# Correspondence Analysis Multivariate Chi Square Goals of CA

• Slides: 17

Correspondence Analysis Multivariate Chi Square

Goals of CA • Produce a picture of multivariate data in one or two dimensions • Analyze rows and columns simultaneously • Plot both on a single scale • Often shows chronological ordering

Data • Counts or presence/absence for a series of cases or observations (rows) by a number of variables (columns) • Composition data: assemblage, pollen, botanical, faunal, trace elements, etc

Dimensions • CA works by extracting orthogonal dimensions from the data table (similarly to principal components) • Typically one or 2 dimensions are extracted but the maximum number of dimensions is min[(rows-1), (columns 1)]

Plotting • CA produces coordinates for each dimension for each row and column in the original data • On the plot, the distance between two row points or two column points reflects their similarity or difference • Row points help to understand the patterns of column points and vice versa

N. C. Nelson. 1916. Chronology of the Tano Ruins, New Mexico. American Anthropologist 18(2): 159 -180. > round(prop. table(as. matrix(Nelson[, 2: 8]), 1)*100, 2) Corrugated Biscuit Type_II_Red Type_II_Yellow Type_II_Gray Type_III 1 36. 77 6. 45 1. 29 15. 48 14. 84 21. 94 3. 23 2 31. 27 4. 58 0. 54 17. 25 24. 26 20. 49 1. 62 3 15. 34 1. 14 5. 68 38. 64 10. 23 27. 27 1. 70 4 21. 37 3. 05 4. 58 39. 69 15. 27 16. 03 0. 00 5 17. 39 4. 35 0. 58 37. 10 15. 94 24. 64 0. 00 6 18. 66 5. 22 1. 99 47. 76 13. 18 12. 94 0. 25 7 23. 14 4. 37 17. 47 39. 74 8. 73 6. 55 0. 00 8 24. 67 0. 88 51. 98 19. 82 0. 44 2. 20 0. 00 9 45. 59 0. 49 52. 45 1. 47 0. 00 10 54. 55 0. 65 44. 81 0. 00

> Ca. Model. 1 <- corresp(Nelson[, 2: 8], nf=2) > Ca. Model. 1 First canonical correlation(s): 0. 6597448 0. 2920078 Row scores: [, 1] 1 0. 46210940 2 0. 60419349 3 0. 61729088 4 0. 53546269 5 0. 79817759 6 0. 66325251 7 -0. 07289875 8 -1. 53206047 9 -1. 89221367 10 -1. 72783895 [, 2] -1. 7012147 -1. 5122232 0. 3932446 0. 4828572 0. 2253562 0. 9763632 1. 0187273 0. 9932521 -0. 4542234 -0. 9356060

Column scores: [, 1] Corrugated -0. 4321891 Biscuit 0. 6712457 Type_I -2. 0277828 Type_II_Red 0. 6086514 Type_II_Yellow 0. 8817724 Type_II_Gray 0. 8845662 Type_III 0. 8539497 [, 2] -0. 9113879 -0. 2200843 0. 5029450 1. 3687118 -0. 8926234 -0. 5461081 -3. 5212105

> str(Ca. Model. 1) List of 4 \$ cor : num [1: 2] 0. 66 0. 292 \$ rscore: num [1: 10, 1: 2] 0. 462 0. 604 0. 617 0. 535 0. 798. . . - attr(*, "dimnames")=List of 2. . \$ : chr [1: 10] "1" "2" "3" "4". . . . \$ : NULL \$ cscore: num [1: 7, 1: 2] -0. 432 0. 671 -2. 028 0. 609 0. 882. . . - attr(*, "dimnames")=List of 2. . \$ : chr [1: 7] "Corrugated" "Biscuit" "Type_I”. . . . \$ : NULL \$ Freq : num [1: 10, 1: 7] 57 116 27 28 60 75 53 56 93 84. . . - attr(*, "dimnames")=List of 2. . \$ Row : chr [1: 10] "1" "2" "3" "4". . . . \$ Column: chr [1: 7] "Corrugated" "Biscuit" "Type_I". . . - attr(*, "class")= chr "correspondence“ > biplot(Ca. Model. 1, xlim=c(-1, . 75)) > plot(Ca. Model. 1\$rscore, type="c") > text(Ca. Model. 1\$rscore, as. character(1: 10))

More Details • Package ca provides more statistics regarding the fit – install. packages("ca") – library(ca) – Ca. Model. 2 <- ca(Nelson[, 2: 8]) – Ca. Model. 2 – summary(Ca. Model. 2) – plot(Ca. Model. 2, xlim=c(-1. 3, . 8))

CA Terminology 1 • Principal Inertias (eigenvalues) – a measure of the inertia (chi square deviation from the mean) explained by each dimension • Mass – The weight of each row/col in the analysis (the proportion of cases in that row/column)

CA Terminology 2 • Chi. Dist – how much a profile (row or column) differs from the mean profile • Inertia –deviation from average for this row/col • Dim. – the scores for each axis

summary() output 1 • mass = Mass*1000 • qlt = (quality) how well the r/c is represented • inr = Inertia*1000 • cor = (relative contribution to inertia) contribution to quality for that dimension

summary() output 2 • ctr = (absolute contribution to inertia) proportion of r/c inertia for that dimension