Can we start by looking at the partial correlation network? I will refer you to my post on Driver Analysis vs. Partial Correlation Analysis and will not repeat that more detailed overview.
Each of the nodes is a variable (e.g., purchase intent is located on the far right). An edge drawn between any two nodes shows the partial correlation between those two nodes after controlling for all the other variables in the network. The color indicates the sign of the partial correlation with green for positive and red for negative. The size of the partial correlation is indicated by the thickest of the edge.
- Northwest - In Love / Romantic / Passionate / Radiant,
- Southwest - Bold / Active / Character / Fulfilled / Trust / Free,
- Mid-South - Classical / Tenacious / Quality / Timeless / High End,
- Mid-North - Wooded / Spiced,
- Center - Chic / Elegant / Rich / Modern,
- Northeast - Sweet / Fruity / Flowery / Fresh, and
- Southeast - Easy to Wear / Please Others / Pleasure.
Unlike the Probabilistic Structural Equation Model (PSEM) in Chapter 8 of BayesiaLab's book, my network is undirected because I can find no justification for assigning causality. Yet, the structure appears to be much the same for the two analyses, for example, compare this partial correlation network with BayesiaLab's Figure 8.2.3.
All this looks very familiar to those of us who have analyzed consumer rating scales. First, we expect negative skew and high collinearity because consumers tend to give ratings in the upper end of the scale and their responses often are highly intercorrelated. In fact, the first principal component accounted for 64% of the total variation, and it would have been higher had Wooded and Spiced been excluded from the battery.
A more cautious researcher might stop with extracting a single dimension and simply concluding that the women either liked or disliked the perfumes they tested and rated everything either uniformly higher or lower. They would speak of halo effects and question whether any more than an overall score could be extracted from the data. Nevertheless, as we see from the above partial correlation network, there is an interpretable local structure even when all the variables are highly interrelated.
I have discussed this issue before in a post about separating global from specific factors. The bifactor model outlined in that post provides another view into the structure of the perfume rating data. What if there were a global factor explaining what we might call the "halo effect" (i.e., uniformly high correlations among all the variables) and then additional specific factors accounting for the extra correlation among different subsets of variables (e.g., the regions in the above partial correlation network map)?
The bifactor diagram shown below may not be pretty with so many variables to be arrayed. However, you can see the high factor loadings radiating out from the global factor g and how the specific factors F1* through F6* provide a secondary level of local structure corresponding to the regions identified in the above network.
I will end with a technical note. The 1321 observations were nested within the 11 perfumes with each respondent seeing only one perfume. Although we would not expect the specific perfume rated to alter the correlations (factorial invariance), mean-level differences between the perfumes could inflate the correlations calculated over the entire sample. In order to test this, I reran the analysis with deviation scores by subtracting the corresponding mean perfume score from each respondent's original ratings. The results were essentially the same.
R Code Needed to Import CSV File and Produce Plots
# Set working directory and import data file setwd("C:/directory where file located") perfume<-read.csv("Perfume.csv", sep=";") apply(perfume, 2, function(x) table(x,useNA="always")) # Calculates Sparse Partial Correlation Matrix library("qgraph") sparse_matrix<-EBICglasso(cor(perfume[,2:48]), n=1321) qgraph(sparse_matrix, layout="spring", label.scale=FALSE, labels=names(perfume)[2:48], label.cex=1, node.width=.5) library(psych) # Purchase Intent Not Included scree(perfume[,3:48]) omega(perfume[,3:48], nfactors=6)