Evaluation of Complementary Numerical and Visual Approaches for Investigating Pairwise Comparisons after Principal Component Analysis

We propose and evaluate numerical and visual methods for investigating paired comparisons after principal component analysis (PCA). PCA results can be visualized to facilitate an understanding of the relationships between the products and the sensory attributes. But identifying and visualizing significant product differences in multiple PCs simultaneously is not straightforward. A benefit of the proposed methods is that they provide a screening tool for evaluating PCA results rapidly. We begin with a real data set which is analyzed and submitted to the truncated total bootstrap (TTB) procedure. This TTB procedure simulates and analyzes results from virtual panels. The TTB-derived results form clouds of uncertainty around each product and paired comparison. Although these clouds can be visualized directly or by plotting the smallest contours that enclose 95% of their kernel-estimated densities, we propose that plotting TTB-derived 95% confidence ellipsoids provide a less cumbersome approach. We show that it is also possible to calculate P values that evaluate whether pairs of products are discriminated in the PCA subspace. The interpretation of these P values coincides with the visual interpretation of the confidence ellipsoids. The volumes of these confidence ellipsoids, which quantify uncertainty, are calculated easily. The confidence ellipsoids, the P values, and the volumes provide a simple and consistent approach for investigating paired comparisons after PCA. We illustrate the methods with two real data sets, one a sensory quantitative-descriptive data set from a trained panel, the other a check-all-that-apply (CATA) data set from a consumer panel. We also conduct a simulation study based on each of these data sets. The results from these simulation studies show that under repetition, the 95% confidence ellipsoids often have coverage of approximately 95%, but in some cases, coverage can be substantially lower. This indicates that the proposed ellipsoids have an approximately frequentist interpretation, but coverage varies. The complementary numerical and visual approaches can be applied to a wide range of data sets from sensory evaluation and to data from other domains.