.E. Est./S.E. P-Value

Y ON

X1 -1.056 0.079 -13.436 0.000

X2 1.444 0.071 20.407 0.000

Residual Variances

Y 0.239 0.029 8.105 0.000

R-SQUARE

Observed Two-Tailed

Variable Estimate S.E. Est./S.E. P-Value

Y 0.761 0.029 25.823 0.000

The result is due to collinearity (.8 correlation between X1 and X2) which results in a suppression effect (X1 increases the "utility" of X2 as a predictor of Y). X1 is not highly correlated with Y (only .1), but the fact that X1 is highly correlated with X2 (.8) makes it useful in the model as a suppressor variable.

The same kinds of effects can occur in path analysis and SEM with more than one outcome (dependent) variable.

Thank you for this example.

Thank you for the input, Christian Geiser

A path coefficient is a standardized partial-regression coefficient. If it goes above 1, it is a strong sign there is an issue in a SEM model—highly likely multicollinearity in the data. The existence of multicollinearity violates a fundamental assumption with SEM, thus rendering it impractical for measurement error. Therefore, a full collinearity check is highly recommended based on the logic that the model suitability comes first and then parameter estimates. The attached direct quote from Grewal et al. (2004, p. 528) reflects the seriousness of measurement error in SEM.

Grewal, R., Cote, J. A., & Baumgartner, H. (2004). Multicollinearity and measurement error in structural equation models: Implications for theory testing. Marketing Science, 23(4), 519–529. https://doi.org/10.1287/mksc.1040.0070

Good luck,

Mohialdeen Alotumi You wrote: "The existence of multicollinearity violates a fundamental assumption with SEM". There is no such assumption in SEM. In fact, regression, path analysis, confirmatory factor analysis (CFA), and structural equation modeling are all designed to appropriately handle multicollinearity (i.e., correlated independent and/or dependent variables). Collinearity, that is, high positive correlations of factor indicators within the same factor, is even desirable in measurement models (CFA) as such high correlations indicate highly reliable indicators (small measurement error variances).

That being said, high collinearity of independent variables in structural (regression, path) models can cause issues with high standard errors and interpretability of regression and path coefficients. Consider, for example, the suppressor situation where coefficients may even change sign relative to the zero-order correlations as in my data example above. But there is no violation of assumptions with collinearity. It is just a reality of life that highly correlated variables can occur in a causal model.

Christian Geiser,

You wrote, “There is no such assumption in SEM.” It seems that Collier (2020) strongly disagrees with you. He listed eight assumptions, the sixth of which is multicollinearity. Please see the attachment, which should be considered a direct quote from Collier (2020, p. 8). On the very same page, he stressed that “with any statistical technique, assumptions are made.” Below is the full citation.

Collier, J. E. (2020). Applied structural equation modeling using AMOS: Basic to advanced techniques. Routledge. https://www.taylorfrancis.com/books/mono/10.4324/9781003018414/applied-structural-equation-modeling-using-amos-joel-collier

Best,

Mohialdeen Alotumi Many of the "assumptions" and statements listed in the highlighted quote are incorrect and/or misleading/problematic:

- SEM does not assume that data need to be normally distributed. Only certain estimators, such as maximum likelihood, require this assumption. If you use a different estimator such as, for example, weighted least squares (WLS), the multivariate normality assumption is not at all required.

- Dependent variables in SEM need not be continuous. Binary and ordinal variables can be used both as indicators of latent variables and otherwise in a CFA or SEM.

- SEM does not necessarily require linear relationships. Nonlinear relationships (e.g., quadratic) can also be modeled.

- Data need not be complete for SEM. You can use multiple imputation or full information maximum likelihood estimation to estimate an SEM with missing data under the missing at random data mechanism.

- I already explained previously that multicollinearity is not a problem per se for SEM.

So yes, I agree that I strongly disagree with the statements made in the reference that you mentioned and I would not recommend this book to anyone who wants to learn about SEM.

Christian Geiser,

The book by Collier (2020) is about CB-SEM. I beg to disagree that the assumptions highlighted in the book are “misleading/problematic.” Collier (2020) raised assumptions about conducting SEM using AMOS—which applies CB-SEM. Such assumptions are also established in relevant research. For instance, you wrote, “SEM does not assume that data need to be normally distributed” which contradicts what Hair et al. (2017, p. 119) highlighted in the attached quote comparing normality in CB-SEM and PLS-SEM.

Collier, J. E. (2020). Applied structural equation modeling using AMOS: Basic to advanced techniques. Routledge. https://www.taylorfrancis.com/books/mono/10.4324/9781003018414/applied-structural-equation-modeling-using-amos-joel-collier

Hair, J. F., Matthews, L. M., Matthews, R. L., & Sarstedt, M. (2017). PLS-SEM or CB-SEM: Updated guidelines on which method to use. International Journal of Multivariate Data Analysis, 1(2). https://doi.org/10.1504/ijmda.2017.087624

Best,

Mohialdeen Alotumi Yes, it is a book about CB-SEM. Somebody writing something in a book does not mean that it is correct. It is incorrect (at the very least imprecise) to state that "CB-SEM assumes normality of data distributions." CB-SEM as such does not make such an assumption. Certain estimation methods (e.g., maximum likelihood) require normality. However, other estimation methods also used in CB-SEM (e.g., ADF estimator, Browne, 1984) do not require normality. Therefore, it is misleading to state that "CB-SEM assumes normality" because it makes people think that they cannot use CB-SEM with non-normal data. This, however, is not true. You can even use (robust) maximum likelihood estimation (i.e., with robust standard errors and corrected/scaled test statistics such as with the Satorra-Bentler correction or various other robust ML estimators such as MLR or MLMV in Mplus) with non-normal data in CB-SEM. These issues have been figured out a long time ago (see references below).

I suspect that some proponents of PLS use this as an argument against CB-SEM, and in "favor" of PLS. However, it is not a good argument because the distributional issues in CB-SEM parameter estimation have been resolved many years ago through development of distribution-free, robust ML, or bootstrapping methods. Most standard text books on CB-SEM provide this information and the relevant references.

Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83.

Finney, S. J., & DiStefano, C. (2006). Non-normal and categorical data in structural equation modeling. In G. R. Hancock & R. O. Mueller (Eds.). Structural equation modeling: a second course. Greenwich, CT: Information Age Publishing.

West, S. G., Finch, J. F., & Curran, P. J. (1995). Structural equation models with nonnormal variables: Problems and remedies. In R. H. Hoyle (Ed.), Structural equation modeling: Concepts, issues, and applications (pp. 56–75). Sage Publications, Inc.

Christian Geiser,

Don’t you think the provided references are a bit old? Henseler (2021) highlighted the alleged claim that CB-SEM does not require normal distribution as a thing of the past. Kindly check out the attachment, which should be deemed as a direct quote from Henseler (2021, p. 13).

Henseler, J. (2021). Composite-based structural equation modeling: Analyzing latent and emergent variables. The Guilford Press. https://www.guilford.com/books/Composite-Based-Structural-Equation-Modeling/Jorg-Henseler/9781462545605

Best,

Mohialdeen Alotumi The fact that something is "old" does not make it untrue. There are plenty of studies that show that covariance-based SEM can be done properly with non-normal data if robust estimation methods are used. I have not seen any "newer" papers that contradict these findings. If you have, then please share them with us.

I think you may be misreading the quote by Henseler that you highlighted in your last post. Notice that he is referring to composite-based SEM, not covariance-based SEM. In other words, he uses the abbreviation CB-SEM for composite SEM (e.g., PLS), not covariance-based SEM.

In fact, what he is saying in the quote is exactly in line with what I was explaining earlier and does not contradict any of what I was saying. He speaks of "alleged" advantages of [composite] methods such as PLS and the fact that "the generality of these ascribed characteristics is limited."

Christian Geiser,

Since there has been controversy on the various approaches to SEM, recent literature has unraveled new insights that refuted many old claims (e.g., Dash et al., 2021; Hair et al., 2021). That is the point I implied when I asked for new references—not as you indicated that an old reference “does not make it untrue.”

In addition, I did not misread the quote for two reasons. First, Henseler (2021) did not use the acronym CB-SEM to refer to composite-based SEM—nor did I. Besides, he indicated that formative and composite models “have been the standard [in CB-SEM] for decades” (see the attachment) (p. 54). He defined composite-based SEM as “those SEM techniques that involve composites in the estimation phase” (p. 11). Such a definition does not agree with what you wrote, “what he is saying … exactly in line with what I was explaining earlier.” In fact, in the first chapter of his book, which is the context of the quote I cited in the previous post, Henseler (2021) drew distinctions between various SEM approaches. For instance, he differentiated between composite-based SEM and factor-based SEM, as well as between variance-based SEM and covariance-based SEM. He also rendered a typology that included variance-based, limited information estimation of composite models (i.e., partial least squares (PLS) path modeling) and covariance-based, limited information estimation of composite models (i.e., piecewise maximum likelihood SEM with emergent variables), among others. In other words, he did not restrict the concept of composite-based SEM to what you referred to as "composite SEM (e.g., PLS), not covariance-based SEM" and composite “methods such as PLS.”

By the way, the IBM SPSS Amos 26 user’s guide by Arbuckle (2019) encompassed clarification on the normality assumption under the subheading “Distribution Assumptions for Amos Models” (see an excerpt attached). Hopefully, such clarification could make the record straight.

Arbuckle, J. L. (2019). IMB SPSS Amos 26. IBM Corp. https://www.ibm.com/docs/SSLVMB_26.0.0/pdf/amos/IBM_SPSS_Amos_User_Guide.pdf

Dash, G., & Paul, J. (2021). CB-SEM vs PLS-SEM methods for research in social sciences and technology forecasting. Technological Forecasting and Social Change, 173. https://doi.org/10.1016/j.techfore.2021.121092

Hair, J. F., Hult, G. T. M., Ringle, C. M., Sarstedt, M., Danks, N. P., & Ray, S. (2021). An introduction to structural equation modeling. In Partial least squares structural equation modeling (PLS-SEM) using r (pp. 1-29). https://doi.org/10.1007/978-3-030-80519-7_1

Henseler, J. (2021). Composite-based structural equation modeling: Analyzing latent and emergent variables. The Guilford Press. https://www.guilford.com/books/Composite-Based-Structural-Equation-Modeling/Jorg-Henseler/9781462545605

Best,

There has been a long discussion on the issue here, which may be somewhat confusing, especially due to the contradictory opinions. I mostly agree with Christian Geiser. I summarize my view here:

1) A standardized coefficient above one does not mean an improper solution, as long as the variance is not negative. So you should look for negative variances in your model.

2) If there is no negative variance, it is very likely that there is colinearity among the predictors (e.g. TotalPos may be highly correlated with TotalNeg). However, the multicollinearity does not make your model incorrect or biased, but it makes the slope highly variable. In other words, the standard error, which shows the sampling variability, becomes highly inflated. This means that if you repeat the research (with a similar statistical model) you may get very different results (on the colinear variables), which can make interpretation difficult. However, the confidence intervals and the p-value are still unbiased. In this situation, it is probably better to find the colinearity and exclude one of the variables. Note that when you exclude one variable, you are not losing much, because the variables are highly correlated (i.e., you had excluded one, but the effect is still in the model, as it was shared with another variable).

I hope this helps.

Hi Christian, "Yes, it is a book about CB-SEM. Somebody writing something in a book does not mean that it is correct." I agree, as an old colleague of mine once said, "Paper has never refused ink".