Load the data and the packages

With the following command

library(R4SPISE2022)
data("sixBeers12Descriptors10Judges", package = "data4PCCAR")
beers <- sixBeers12Descriptors10Judges$ratingsIntensity
beerColors <- sixBeers12Descriptors10Judges$color4Products
descColors <- sixBeers12Descriptors10Judges$color4Descriptors

The Beers

Bitter Acid Sweet Astringent Alcoholic Hop Cereal Toasted Yeast Hay Malt Fermented
Bruxelles 3.190103 2.250000 2.002791 2.486309 3.237597 3.813985 3.341958 2.858587 3.305860 2.774995 3.693717 3.314378
FranzDunkel 3.898867 2.116649 2.690365 3.419777 4.316269 4.480046 3.718364 2.958749 3.419777 2.970115 3.685813 3.505336
FranzNatur 4.277778 2.075397 3.085289 3.426917 4.407831 4.536122 4.138244 3.895192 3.496122 3.261656 3.940780 2.805860
Hofbrau 4.399007 3.087549 0.000000 3.158053 3.984467 4.674603 3.706632 3.395192 3.630654 3.616398 4.023809 3.731262
Paulaner 3.697283 2.404388 1.028142 3.326382 3.910490 4.749270 3.786054 3.144253 3.444692 3.278671 3.792303 3.281101
BlueMoon 3.200000 3.300000 4.600000 3.300000 5.200000 5.200000 2.500000 2.500000 2.100000 2.900000 4.200000 4.100000

The ratings

Do the Principal Component Analysis (PCA)

res.pca <- epPCA(beers, 
                 center = TRUE, # Center the data
                 scale = FALSE, # **Do not** scale the variables
                 graphs = FALSE)

The default of this function will center (to have means equal 0) and scale (to have the standard deviations equal 1) all variables.

Plot the results

res.plot.pca <- OTAplot(
    resPCA = res.pca,
    data = beers
    )

Correlations of the variables

We can check the data by plotting the correlation matrix between the variables. This correlation matrix is where the components are extracted.

res.plot.pca$results.graphs$correlation

Scree plot

The scree plot shows the eigenvalues of each component. These eigenvalues give the variance of each component (or called dimension in the figure). In other words, the singular values, which are the square root of the eigenvalues, give the standard deviations of these components. The sum of the eigenvalues is equal to the total inertia of the data.

res.plot.pca$results.graphs$scree

According to an elbow test, there are 4 dimensions in this data.

Factor scores (rows)

Here, the observation map shows how the observations (i.e., rows) are represented in the component space. If center = TRUE when running PCA, the origin should be at the center of all observations. (If it’s not, something is off.)

res.plot.pca$results.graphs$factorScoresI12

The result in this plot showed that the first dimension separates BlueMoon from Hofbrau, and the second dimension separates BlueMoon and Hofbrau from FranzDunkel and FranzNatur.

Circle of correlation

The circle of correlations illustrate how the variables are correlated with each other and with the dimensions. From this figure, the length of an arrow indicates how much this variable is explained by the two given dimensions. The cosine between any two arrows gives their correlation. The cosine between a variable and an axis gives the correlation between that variable and the corresponding dimension.

In this figure, an angle closer to 0° indicates a correlation close to 1; an angle closer to 180° indicates a correlation close to -1; and an 90° angle indicates 0 correlation. However, it’s worth noted that this implication of correlation might only be true within the given dimensions. When a variable is far away from the circle, it is not fully explained by the dimensions, and other dimensions might be characterized by other pattern of relationship between this and other variables.

res.plot.pca$results.graphs$cosineCircleArrowJ12

This circle of correlation plot illustrates that Dimension 1 is Sweet and Alcoholic versus Yeast, Hay, Bitter, Cereal, and Toasted; Dimension 2 is Acid and Fermented versus Cereal and Toasted. Interestingly, Acid and Fermented are orthogonal to Sweet and Hay (which are anti-correlated from each other) on the first two dimension. On these two dimensions, Cereal and Toasted are negatively correlated with Malt and Hop.

Inference plots and results

The inference analysis of PCA (performed by OTAplotInference) includes bootstrap test of the proportion of variance explained, permutation tests of the eigenvalues, the bootstrap tests of the column factor scores \(\mathbf{G}\) (i.e., the right singular vectors scaled to have the variance of each dimension equals the corresponding singular value; \(\mathbf{G} = \mathbf{Q}\boldsymbol{\Delta}\) with \(\mathbf{X = P}\boldsymbol{\Delta}\mathbf{Q}^\top\)). These column factor scores are stored as fj in the output of epPCA.

res.plot.pca.inference <- OTAplotInference(
    resPCA = res.pca,
    data = beers
    ) %>% suppressMessages()

Bootstrap and permutation tests on the eigenvalues

The inference scree plot illustrates the 95% bootstrap confidence intervals of the percentage of variance explained by each eigenvalue. If an interval includes 0, the component explains reliably larger than 0% of the variance.

res.plot.pca.inference$results.graphs$scree

The bootstrap test identifies one significant dimension explains the variance reliably larger than 0%. The permutation test identifies one significant dimension with an eigenvalue significantly larger than 0.

Bootstrap test on the column factor scores

The bar plot illustrates the bootstrap ratios which equals \[\frac{M_{g_{j}boot}}{SD_{g_{j}boot}},\] where \(M_{g_{j}boot}\) is the mean of the bootstrapped sample of the jth column factor score and \(SD_{g_{j}boot}\) is the bootstrapped standard deviation of the factor score. A bootstrap ratio is equivalent to a t-statistics for the column factor score with the \(\mathrm{H}_0: g_j = 0\). The threshold is set to 2 to approximate the critical t-value of 1.96 at \(\alpha\) = .05.

res.plot.pca.inference$results.graphs$BR1

res.plot.pca.inference$results.graphs$BR2

The results show no significant factor scores for both dimensions that is reliably different from 0.