In short
Panel data are a table: samples in rows, descriptors in columns. Regression asks how one column predicts another; principal component analysis (PCA) finds the few directions along which the samples differ most, so a dozen descriptors can be seen as a map. Both are easy to compute and easy to over-read. With few samples, unstandardized scales or an untrained panel, the tidiest map can describe the panel's habits rather than the fragrances.
What you will learn
- Compute a covariance matrix, its eigenvalues and the first principal component for two descriptors
- Interpret loadings, scores and the share of variance a component explains
- Fit a least-squares regression and state what its correlation does not establish
- Recognize the conditions under which a sensory map misleads: few samples, unstandardized scales, panel effects
Before this lesson: Discrimination Tests and Thurstonian d′, Mixture Designs for Accords, A Framework for Structured Critique
Before you start
This assumes the statistics of Thurstonian d′, the panel work of mixture designs and a structured descriptor vocabulary from the critique framework.
A worked two-descriptor case
Five fragrances, panel means on "woody" (x) and "smoky" (y), invented for the arithmetic: x = 2, 4, 6, 8, 10 and y = 3, 4, 7, 7, 9.
- Center. Both means are 6.
- Covariance matrix (dividing by n − 1 = 4): var(x) = 10, var(y) = 6, cov(x, y) = 7.5.
- Eigenvalues of [[10, 7.5], [7.5, 6]]: λ = 8 ± √(4 + 56.25), so 15.76 and 0.24.
- First component. It explains 15.76/16 = 98.5% of the variance; its loadings are (0.793, 0.609). Woody and smoky rise together, woody slightly more.
Each sample's score on PC1 is 0.793(x − 6) + 0.609(y − 6). Plotted, the five samples fall almost on a line: in this set, one "woody-smoky intensity" dimension describes them.
Regression. The least-squares line of y on x has slope cov/var(x) = 0.75 and intercept 6 − 0.75 × 6 = 1.5; the correlation is 7.5/√(10 × 6) = 0.968. A strong correlation in five samples you chose says the descriptors moved together in those samples. It does not say woody materials produce smokiness, and a sixth sample — a clean cedar — could break it.
More descriptors
With p descriptors, PCA diagonalizes the p × p covariance (or correlation) matrix. In practice you use software, and the decisions that matter are yours:
- Covariance or correlation. PCA on correlations is PCA on standardized descriptors. Unless every descriptor was rated on the same scale with comparable use, standardize.
- How many components. Report the share each explains. Look for a clear drop, and do not interpret components that explain little.
- Samples versus descriptors. With six samples, at most five components can carry information, whatever the number of descriptors. Many descriptors and few samples guarantee tidy-looking but unstable maps.
Where it misleads
- Panel effects. Assessors use scales differently. Analyze panel means only after checking agreement, and look at assessor-by-sample interaction.
- Halo and dumping. If the scoresheet lacks a descriptor for what people smell, they put it in a nearby one, which manufactures correlations.
- Selection. The map reflects the samples chosen. Six woody fragrances give a woody map.
- PCA finds variance, not meaning. A component is a mathematical direction; naming it is interpretation, to be checked against the loadings.
In professional practice
Descriptive analysis with trained panels and multivariate maps is how evaluation departments summarize where a submission sits relative to competitors and to a brief. The perfumer's question is usually narrower: did my modification move the sample in the direction the brief asked, and by more than the panel's noise?
Open questions
How stable are fragrance sensory maps across panels, cultures and time after application? Published replication across panels is limited, and it bears directly on whether a map built in one market should guide development for another.
Practical exercise: Map six fragrances from your own panel
- Choose six fragrances or accords that differ along at least two directions you can name.
- Define five descriptors with reference smells, and have at least five people rate each coded sample on the same 0–9 scale, in a randomized order, at one stated time.
- Check agreement: for each descriptor, compare assessors' rankings of the samples.
- Run a PCA on the standardized panel means in a spreadsheet or Python. Report each component's share of variance and the loadings of the first two.
- Plot the scores and write what the map shows, then what it cannot show given six samples and your panel.
You have done this well if descriptors were defined before rating, samples were coded and randomized, you standardized and said why, you interpret only components with a meaningful share, and your limitations name the sample count and panel agreement explicitly.
Common mistakes
- Interpreting the third component of a six-sample map. It is mostly noise.
- Naming a component without reading its loadings.
- Treating correlation as mechanism.
Safety
Samples are smelled on strips. Brief assessors as in the ethics lesson, and record no personal data you do not need.
Key terms
- Principal component
- A direction in descriptor space along which the samples vary most, uncorrelated with the components before it. The first explains the largest share of variance.
- Loading
- The weight of each original descriptor in a principal component. Large loadings of the same sign mean the descriptors rise together.
- Score
- A sample's coordinate on a principal component: its centered descriptor values multiplied by the loadings.
- Standardization
- Rescaling each descriptor to unit variance before PCA, so that a descriptor rated on a wider range does not dominate merely because of its scale.
Knowledge check
Written and reviewed by the RUŌOD Lab team. This is general education, not legal, regulatory, medical or safety advice. Example dilutions and exercises are for learning, are not validated products, and have not been safety-assessed. Completing a lesson confers no qualification. How we check these lessons.