Two-way tables: biplots associated with correspondence analysis - Understanding Biplots - page 314

Ta b l e 7 . 5 The matrix X :9 × 14 representing the 2007/08 crime data two-way
contingency table.

Arsn AGBH AtMr BNRs BRs CrJk CmAs CmRb DrgR InAs Mrd PubV Rape RAC

Ecpe 1235 34479 2160 5946 29508 604 19875 7086 7929 764 3515 88 5499 8939

FrSt 432 16833 939 4418 15705 156 19885 4193 4525 427 879 59 2628 4501

Gaut 1824 46993 5257 15117 62703 7466 57153 22152 12348 1501 3674 174 8073 50970

KZN 1322 30606 4946 10258 37203 3889 29410 9264 24174 1197 4713 76 6502 24290

Limp 573 13670 722 5401 11857 203 11024 3760 3198 215 696 31 2816 2447

Mpml 588 16849 1271 4273 18855 664 12202 4752 1770 251 835 61 2635 5907

NWst 624 15861 881 4987 14722 291 10406 3863 7004 345 917 117 3017 5528

NCpe 169 9898 775 1956 4924 5 5431 1337 2201 213 422 32 1020 1175

WCpe 629 24915 1844 10639 42376 923 32663 8578 45985 1850 2836 257 4000 14555

Ta b l e 7 . 6 The transposed version of X :9 × 14 with the row and column sums of X

added.

ECpe FrSt Gaut KZN Limp Mpml NWst NCpe WCpe Total

Arsn 1235 432 1824 1322 573 588 624 169 629 7396

AGBH 34479 16833 46993 30606 13670 16849 15861 9898 24915 210104

AtMr 2160 939 5257 4946 722 1271 881 775 1844 18795

BNRs 5946 4418 15117 10258 5401 4273 4987 1956 10639 62995

BRs 29508 15705 62703 37203 11857 18855 14722 4924 42376 237853

CrJk 604 156 7466 3889 203 664 291 5 923 14201

CmAs 19875 19885 57153 29410 11024 12202 10406 5431 32663 198049

CmRb 7086 4193 22152 9264 3760 4752 3863 1337 8578 64985

DrgR 7929 4525 12348 24174 3198 1770 7004 2201 45985 109134

InAs 764 427 1501 1197 215 251 345 213 1850 6763

Mrd 3515 879 3674 4713 696 835 917 422 2836 18487

PubV 88 59 174 76 31 61 117 32 257 895

Rape 5499 2628 8073 6502 2816 2635 3017 1020 4000 36190

RAC 8939 4501 50970 24290 2447 5907 5528 1175 14555 118312

Total 127627 75580 295405 187850 56613 70913 68563 29558 192050 1104159

subscript that has been replaced by the dot. Similarly, the column sums are the diagonal
elements of the matrix C , namely 7396, 210 104, ... , 118 312, written symbolically as
x . 1 , x . 2 ,

...

, x . 14 , respectively. The sum of the elements in X , n = x .. = 1 104 159, is the

total number of the 14 categories of crime reported in the country as a whole.
The meaning of the independence model for the contingency table given in Table 7.5
is that the type of crime reported is independent of the province in which it occurs. The
matrix E :9 × 14 of the independence model is given in Table 7.7, while the matrix of
deviations from independence, (7.1), is given in Table 7.8.
The hypothesis that the independence model E = R11 C / n describes the observed
frequencies in Table 7.5 satisfactorily is most implausible. Unsurprisingly, a formal Pear-
son's chi-squared test (7.4) gives 123 183 with 8

13 degrees of freedom. The natural
question that arises is whether a biplot can be constructed that provides information on
how the different cells in X contribute to the lack of fit of the independence model.
Table 7.9 shows the data in weighted deviation form R − 1 / 2

×

( X − E ) C − 1 / 2 .