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MULTIPLE CORRESPONDENCE

ANALYSIS (MCA)

• A New Platform for Categorical Variables, Ready to be

explored

• Jianfeng Ding

• JMP R&D

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COVERED TOPICS

• What is CA and MCA

• Who uses MCA

• How to use MCA (Live Demos of JMP)

• Conclusions

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WHAT IS CORRESPONDENCE

ANALYSIS(CA)

• Scenario: new sales person who has the car sale information from

the last quarter

• Challenge: how to sell cars efficiently

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WHAT IS CORRESPONDENCE

ANALYSIS(CA)

• Look at two categorical variables each time

• Determine the association between two categorical variables

SVD

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WHAT IS MULTIPLE CORRESPONDENCE

ANALYSIS (MCA)

• Look at multiple categorical variables at the same time

• Determine the association among all categorical variables

Design matrix

SV

D

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WHAT IS MULTIPLE CORRESPONDENCE

ANALYSIS (MCA)

Only in JMP12 Available in JMP11

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ALGORITHM AND NOTATION OF CA

• Two design matrices Z1, Z2

• Scale vectors s1, s2

• Canonical correlation s12=(1/n) st1z

t1z2s2=st

1p12s2

• Find the scale vectors that maximize the correlation

• Do SVD to the matrix to reveal the association

structure

• S= 𝐷−1

2𝑟(𝑃 − 𝑟𝑐𝑇)𝐷

−1

2𝑐

=UΣVT

• Row coordinates and column coordinates

• S1=𝐷−1

2𝑟

U Σ s2=𝐷−1

2𝑐

V Σ

• Trace(sst) is the total inertia that quantifies the total

variances in the cross-table.

Element of

correspondence

matrix pij

Row masses ri

Column masses cj

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ALGORITHM AND NOTATION OF MCA

• Two ways to perform MCA

• CA of the design matrix Z = [ Z1 Z2,…,ZQ ], do SVD

to

• 𝑛𝑍

𝑄𝑛−1

𝑛11𝑇𝐷 𝐷−1/2

• Ca of the burt matrix c = ztz, do svd to

• 𝐷−1/2𝐶

𝑄2𝑛− 𝐷11𝑇𝐷 𝐷−1/2

• Where D=(1/Q)diag(D1 D2,…,Dq )

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WHO USES MCA?

• MCA is popular in Europe and Japan

• Social science research

• Health in psychology

• Market data, product preferences, and food preferences

• Elections in political science and credit scoring

• Epidemiology of animal health and microarray studies in genetics

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STEPS OF ANALYZING DATA WITH MCA

• Preparation of the data for MCA

• Elementary statistical analysis

• Examination of the basic results such as eigenvalues(inertia), coordinates,

contributions, and clouds

• Informal inspection of the clouds in several principal planes

• Interpretation of axes

• Inspecting and “dressing up” the cloud of individuals

• Supplementary elements: individuals and variables

• Deep investigation of the cloud of individuals

• Statistical inference

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CLOUDS OF CATEGORIES AND INDIVIDUALS

𝑦𝑖 =1

λ

𝑘∈𝐾𝑖

𝑦𝑘 𝑄

𝑦𝑘 =1

λ

𝑖∈𝐼𝑘

𝑦𝑖 𝑛𝑘

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CONCLUSIONS

• Support CA and MCA

• Support supplementary elements

• Visualizing the output statistics

• The interactivity makes it easy and fun to analyze and interpret data