Multiple correspondence analysis

Key takeaways

Multiple correspondence analysis belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Multiple correspondence analysis to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Multiple correspondence analysis from memory before moving on to harder problems.

Reference excerpt

In statistics, multiple correspondence analysis (MCA) is a data analysis technique for nominal categorical data, used to detect and represent underlying structures in a data set. It does this by representing data as points in a low-dimensional Euclidean space. The procedure thus appears to be the counterpart of principal component analysis for categorical data. MCA can be viewed as an extension of simple correspondence analysis (CA) in that it is applicable to a large set of categorical variables.

As an extension of correspondence analysis MCA is performed by applying the CA algorithm to either an indicator matrix (also called complete disjunctive table – CDT) or a Burt table formed from these variables. An indicator matrix is an individuals × variables matrix, where the rows represent individuals and the columns are dummy variables representing categories of the variables. Analyzing the indicator matrix allows the direct representation of individuals as points in geometric space. The Burt table is the symmetric matrix of all two-way cross-tabulations between the categorical variables, and has an analogy to the covariance matrix of continuous variables. Analyzing the Burt table is a more natural generalization of simple correspondence analysis, and individuals or the means of groups of individuals can be added as supplementary points to the graphical display. In the indicator matrix approach, associations between variables are uncovered by calculating the chi-square distance between different categories of the variables and between the individuals (or respondents). These associations are then represented graphically as "maps", which eases the interpretation of the structures in the data. Oppositions between rows and columns are then maximized, in order to uncover the underlying dimensions best able to describe the central oppositions in the data. As in factor analysis or principal component analysis, the first axis is the most important dimension, the second axis the second most important, and so on, in terms of the amount of variance accounted for. The number of axes to be retained for analysis is determined by calculating modified eigenvalues.

Details Since MCA is adapted to draw statistical conclusions from categorical variables (such as multiple choice questions), the first thing one needs to do is to transform quantitative data (such as age, size, weight, day time, etc) into categories (using for instance statistical quantiles). When the dataset is completely represented as categorical variables, one is able to build the corresponding so-called complete disjunctive table. We denote this table X {\displaystyle X} . If I {\displaystyle I} persons answered a survey with J {\displaystyle J} multiple choices questions with 4 answers each, X {\displaystyle X} will have I {\displaystyle I} rows and 4 J {\displaystyle 4J} columns. More theoretically, assume X {\displaystyle X} is the completely disjunctive table of I {\displaystyle I} observations of K {\displaystyle K} categorical variables. Assume also that the k {\displaystyle k} -th variable have J k {\displaystyle J_{k}} different levels (categories) and set J = ∑ k = 1 K J k {\displaystyle J=\sum _{k=1}^{K}J_{k}} . The table X {\displaystyle X} is then a I × J {\displaystyle I\times J} matrix with all coefficient being 0 {\displaystyle 0} or 1 {\displaystyle 1} . Set the sum of all entries of X {\displaystyle X} to be N {\displaystyle N} and introduce Z = X / N {\displaystyle Z=X/N} . In an MCA, there are also two special vectors: first r {\displaystyle r} , that contains the sums along the rows of Z {\displaystyle Z} , and c {\displaystyle c} , that contains the sums along the columns of Z {\displaystyle Z} . Note D r = diag ( r ) {\displaystyle D_{r}={\text{diag}}(r)} and D c = diag ( c ) {\displaystyle D_{c}={\text{diag}}(c)} , the diagonal matrices containing r {\displaystyle r} and c {\displaystyle c} respectively as diagonal. With these notations, computing an MCA consists essentially in the singular value decomposition of the matrix:

M = D r − 1 / 2 ( Z − r c T ) D c − 1 / 2 {\displaystyle M=D_{r}^{-1/2}(Z-rc^{T})D_{c}^{-1/2}}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Multiple correspondence analysis

Start with the simplest possible case. Write down what Multiple correspondence analysis claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Multiple correspondence analysis before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Multiple correspondence analysis ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Multiple correspondence analysis

In research: Multiple correspondence analysis appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Multiple correspondence analysis in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Multiple correspondence analysis is common in secondary-school and first-year university syllabi. It links to neighbouring topics Dimension reduction, so understanding it makes those chapters shorter.
In everyday life: Look for Multiple correspondence analysis outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Multiple correspondence analysis in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Multiple correspondence analysis means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Multiple correspondence analysis out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Multiple correspondence analysis in simple terms?

In statistics, multiple correspondence analysis (MCA) is a data analysis technique for nominal categorical data, used to detect and represent underlying structures in a data set. It does this by representing data as points in a low-dimensional Euclidean space.

Why does Multiple correspondence analysis matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Multiple correspondence analysis?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Multiple correspondence analysis.