Exercise 6.2 The data set criminal in the package logmult gives the 4 × 5 table below of the number of men aged 15–19 charged with a criminal case for whom charges were dropped in Denmark from 1955–1958.

Carry out a simple correspondence analysis on this table. (a)What percentages of the Pearson χ2 for association are explained by the various dimensions?

library(logmult)
library(vcd)
library(MASS)
library(ca)
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
cr.ca <- ca(criminal)
summary(cr.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 |

Dimension 1 explains 90.3% Pearson χ2 while dimension 2 explains 9.0% of Pearson χ2. Dimension 3 accounts for the rest (0.7%)

(b)Plot the 2D correspondence analysis solution. Describe the pattern of association between year and age.

plot(cr.ca)

Along dimension 1, year could be considered roughly equally spaced, but for age, 17 is quite different in terms of its frequency profile

Dimension 2 contrasts years 1955 and 1956 with 1957 and 1958

Exercise 6.11 The data set Vietnam in vcdExtra gives a 2 × 5 × 4 contingency table in frequency form reflecting a survey of student opinion on the Vietnam War at the University of North Carolina in May 1967. The table variables are sex, year in school, and response, which has categories: (A) Defeat North Vietnam by widespread bombing and land invasion; (B) Maintain the present policy; (C) De-escalate military activity, stop bombing and begin negotiations; (D) Withdraw military forces immediately

  1. 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 and sex.
data("Vietnam", package = "vcdExtra")
#Vietnam
str (Vietnam)
## 'data.frame':    40 obs. of  4 variables:
##  $ sex     : Factor w/ 2 levels "Female","Male": 1 1 1 1 1 1 1 1 1 1 ...
##  $ year    : int  1 1 1 1 2 2 2 2 3 3 ...
##  $ response: Factor w/ 4 levels "A","B","C","D": 1 2 3 4 1 2 3 4 1 2 ...
##  $ Freq    : int  13 19 40 5 5 9 33 3 22 29 ...
Vietnam <- within(Vietnam, {year_sex <- paste(year, toupper(substr(sex,1,1)))})
Vn.year_sex <- xtabs(Freq ~ year_sex + response, data=Vietnam)
Vn.year_sex
##         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
Vn.ca <- ca(Vn.year_sex)
summary(Vn.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 |
  1. Construct an informative 2D plot of the solution, and interpret in terms of how the response varies with year for males and females.
plot(Vn.ca)

Males with exception of year=5 are mostly associated with response group A and B. Females are mostly associated with response C.

  1. 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.
vn.mca <- mjca(Vn.year_sex)
plot(vn.mca)

Men in year 1 and 2 are associated with response group A, while Men in year 3 and 4 are associated with response B. Women in year 1 and 4 are associated with response group C