Correspondence analysis

Lecture 15 · room 204

Prof. Niccolò Salvini

HEMA · Università Cattolica del Sacro Cuore, Rome

Tuesday 17 November 2026

Last time

Lecture 14 · Principal components · Mon 16 Nov

  • A component is the direction of greatest variance — loadings say what it is made of, scores where each row sits
  • Scale again: correlation, not covariance, when the units differ; stop where the scree flattens
  • Naming a component is your job, not the algorithm’s

Today

Time Min
10:00 Last time · Set 12, the In class questions 15’
10:15 From a table to distances: chi-square 45’
11:00 Reading the map — and what not to read 30’
11:30 Break 15’
11:45 Lab: from a survey file to the map 65’
12:50 Set 13 10’

Objectives

  • Take a contingency table and turn it into a map
  • Understand what the map shows — and the one thing it does not
  • Build a brand perception map from survey data, and read it aloud
  • Lab: from a raw survey file to the map, in the second half

Why a manager cares

“Where does our brand sit in the customer’s head?”

  • Ask 1 200 people which attributes they associate with each of six brands: reliable, cheap, innovative, premium, for young people, …
  • You get a table: brands in rows, attributes in columns, counts in the cells
  • The question is not which attribute is most common — it is which brands are close to which attributes, and to each other

Correspondence analysis draws that table as a picture. It is PCA for counts, and it is the standard tool of brand positioning.

Section 1 From a table to distances

The table, and what “close” should mean

Reliable Cheap Innovative Premium
Brand A 320 40 210 180
Brand B 90 410 60 20
Brand C 150 60 380 150
Brand D 260 30 110 350

Two brands are similar if their row profiles — the counts divided by the row total — are similar. Not the raw counts: B was mentioned more often than D, and that alone means nothing.

A: (.43, .05, .28, .24). D: (.35, .04, .15, .47). Similar on reliable, apart on premium.

The chi-square distance

Between two row profiles r_i and r_{i'}:

d^2(i, i') \;=\; \sum_{j} \frac{(r_{ij} - r_{i'j})^2}{c_j}

where c_j is the column’s overall share.

  • Ordinary Euclidean distance between profiles, weighted by 1/c_j: a difference on a rare attribute counts more than the same difference on a common one
  • Being associated with innovative when few brands are is informative. Being associated with reliable when every brand is, is not.

Note

Total inertia — the table’s whole spread — is \chi^2 / n from the independence test. CA decomposes exactly the departure from independence.

Then PCA, on those distances

  • Rows placed in a space where distances are chi-square distances
  • Find the directions of greatest spread — the axes, as in PCA
  • Do the same for columns
  • Draw both on the first two axes: the map

Rows near each other: similar profiles. Columns near each other: attributes that go together. That much is exact.

Section 2 Reading the map

What the map shows

  • Brand–brand distance: real. Two brands close together are perceived alike.
  • Attribute–attribute distance: real. Cheap and for students near each other are associated in people’s minds.
  • Direction from the origin: a brand out along the premium side is over-associated with premium relative to average.
  • The origin is the average profile — a brand at the centre is undifferentiated, which is usually the worst place to be.

What the map does not show

The misreading in every third marketing deck

  • Brand-to-attribute distance is not a distance. Rows and columns live in different spaces, overlaid for convenience. Brand A “next to” innovative is suggestive, not measured.
  • The honest version: look at the angle from the origin. A brand and an attribute in the same direction are associated; at 90°, unrelated; opposite, anti-associated.
  • Two axes may explain 60% of inertia. The other 40% is not on the page.

Attention

Never present a CA map without the percentage of inertia on each axis. A map that explains 35% of the table is a sketch, and should be called one.

In R

library(FactoMineR); library(factoextra)

tab <- as.table(rbind(A = c(320, 40, 210, 180), B = c(90, 410, 60, 20),
                      C = c(150, 60, 380, 150), D = c(260, 30, 110, 350)))
dimnames(tab)[[2]] <- c("Reliable", "Cheap", "Innovative", "Premium")

chisq.test(tab)                    # is there anything to map at all?

ca <- CA(tab, graph = FALSE)

fviz_ca_biplot(ca, repel = TRUE)   # the perception map

ca$eig gives inertia per axis. Put the two percentages on the plot’s axis labels before anyone sees it.

Reading our four brands

  • Axis 1 separates Cheap (B, alone) from everything else — the value/premium divide. It carries most of the inertia.
  • Axis 2 separates Innovative (C) from Premium (D) — the two ways of not being cheap
  • A sits between C and D and near Reliable: solid, differentiated on nothing in particular
  • B owns cheap and nothing else — which is a position, and a fragile one

Those four sentences are the analysis. The map exists so a marketing director can check them in ten seconds.

Section 3 Lab

From a survey file to the map

brands.csv on Blackboard: one row per respondent, columns brand and attribute — each respondent named up to three.

library(tidyverse); library(FactoMineR); library(factoextra); library(here)
resp <- read_csv(here("data", "brands.csv"))

tab <- table(resp$brand, resp$attribute)       # the contingency table

chisq.test(tab)$p.value                        # < 0.05 or stop here

ca <- CA(tab, graph = FALSE); round(ca$eig[1:3, 2], 1)
fviz_ca_biplot(ca, repel = TRUE) + labs(title = "Brand perception, 1 200 respondents")

Then, in pairs: write the four sentences for this map. Swap with the pair next to you. Where do you disagree, and is it the map or the reading?

A perception map does not tell you where to go. It tells you where you are, and who is standing next to you.

Practice: Set 13

Correspondence analysis · about 40 minutes · before Mon 23 Nov

  1. Do it in your own script, with the seed the set gives you.
  2. Check your numbers on Blackboard: test Set 13. Feedback the moment you submit — as many attempts as you like.
  3. Bring the sentences. Questions marked In class have no number to check: we discuss them on Mon 23 Nov, at the start of lecture 16.

Not graded. The test is a mirror, not an exam.

Reading

  • ISLR has no CA chapter. Read Greenacre, Correspondence Analysis in Practice, chapters 1–4 — on Blackboard, and the standard reference
  • Provost & Fawcett chapter 6 — positioning as a similarity problem
  • Reference: ?FactoMineR::CA, factoextra::fviz_ca_biplot

Statistics & Big Data · HEMA, Università Cattolica del Sacro Cuore