Analyse des Correspondence Multiples (ACM) Multiple Correspondence Analysis (MCA) Hervé Abdi. Refresher! and Menu MCA... MCA... Home Page.

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1 u u Analyse des Correspondence Multiples (ACM) Multiple Correspondence Analysis (MCA) Hervé Abdi herve herve@utdallas.edu Page of 64

2 . u u Some more eamples of simple CA What to do with more than nominal variables? Answer: Multiple Correspondence Analysis. Back to punctuation... Another look at the individual / patterns MCA for nominal variables and CA! The eigenvalue problem Correction formula... More than variables MCA and ordinal data (ranks) Page of 64

3 . u u Science Diplomas in the US from 966 to 000 Page 3 of 64

4 Table : Correspondence Analysis. All Diplomas in the US ( ). (Bachelor, Master, Ph.D., Men and Women) The Data u u Engineer Physics Earth Math Biology Psychology Social NonSc Page 4 of 64

5 PHYSICS λ =.009 τ =3% u u SOCIAL SCIENCES EARTH Non Sciences BIOLOGY PSYCHOLOGY ENGINEERING MATHEMATICS Figure : Diplomas in the US: Correspondence Analysis. λ =.0044 τ = 48% Page 5 of 64

6 PHYSICS λ =.009 τ =3% u u SOCIAL SCIENCES EARTH Non Sciences BIOLOGY PSYCHOLOGY ENGINEERING MATHEMATICS λ =.0044 τ = 48% Figure : Diplomas in the US: Correspondence Analysis (with frills). Page 6 of 64

7 u u PHYSICS λ =.009 τ =3% MEN SOCIAL SCIENCES 970 ENGINEERING 97 EARTH Non Sciences BIOLOGY PSYCHOLOGY MATHEMATICS λ =.0044 τ = 48% Page 7 of 64 WOMEN

8 u u PHYSICS λ =.009 τ =3% SOCIAL SCIENCES 970 ENGINEERING 97 EARTH MATHEMATICS λ 975 Non Sciences = τ = 48% BIOLOGY PSYCHOLOGY Page 8 of 64 Figure 4: Diplomas in the US: Correspondence Analyss. By Diploma (Green: Bachelor. Red: Master. Gold: Ph.D.).

9 u u Table : Correspondence Analysis. Diploma Global Analysis. J-set (Disciplines). The Projections, Contributions, Squared distance to barycenter, and Cosines F F ctr ctr m d cos cos Qual Engineer Physics Earth Math Biology Psychology Social NonSc Page 9 of 64

10 Table 3: Correspondence Analysis. Diploma Global Analysis. I-set (Disciplines). The Projections, Contributions, Squared distance to barycenter, and Cosines u u 0 3 F F ctr ctr m d cos cos Qual Page 0 of 64

11 The Causes of Death in the USA in 00. u u Table 4: The Causes of Death in the USA (00) Septicemia Hepatitis hiv Cancer tumor Diabetes Parkinson Alheimer Heart Pneumonia Lower Respiratory Liver Kidney Unknown Other Diseases Accident Car Accident NonCar Other Accidents Suicide Homicide Page of 64

12 First have a look at the data (log transformed): u u Color Key 00 USA. Causes of death (log) Value Septicemia Hepatitis HIV Cancer tumor Diabetes Parkinson/Alheimer Heart Pneumonia Lower Respiratory Liver Kidney Unknown Other Diseases Accident Car Accident NonCar Other Accident Page of 64 Suicide Homicide Figure 5: Causes of Death in the US: A picture of the data (log transformed).

13 Correspondence analysis: The Scree. u u Page 3 of 64

14 Eplained Variance per Dimension u u Percentage of Eplained Variance Page 4 of Dimensions Figure 6: Causes of Death in the US: The Scree (How many dimensions?).

15 A (symmetric) plot u u Hepatitis λ τ =.083 =5% λ =.373 τ = 67% 5 34 Homicide 5 4 Accident Car Accident Non Car HIV Suicide Liver Cancer Diabetes Lower Respiratory Septicemia Kidney Other Accidents Other Diseases 85+ Pneumonia Unknown Parkinson/Alheimer < Page 5 of 64 Figure 7: Causes of Death in the US: Correspondence Analysis. Symmetric plot

16 A (symmetric) plot u u Hepatitis λ τ =.083 =5% λ =.373 τ = 67% 5 34 Homicide 5 4 Accident Car Accident Non Car HIV Suicide Liver Cancer Diabetes Lower Respiratory Septicemia Kidney Other Accidents Other Diseases 85+ Pneumonia Unknown Parkinson/Alheimer < Page 6 of 64 Figure 8: Causes of Death in the US: Correspondence Analysis. Symmetric plot with frills. There are three periods in life (or death!).

17 An asymmetric plot from R Hepatitis HIV Liver Cancer Suicide Diabetes Other Diseases Tumor Other Accident Kidney Lower Respiratory Accident Car Accident NonCar Heart Septicemia Homicide Pneumonia Unknown Parkinson/Alheimer 85+ u u Page 7 of 64 Figure 9: Causes of Death in the US: Correspondence Analysis. Asymmetric plot (From R). There are three periods in life (or death!).

18 An asymmetric plot (with frills) u u Hepatitis HIV Liver Cancer Suicide Diabetes Other Diseases Tumor Other Accident Kidney Lower Respiratory Accident Car Accident NonCar Heart Septicemia Homicide Pneumonia Unknown Parkinson/Alheimer Page 8 of 64 Figure 0: Causes of Death in the US: Correspondence Analysis. Asymmetric plot with frills (From R). There are three periods in life (or death!)

19 u u F F ctr ctr m d cos cos Qual Table 5: Correspondence Analysis. Causes of Death in the US. Global Analysis. J-set (Disciplines). The Projections, Contributions, Squared Distances to Barycenter, and (squared) Cosines. Page 9 of 64

20 F F ctr ctr m d cos cos Qual Septicemia Hepatitis HIV Cancer Tumor Diabetes Parkinson/Alheimer Heart Pneumonia Lower Respiratory Liver Kidney Unknown Other Diseases Car Accidents Non-car Accidents Other Accidents Suicide Homicide u u Page 0 of 64 Table 6: Correspondence Analysis. Cause of Death in the US Global Analysis. I-set (Causes). The Projections, Contributions, Squared distance to barycenter, and Cosines

21 3. u u Page of 64

22 u u.,!?... R C H Z P G mark #. Period Rousseau marks later mark # Comma Rousseau marks later last mark: Other Giraudou Page of 64 Table 7: Multiple Correspondence Analysis, Punctuation eample. The table of patterns.

23 .,!?... R C H Z P G P-R P-C P-H P-Z P-P P-G C-R C-C C-H C-Z C-P C-G O-R O-C O-H O-Z O-P O-G Page 3 of 64 u u Table 8: Multiple Correspondence Analysis, Punctuation eample. The table of patterns.

24 The magic of the distributional equivalence!.,!?... R C H Z P G P-R P-C P-H P-Z P-P P-G C-R C-C C-H C-Z C-P C-G O-R O-C O-H O-Z O-P O-G u u Page 4 of 64 Table 9: Multiple Correspondence Analysis, Punctuation eample. The table of patterns.

25 u u Table 0: Multiple Correspondence Analysis. Punctuation eample. The Projections, Contributions, Squared Distances to Barycenter, and (Squared) Cosines. F F Ctr Ctr m d cos cos Qual PERIOD COMMA OTHER Rousseau Chateaubriand Hugo Zola Proust Giraudou Page 5 of 64

26 MCA Punctuation! Chateaubriand OTHER Rousseau Hugo COMMA Zola Proust u u PERIOD Page 6 of 64 Giraudou Figure : Multiple Correspondence Analysis: Punctuation

27 MCA Punctuation! With Frills Chateaubriand OTHER Rousseau Hugo COMMA Zola Proust u u PERIOD Page 7 of 64 Giraudou Figure : Multiple Correspondence Analysis: Punctuation

28 Compare with CA Punctuation! u u Chateaubriand Rousseau OTHER COMMA Proust Hugo PERIOD Zola Giraudou Page 8 of 64 Figure 3: Punctuation: Correspondence Analysis

29 MCA Punctuation! With Frills u u Chateaubriand Rousseau COMMA Proust OTHER Hugo PERIOD Zola Giraudou Figure 4: Punctuation Correspondence Analysis Page 9 of 64

30 The Eigenvalue Problem: Fictitious Eigenvalues (the hypercube problem) Compare u u λ λ τ τ 73 4 With M λ M λ M λ 3 M λ 4 M λ 5 M λ 6 M λ τ τ τ 3 τ 4 τ 5 τ 6 τ Page 30 of 64 The Factors are the same but not the eigenvalues! in MCA, the M λ /K code the hypercube!

31 Correcting for The Eigenvalue Problem: Keep only the M λ > ( ) λ = 4 Mλ u u Eample λ = 4 ( Mλ ) = 4 (0.0667) = =.078 Page 3 of 64

32 And Vice-Versa u u Mλ = (λ + ) Page 3 of 64

33 u u The Burt Table Table : Multiple Correspondence Analysis, Punctuation eample. The Burt Table..,!?... R C H Z P G , !? R C H Z P G Page 33 of 64

34 u u The Burt Table Table : Multiple Correspondence Analysis, Punctuation eample. The Burt Table..,!?... R C H Z P G , !? R C H Z P G Page 34 of 64

35 Add something on the Burt Table and the average inertia (From Greenacre 007) u u Page 35 of 64

36 u u Table 3: Multiple Correspondence Analysis. Burt Table Analysis. Punctuation eample. The Projections, Contributions, Squared distance to barycenter, and Cosines F F ctr ctr m d cos cos Qual , R C H Z P G Page 36 of 64

37 Compare u u λ λ τ τ 73 4 With M λ M λ M λ 3 M λ 4 M λ 5 M λ 6 M λ τ τ τ 3 τ 4 τ 5 τ 6 τ With Page 37 of 64 Bλ B λ B λ 3 B λ 4 B λ 5 B λ 6 B λ τ τ τ 3 τ 4 τ 5 τ 6 τ Bλ = M λ M λ = B λ

38 u u CA MCA Burt Normalied F F F F F F F F , , R C H Z P G Table 4: Comparison between three ways of analying the same contingency table. Correspondence analysis Multiple correspondence analysis, and Burt Table Analysis. Punctuation eample. The last two rows are normalied to. They can be obtained by normaliation of any of the analyses. The normaliation is obtained by dividing each entry of a column by the square root of its eigenvalue. Page 38 of 64

39 A Small Eample u u Effect of Oak Species on Pinot Noirs Aged 3 Pinot with Oak Type I and 6 with Oak Type II. Three Assessors choose their own variables Answers coded as binary or ternary (0/) Page 39 of 64

40 Table : Data for the barrel-aged red burgundy wines eample. Oak Type" is an illustrative (supplementary) variable, The wine W? is an unknown wine treated as a supplementary observation. Epert Epert Epert 3 Oak red Wine Type fruity woody coffee fruit roasted vanillin woody fruity butter woody W W W W W W W?? Figure 5: Wines. The Data.. Valentin: Multiple Correspondence Analysis u u h \paperheight: pt (40mm) Page 40 of 64 -MCA007-pretty.te - page 4 \paperwidth: pt (60mm)

41 corrections for the eigenvalues for MCA u u First: Benécri: Keep eigenvalue K λ l : eigenvalues from the analysis indicator matri. cλ l corrected eigenvalues [( ) ( K λ l )] if λ l > K K K cλ l = 0 if λ l K Over-correct (but not much)!. () Page 4 of 64

42 corrections for the eigenvalues for MCA u u Greenacre: Keep Benécri s correction: Correct the Inertia λ l : eigenvalues from indicator matri cλ l corrected eigenvalues Use average inertia, denoted I: I = K K Percentage inertia: τ c = c λ I ( l instead of cλ λ l J K K ) () cλ l. (3) Page 4 of 64

43 Table 5: Eigenvalues, corrected eigenvalues, proportion of eplained inertia and corrected proportion of eplained inertia. The eigenvalues of the Burt matri are equal to the squared eigenvalues of the indicator matri; The corrected eigenvalues for Benécri and Greenacre are the same, but the proportion of eplained variance differ. Eigenvalues are denoted by λ, proportions of eplained inertia by τ (note that the average inertia used to compute Greenacre s correction is equal to I =.7358). u u Indicator Burt Benécri Greenacre Matri Matri Correction Correction Factor Iλ τ I B λ τ B Z λ τ Z cλ τ c Page 43 of 64

44 u u Table 6: Factor scores, squared cosines, and contributions for the observations (I-set). The eigenvalues and proportions of eplained inertia are corrected using Benécri/Greenacre formula. Contributions corresponding to negative scores are in italic. The mystery wine (Wine?) is a supplementary observation. Only the first two factors are reported. Wine Wine Wine 3 Wine 4 Wine 5 Wine 6 Wine? F cλ % c Factor Scores F Squared Cosines F Contributions Page 44 of 64

45 u u 4 3 λ =.03 τ = % 5? 6 λ =.7004 τ = 95% Page 45 of 64 Figure 6: Wines. Map of the wines.

46 b Figure 7: Wines. Map of the Variables (Assessors Scales). tions on the rst dimensions. The eigenvalues (λ) and ed with Benécri/Greenacre formula. (a) The I set: rows The J set: columns (i.e., adjectives). Oak and Oak are been slightly moved to increase readability). (Projections H. Abdi & D. Valentin: Multiple Correspondence An u u Page 46 of 64 EX document: Abdi-MCA007-pretty.te - page 0 gust, 009-5h \paperhe \paperw

47 TAKS in TEXAS u u A Bigggggggggggg Analysis: Effect of Correction on Big Data Eemple: Math Test: 5th Grade (all of DISD). (44 Questions 4 answers each = 7 Columns). Page 47 of 64

48 u u Principal Component Number MCA Taks Grade 5 o o o o o o o o o o o o o o o o o o o o o o o o * o o o o o o o o o o * * * o o o * * o o * * o o o * o o o * * o o o o o o o * o o o * o o o o oo o o * * o * * ** o o o o o o o o o Principal Component Number Page 48 of 64

49 Correcting for The Eigenvalue Problem: With K Variables: Keep only the M λ > K λ = Mλ =.884 ( ) ( K Mλ ) K K Eample: TAKS ( ) ( 44 λ =.884 ) =.087 (Goes from 7% to 96%!) u u Page 49 of 64

50 Ordinal Data u u Five judges Each judge rates five beer A Beer eample On a 6-point rating scale (0 to 5) Page 50 of 64

51 u u Cain Sims Guy Kinchen Schalbs Miller 3 RollRock Lowenbrau Coors Budweiser Table 7: Correspondence Analysis. Scoring Beers. The Original data. Page 5 of 64

52 Thermometer Coding: Use both sides of the scale First Idea: Use the ratings Not a very good idea! Use Thermometer Coding. u u Page 5 of 64

53 Thermometer Coding: The Leverage Principle u u Page 53 of 64

54 u u Cain Sims Guy Kinchen Schalbs Miller RollRock Lowenbrau Coors Budweiser Table 8: Correspondence Analysis. Scoring Beers. The data with thermometer coding. Page 54 of 64

55 u u 0. Guy $ $ Beer Ranks I and J Cain $ $ Coors Principal Component Number Sims $ $ Miller Schalbs $ $ Budweiser Kinchen $ $ RollRock Kinchen $+$ Schalbs $+$ Cain $+$ Guy $+$ Sims $+$ Lowenbrau Principal Component Number Figure 8: Ranking Beers: Correspondence Analysis. The Ugly plot: no frill at all! (Don t show that ecept to your close friends) Page 55 of 64

56 u u Coors Miller Budweiser RollRock Lowenbrau Page 56 of 64 Figure 9: Ranking Beers: Correspondence Analysis (no frill, but prettier). The Beer Plot

57 u u Guy Cain Schalbs Coors Sims Sims Miller Kinchen Budweiser Kinchen RollRock Schalbs Cain Guy Lowenbrau Figure 0: Ranking Beers: Correspondence Analysis. Beers and Raters (+ and -) Page 57 of 64

58 u u Guy Cain Schalbs Coors Sims Sims Miller Kinchen Budweiser Kinchen RollRock Schalbs Cain Guy Lowenbrau Figure : Ranking Beers: Correspondence Analysis. Beers and Raters (+ and -). Raters are lines Page 58 of 64

59 u u Guy Cain Schalbs Coors Sims Sims Miller Kinchen Budweiser Kinchen RollRock Schalbs Cain Guy Lowenbrau Figure : Ranking Beers: Correspondence Analysis. Projecting a Beer on a Rater. Page 59 of 64

60 u u Guy Cain Schalbs Coors Sims Sims Miller Kinchen Budweiser Kinchen RollRock Schalbs Cain Guy Lowenbrau Figure 3: Ranking Beers: Correspondence Analysis. Projecting a Beer on another Rater. Page 60 of 64

61 u u Guy Miller Cain Schalbs Coors Kinchen Sims Kinchen RollRock Lowenbrau Sims Budweiser Schalbs Cain Guy Page 6 of 64 Figure 4: Ranking Beers: Correspondence Analysis. More projections.

62 u u Guy Cain Schalbs Coors Sims Sims Miller Kinchen Budweiser Kinchen RollRock Schalbs Cain Guy Lowenbrau Figure 5: Ranking Beers: Correspondence Analysis. Even ore projections Page 6 of 64

63 u u F F ctr ctr m d cos cos Qual Cain Cain Sims Sims Guy Guy Kinchen Kinchen Schalbs Schalbs Page 63 of 64 Table 9: Correspondence Analysis. Beers. J-set (Raters). The Projections, Contributions, Squared distance to barycenter, and (Squared) Cosines

64 u u F F ctr ctr m d cos cos Qual Miller RollRock Lowenbrau Coors Budweiser Table 0: Correspondence Analysis. Beers: The I-set (Beers). The Projections, Contributions, Squared distance to barycenter, and Cosines Page 64 of 64