data("criminal", package = "logmult")
criminal
## Age
## Year 15 16 17 18 19
## 1955 141 285 320 441 427
## 1956 144 292 342 441 396
## 1957 196 380 424 462 427
## 1958 212 424 399 442 430
(a)What percentages of the Pearson X^2 for association are explained by the various dimensions?
library(ca)
criminal.ca <- ca(criminal)
summary(criminal.ca)
##
## Principal inertias (eigenvalues):
##
## dim value % cum% scree plot
## 1 0.004939 90.3 90.3 ***********************
## 2 0.000491 9.0 99.3 **
## 3 3.8e-050 0.7 100.0
## -------- -----
## Total: 0.005468 100.0
##
##
## Rows:
## name mass qlt inr k=1 cor ctr k=2 cor ctr
## 1 | 1955 | 230 996 347 | 88 939 361 | -22 58 223 |
## 2 | 1956 | 230 978 157 | 58 908 157 | 16 71 124 |
## 3 | 1957 | 269 984 111 | -39 669 82 | 27 315 391 |
## 4 | 1958 | 271 999 385 | -85 938 399 | -22 61 262 |
##
## Columns:
## name mass qlt inr k=1 cor ctr k=2 cor ctr
## 1 | 15 | 99 998 185 | -101 992 203 | -7 5 11 |
## 2 | 16 | 197 996 312 | -91 959 331 | -18 37 128 |
## 3 | 17 | 211 991 75 | -23 281 23 | 37 710 594 |
## 4 | 18 | 254 989 235 | 70 980 255 | 7 9 24 |
## 5 | 19 | 239 990 194 | 62 877 188 | -22 112 243 |
90.3 percentages of the Pearson X^2 for association are explained by 1 dimension and 99.3 percentages of the Pearson X^2 for association are explained by 2 dimensions.
(b)Plot the 2D correspondence analysis solution. Describe the pattern of association between year and age.
plot(criminal.ca)
The most crime in year 1958 are associated with people in age 15 and 16s and in year 1957 the most crime are associated with people in age 17.In year 1958 people with age 18 tend to do crime and in 1955 people with age 19 had done more crime.
data("Vietnam", package = "vcdExtra")
Vietnam
## sex year response Freq
## 1 Female 1 A 13
## 2 Female 1 B 19
## 3 Female 1 C 40
## 4 Female 1 D 5
## 5 Female 2 A 5
## 6 Female 2 B 9
## 7 Female 2 C 33
## 8 Female 2 D 3
## 9 Female 3 A 22
## 10 Female 3 B 29
## 11 Female 3 C 110
## 12 Female 3 D 6
## 13 Female 4 A 12
## 14 Female 4 B 21
## 15 Female 4 C 58
## 16 Female 4 D 10
## 17 Female 5 A 19
## 18 Female 5 B 27
## 19 Female 5 C 128
## 20 Female 5 D 13
## 21 Male 1 A 175
## 22 Male 1 B 116
## 23 Male 1 C 131
## 24 Male 1 D 17
## 25 Male 2 A 160
## 26 Male 2 B 126
## 27 Male 2 C 135
## 28 Male 2 D 21
## 29 Male 3 A 132
## 30 Male 3 B 120
## 31 Male 3 C 154
## 32 Male 3 D 29
## 33 Male 4 A 145
## 34 Male 4 B 95
## 35 Male 4 C 185
## 36 Male 4 D 44
## 37 Male 5 A 118
## 38 Male 5 B 176
## 39 Male 5 C 345
## 40 Male 5 D 141
(a) Using the stacking approach, carry out a correspondence analysis corresponding to the loglinear model [R][YS], which asserts that the response is independent of the combinations of year an sex.
Vietnam1 <- Vietnam
Vietnam1$year_sex <- interaction(Vietnam$year, Vietnam$sex, sep='.')
Vietnam1$year_sex
## [1] 1.Female 1.Female 1.Female 1.Female 2.Female 2.Female 2.Female
## [8] 2.Female 3.Female 3.Female 3.Female 3.Female 4.Female 4.Female
## [15] 4.Female 4.Female 5.Female 5.Female 5.Female 5.Female 1.Male
## [22] 1.Male 1.Male 1.Male 2.Male 2.Male 2.Male 2.Male
## [29] 3.Male 3.Male 3.Male 3.Male 4.Male 4.Male 4.Male
## [36] 4.Male 5.Male 5.Male 5.Male 5.Male
## 10 Levels: 1.Female 2.Female 3.Female 4.Female 5.Female 1.Male ... 5.Male
Vietnam1$year_sex <- paste(Vietnam$year, Vietnam$sex, sep=':')
Vietnam1$year_sex
## [1] "1:Female" "1:Female" "1:Female" "1:Female" "2:Female" "2:Female"
## [7] "2:Female" "2:Female" "3:Female" "3:Female" "3:Female" "3:Female"
## [13] "4:Female" "4:Female" "4:Female" "4:Female" "5:Female" "5:Female"
## [19] "5:Female" "5:Female" "1:Male" "1:Male" "1:Male" "1:Male"
## [25] "2:Male" "2:Male" "2:Male" "2:Male" "3:Male" "3:Male"
## [31] "3:Male" "3:Male" "4:Male" "4:Male" "4:Male" "4:Male"
## [37] "5:Male" "5:Male" "5:Male" "5:Male"
vietnamtab <- xtabs(Freq ~ year_sex + response, data=Vietnam1)
vietnamtab
## response
## year_sex A B C D
## 1:Female 13 19 40 5
## 1:Male 175 116 131 17
## 2:Female 5 9 33 3
## 2:Male 160 126 135 21
## 3:Female 22 29 110 6
## 3:Male 132 120 154 29
## 4:Female 12 21 58 10
## 4:Male 145 95 185 44
## 5:Female 19 27 128 13
## 5:Male 118 176 345 141
vietnamtab.ca <- ca(vietnamtab)
summary(vietnamtab.ca)
##
## Principal inertias (eigenvalues):
##
## dim value % cum% scree plot
## 1 0.085680 73.6 73.6 ******************
## 2 0.027881 23.9 97.5 ******
## 3 0.002854 2.5 100.0 *
## -------- -----
## Total: 0.116415 100.0
##
##
## Rows:
## name mass qlt inr k=1 cor ctr k=2 cor ctr
## 1 | 1Fml | 24 818 13 | -167 452 8 | -150 367 20 |
## 2 | 1Mal | 139 997 181 | 386 986 242 | -41 11 8 |
## 3 | 2Fml | 16 995 35 | -407 647 31 | -299 349 51 |
## 4 | 2Mal | 140 984 131 | 326 982 175 | -15 2 1 |
## 5 | 3Fml | 53 999 112 | -334 453 69 | -367 547 256 |
## 6 | 3Mal | 138 904 40 | 175 904 49 | -4 0 0 |
## 7 | 4Fml | 32 982 37 | -344 887 44 | -113 95 15 |
## 8 | 4Mal | 149 383 23 | 81 372 11 | 14 11 1 |
## 9 | 5Fml | 59 994 153 | -453 686 143 | -304 309 197 |
## 10 | 5Mal | 248 1000 276 | -281 608 228 | 225 391 451 |
##
## Columns:
## name mass qlt inr k=1 cor ctr k=2 cor ctr
## 1 | A | 255 985 381 | 414 985 509 | -1 0 0 |
## 2 | B | 235 720 60 | 135 608 50 | 58 112 28 |
## 3 | C | 419 999 283 | -247 773 298 | -133 226 267 |
## 4 | D | 92 995 276 | -366 383 143 | 463 612 705 |
OR
Vietnam_new <- within(Vietnam, {year_sex <- paste(year, toupper(substr(sex,1,1)))})
Vietnam_new
## sex year response Freq year_sex
## 1 Female 1 A 13 1 F
## 2 Female 1 B 19 1 F
## 3 Female 1 C 40 1 F
## 4 Female 1 D 5 1 F
## 5 Female 2 A 5 2 F
## 6 Female 2 B 9 2 F
## 7 Female 2 C 33 2 F
## 8 Female 2 D 3 2 F
## 9 Female 3 A 22 3 F
## 10 Female 3 B 29 3 F
## 11 Female 3 C 110 3 F
## 12 Female 3 D 6 3 F
## 13 Female 4 A 12 4 F
## 14 Female 4 B 21 4 F
## 15 Female 4 C 58 4 F
## 16 Female 4 D 10 4 F
## 17 Female 5 A 19 5 F
## 18 Female 5 B 27 5 F
## 19 Female 5 C 128 5 F
## 20 Female 5 D 13 5 F
## 21 Male 1 A 175 1 M
## 22 Male 1 B 116 1 M
## 23 Male 1 C 131 1 M
## 24 Male 1 D 17 1 M
## 25 Male 2 A 160 2 M
## 26 Male 2 B 126 2 M
## 27 Male 2 C 135 2 M
## 28 Male 2 D 21 2 M
## 29 Male 3 A 132 3 M
## 30 Male 3 B 120 3 M
## 31 Male 3 C 154 3 M
## 32 Male 3 D 29 3 M
## 33 Male 4 A 145 4 M
## 34 Male 4 B 95 4 M
## 35 Male 4 C 185 4 M
## 36 Male 4 D 44 4 M
## 37 Male 5 A 118 5 M
## 38 Male 5 B 176 5 M
## 39 Male 5 C 345 5 M
## 40 Male 5 D 141 5 M
Vietnam.tab <- xtabs(Freq ~ year_sex + response, data=Vietnam_new)
Vietnam.tab
## response
## year_sex A B C D
## 1 F 13 19 40 5
## 1 M 175 116 131 17
## 2 F 5 9 33 3
## 2 M 160 126 135 21
## 3 F 22 29 110 6
## 3 M 132 120 154 29
## 4 F 12 21 58 10
## 4 M 145 95 185 44
## 5 F 19 27 128 13
## 5 M 118 176 345 141
Vietnam.tab.ca <- ca(Vietnam.tab)
summary(Vietnam.tab.ca)
##
## Principal inertias (eigenvalues):
##
## dim value % cum% scree plot
## 1 0.085680 73.6 73.6 ******************
## 2 0.027881 23.9 97.5 ******
## 3 0.002854 2.5 100.0 *
## -------- -----
## Total: 0.116415 100.0
##
##
## Rows:
## name mass qlt inr k=1 cor ctr k=2 cor ctr
## 1 | 1F | 24 818 13 | -167 452 8 | -150 367 20 |
## 2 | 1M | 139 997 181 | 386 986 242 | -41 11 8 |
## 3 | 2F | 16 995 35 | -407 647 31 | -299 349 51 |
## 4 | 2M | 140 984 131 | 326 982 175 | -15 2 1 |
## 5 | 3F | 53 999 112 | -334 453 69 | -367 547 256 |
## 6 | 3M | 138 904 40 | 175 904 49 | -4 0 0 |
## 7 | 4F | 32 982 37 | -344 887 44 | -113 95 15 |
## 8 | 4M | 149 383 23 | 81 372 11 | 14 11 1 |
## 9 | 5F | 59 994 153 | -453 686 143 | -304 309 197 |
## 10 | 5M | 248 1000 276 | -281 608 228 | 225 391 451 |
##
## Columns:
## name mass qlt inr k=1 cor ctr k=2 cor ctr
## 1 | A | 255 985 381 | 414 985 509 | -1 0 0 |
## 2 | B | 235 720 60 | 135 608 50 | 58 112 28 |
## 3 | C | 419 999 283 | -247 773 298 | -133 226 267 |
## 4 | D | 92 995 276 | -366 383 143 | 463 612 705 |
Females in year 1 through 5 chose answer c. Males in year 1&2 chose answer A rather than males in year 3 and 4 which chose answer B. Males in year 5 chose answer D.
(c) Use mjca () to carry out an MCA on the three-way table. Make a useful plot of the solution and interpret in terms of the relationship of the response to year and sex.
Vietnam.tab1 <- xtabs(Freq ~ year_sex + response, data=Vietnam1)
Vietnam.mca <- mjca(Vietnam.tab1)
Vietnam.mca
##
## Eigenvalues:
## 1 2 3 4 5 6 7
## Value 0.08568 0.027881 0.002854 0 0 0 0
## Percentage 73.6% 23.95% 2.45% 0% 0% 0% 0%
##
##
## Columns:
## year_sex:1:Female year_sex:1:Male year_sex:2:Female
## Mass 0.012234 0.069749 0.007944
## ChiDist 4.468307 1.777556 5.576556
## Inertia 0.244259 0.220386 0.247045
## Dim. 1 0.568864 -1.317425 1.389893
## Dim. 2 -0.898603 -0.245496 -1.789350
## year_sex:2:Male year_sex:3:Female year_sex:3:Male
## Mass 0.070226 0.026533 0.069113
## ChiDist 1.764712 3.007515 1.770337
## Inertia 0.218697 0.239997 0.216608
## Dim. 1 -1.115378 1.139930 -0.596323
## Dim. 2 -0.089188 -2.196323 -0.021487
## year_sex:4:Female year_sex:4:Male year_sex:5:Female
## Mass 0.016047 0.074515 0.029711
## ChiDist 3.891772 1.692276 2.839791
## Inertia 0.243046 0.213397 0.239600
## Dim. 1 1.174839 -0.276343 1.549307
## Dim. 2 -0.674687 0.083233 -1.822541
## year_sex:5:Male response:A response:B response:C response:D
## Mass 0.123928 0.127264 0.117255 0.209565 0.045917
## ChiDist 1.257824 1.245573 1.283375 0.855751 2.262624
## Inertia 0.196068 0.197444 0.193124 0.153466 0.235069
## Dim. 1 0.959368 -1.414434 -0.460340 0.842639 1.250004
## Dim. 2 1.348601 -0.004013 0.346670 -0.798445 2.769966
summary(Vietnam.mca)
##
## Principal inertias (eigenvalues):
##
## dim value % cum% scree plot
## 1 0.085680 73.6 73.6 ******************
## 2 0.027881 23.9 97.5 ******
## 3 0.002854 2.5 100.0 *
## 4 00000000 0.0 100.0
## 5 00000000 0.0 100.0
## 6 00000000 0.0 100.0
## 7 00000000 0.0 100.0
## -------- -----
## Total: 0.116415
##
##
## Columns:
## name mass qlt inr k=1 cor ctr k=2 cor ctr
## 1 | year_sex:1:Female | 12 818 80 | 167 452 4 | -150 367 10 |
## 2 | year_sex:1:Male | 70 997 72 | -386 986 121 | -41 11 4 |
## 3 | year_sex:2:Female | 8 995 81 | 407 647 15 | -299 349 25 |
## 4 | year_sex:2:Male | 70 984 72 | -326 982 87 | -15 2 1 |
## 5 | year_sex:3:Female | 27 999 78 | 334 453 34 | -367 547 128 |
## 6 | year_sex:3:Male | 69 904 71 | -175 904 25 | -4 0 0 |
## 7 | year_sex:4:Female | 16 982 79 | 344 887 22 | -113 95 7 |
## 8 | year_sex:4:Male | 75 383 70 | -81 372 6 | 14 11 1 |
## 9 | year_sex:5:Female | 30 994 78 | 453 686 71 | -304 309 99 |
## 10 | year_sex:5:Male | 124 1000 64 | 281 608 114 | 225 391 225 |
## 11 | response:A | 127 985 65 | -414 985 255 | -1 0 0 |
## 12 | response:B | 117 720 63 | -135 608 25 | 58 112 14 |
## 13 | response:C | 210 999 50 | 247 773 149 | -133 226 134 |
## 14 | response:D | 46 995 77 | 366 383 72 | 463 612 352 |
plot(Vietnam.mca)
Females in year 1 through 5 chose answer c. Males in year 1&2 chose answer A rather than males in year 3 and 4 which chose answer B. Males in year 5 chose answer D.
