Tao Wang's contribution to the Discussion of ‘Vintage Factor Analysis with Varimax Performs Statistical Inference’ by Rohe & Zeng

I congratulate the authors on their excellent paper studying the fundamental mechanism of PCA with the Varimax rotation. The paper thoughtfully establishes theorems to demonstrate that the Varimax rotation can supply an unified estimating technique for a wide range of semiparametric factor models. My discussion will be primarily centred on clarification and potential extensions. The paper illustrates that if PCA is performed on a matrix with independent elements, PCA with the Varimax rotation can be utilized to fit the semiparametric model under certain assumptions. Is the proposed technique applicable in the case of Quartimax or Equamax rotation? In comparison to oblique rotation, the interpretability of the resulting components from orthogonal rotation is not always satisfied (may oversimplify the data). Although the paper reveals that the developed technique can handle correlated factors, to broaden the scope of the paper’s application, I wonder if the built theorems apply to oblique rotations such as Oblimin and Promax. Also, the paper employs the scree plot (better if using parallel analysis scree plot), which is somewhat subjective, to argue the number of PCs to extract in order to obtain the most parsimonious factor structure. According to the paper, ‘there is not a single correct answer for the choice of k’. Provided that the primary purpose of the paper is to lay the theoretical groundwork for a widely used model, I am intrigued whether the authors can deliver a more theoretically justified method for selecting which factors to retain. In addition, the paper demonstrates that the procedure as a whole is effective on condition that the principal component matrix entries are reasonably nonnormal. Practically, we may assume that the original dataset is normally distributed to ensure that the PCs are independent and the results are more robust. In this case, I am concerned with the implications of the nonnormal condition on factor analysis using a normally distributed dataset. Given the leptokurtic condition on the elements of Z that must be satisfied for Varimax rotation to function, the validity of the multivariate normality assumption regarding the distribution of observable variables or latent constructs appears to be compromised. Aside from that, the variation around the mean for symmetric distributions is a useful measure of dispersion. Nonetheless, it can fail when dealing with skewed or asymmetric distributions; see (Tran et al., 2019). Is the developed method capable of accommodating asymmetric distributions or distributions lacking moments (or with undefined or infinite variance)? There has been substantial research towards the robustification of PCA in the field of robust statistics; see (Candès et al., 2011). Will the Varimax rotation work if we consider a more robust PCA method or are interested in capturing the tail behaviour of the data such as quantile estimation? Data sharing not applicable–no new data generated.

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