Premature conclusions about the signal-to-noise ratio in structural equation modeling research: A commentary on Yuan and Fang (2023)

In a recent article published in this journal, Yuan and Fang (British Journal of Mathematical and Statistical Psychology, 2023) suggest comparing structural equation modeling (SEM), also known as covariance-based SEM (CB-SEM), estimated by normal-distribution-based maximum likelihood (NML), to regression analysis with (weighted) composites estimated by least squares (LS) in terms of their signal-to-noise ratio (SNR). They summarize their findings in the statement that “[c]ontrary to the common belief that CB-SEM is the preferred method for the analysis of observational data, this article shows that regression analysis via weighted composites yields parameter estimates with much smaller standard errors, and thus corresponds to greater values of the [SNR].” In our commentary, we show that Yuan and Fang have made several incorrect assumptions and claims. Consequently, we recommend that empirical researchers not base their methodological choice regarding CB-SEM and regression analysis with composites on the findings of Yuan and Fang as these findings are premature and require further research.     

An alternating least squares method for the weighted approximation of a symmetric matrix

Bailey and Gower examined the least squares approximationC to a symmetric matrixB, when the squared discrepancies for diagonal elements receive specific nonunit weights. They focussed on mathematical properties of the optimalC, in constrained and unconstrained cases, rather than on how to obtainC for any givenB. In the present paper a computational solution is given for the case whereC is constrained to be positive semidefinite and of a fixed rankr or less. The solution is based on weakly constrained linear regression analysis.The authors are obliged to John C. Gower for stimulating this research.     

A generalized family of coefficients of relational agreement for numerical scales

A family of coefficients of relational agreement for numerical scales is proposed. The theory is a generalization to multiple judges of the Zegers and ten Berge theory of association coefficients for two variables and is based on the premise that the choice of a coefficient depends on the scale type of the variables, defined by the class of admissible transformations. Coefficients of relational agreement that denote agreement with respect to empirically meaningful relationships are derived for absolute, ratio, interval, and additive scales. The proposed theory is compared to intraclass correlation, and it is shown that the coefficient of additivity is identical to one measure of intraclass correlation.The author thanks the Editor and anonymous reviewers for helpful suggestions.     

Interpreting canonical correlation analysis through biplots of structure correlations and weights

This paper extends the biplot technique to canonical correlation analysis and redundancy analysis. The plot of structure correlations is shown to the optimal for displaying the pairwise correlations between the variables of the one set and those of the second. The link between multivariate regression and canonical correlation analysis/redundancy analysis is exploited for producing an optimal biplot that displays a matrix of regression coefficients. This plot can be made from the canonical weights of the predictors and the structure correlations of the criterion variables. An example is used to show how the proposed biplots may be interpreted.     

New lower and upper bounds for communality in factor analysis

We derive several relationships between communalities and the eigenvalues for ap ×p correlation matrix under the usual factor analysis model. For suitable choices ofj, _j (), where _j () is thej-th largest eigenvalue of , provides either a lower or an upper bound to the communalities for some of the variables. We show that for at least one variable, 1 - _p () improves on the use of squared mulitiple correlation coefficient as a lower bound.This research was done while the second author was at Tokyo Institute of Technology.     

The logics of orthoalgebras

The notion of unsharp orthoalgebra is introduced and it is proved that the category of unsharp orthoalgebras is isomorphic to the category of D-posets. A completeness theorem for some partial logics based on unsharp orthoalgebras, orthoalgebras and orthomodular posets is proved.     

Estimating the polyserial correlation coefficient

We develop simple noniterative estimators of the polyserial correlation coefficient. A general relationship between the polyserial correlation and the point polyserial correlation is exploited to give extensions of Pearson's, Brogden's, and Lord's biserial estimators to the multicategory setting. The small sample and asmptotic properties of these estimators are studied in some detail. A comparison with maximum likelihood estimates shows that Lord's polyserial estimator is fairly efficient across three probability models.The authors would like to thank the referees for suggestions that improved the presentation of the paper.     

Delayed reinforcement of fixed-ratio performance without mediating exteroceptive conditioned reinforcement

The performance of pigeons was studied under conditions in which the completion of a fixed-ratio requirement was not contiguous with the presentation of a reinforcer. Timein and timeout periods alternated throughout the experimental sessions. Responses made by an experimental bird during the timein period were accumulated, and when a fixed-ratio requirement had been met, grain was presented to the experimental bird and a yoked control following their first response in the next timein period. Across most manipulations of the fixed-ratio requirement and of the duration of the timeout period, the response rates of the experimental birds were considerably higher than those of their controls, suggesting that the response-reinforcer dependency controlled the behavior of the experimental bird in the absence of a close temporal association between responding on the ratio schedule and reinforcer presentations.     

Problems with bootstrapping pearson correlations in very small bivariate samples

Efron'sMonte Carlo bootstrap algorithm is shown to cause degeneracies in Pearson'sr for sufficiently small samples. Two ways of preventing this problem when programming the bootstrap ofr are considered.