Three-way Contingency Tables

We are using here the Arthritis dataset from the package vcd.
The data represent a double-blind clinical trial of new treatments for rheumatoid arthritis.

library(vcd)
head(Arthritis)   # top few observations from the data

  ID Treatment  Sex Age Improved
1 57   Treated Male  27     Some
2 46   Treated Male  29     None
3 77   Treated Male  30     None
4 17   Treated Male  32   Marked
5 36   Treated Male  46   Marked
6 23   Treated Male  58   Marked

xtabs() can be used to generate multidimensional tables based on three or more categoricals variables.

mytable <- xtabs(~ Treatment+Sex+Improved, data=Arthritis)
mytable

Also, we can use table() function to generate multidimensional tables.

Here we get the cell frequencies for the three-way classification. Here the ftable() function can be used to print a more compact and attractive version of the table.

Output shown in next slide.

, , Improved = None

         Sex
Treatment Female Male
  Placebo     19   10
  Treated      6    7

, , Improved = Some

         Sex
Treatment Female Male
  Placebo      7    0
  Treated      5    2

, , Improved = Marked

         Sex
Treatment Female Male
  Placebo      6    1
  Treated     16    5

The ftable() function can be used to print multidimensional tables in a compact and attarctive manner.

ftable(mytable) 

                 Improved None Some Marked
Treatment Sex                             
Placebo   Female            19    7      6
          Male              10    0      1
Treated   Female             6    5     16
          Male               7    2      5

The margin.table(), prop.table(), and addmargins() functions extend naturally to more than two dimensionals.

margin.table(mytable, 1)

Treatment
Placebo Treated 
     43      41 

Here we get the marginal frequencies for the Treatment.

• Treatment is referred to by index 1

margin.table(mytable, 2)

Sex
Female   Male 
    59     25 

margin.table(mytable, 2)

Sex
Female   Male 
    59     25 

Here we get the marginal frequencies for the Sex and Improved.

• Sex is referred to by index 2
• Improved is referred to by index 3

margin.table(mytable, c(1,3))

         Improved
Treatment None Some Marked
  Placebo   29    7      7
  Treated   13    7     21

Here we get the marginal frequencies for the Treatment \( \times \) Improved classification.

Improved proportions for Treatment \( \times \) Sex

ftable(prop.table(mytable, c(1,2)))

                 Improved       None       Some     Marked
Treatment Sex                                             
Placebo   Female          0.59375000 0.21875000 0.18750000
          Male            0.90909091 0.00000000 0.09090909
Treated   Female          0.22222222 0.18518519 0.59259259
          Male            0.50000000 0.14285714 0.35714286

The proportion of patients with None, Some and Marked improvement for each Treatment \( \times \) Sex combination.

ftable(addmargins(prop.table(mytable, c(1, 2)), 3))

                 Improved       None       Some     Marked        Sum
Treatment Sex                                                        
Placebo   Female          0.59375000 0.21875000 0.18750000 1.00000000
          Male            0.90909091 0.00000000 0.09090909 1.00000000
Treated   Female          0.22222222 0.18518519 0.59259259 1.00000000
          Male            0.50000000 0.14285714 0.35714286 1.00000000

Here we add a sum margin over the third index.

We can get the percentages instead of proportions, simply multiply the resulting table by 100.

ftable(addmargins(prop.table(mytable, c(1, 2)), 3))*100

                 Improved       None       Some     Marked        Sum
Treatment Sex                                                        
Placebo   Female           59.375000  21.875000  18.750000 100.000000
          Male             90.909091   0.000000   9.090909 100.000000
Treated   Female           22.222222  18.518519  59.259259 100.000000
          Male             50.000000  14.285714  35.714286 100.000000