3.1 Two-Way Tables

Previously, we considered how to tabulate one categorical variable (into a one-dimensional table). Now we consider a table with two categorical variables, resulting in a two-dimensional table, also called a matrix or a two-dimensional array.

3.1.1 Matrices

You can create a matrix with the matrix() function. Unfortunately, the syntax is somewhat unintuitive. You must supply all of the values in a vector, along with the number of rows or columns, like so,

matrix(1:6, nrow = 3)
##      [,1] [,2]
## [1,]    1    4
## [2,]    2    5
## [3,]    3    6

Notice that the numbers filled in the first column first, which is unintuitive to some people. If you wish to fill in the matrix by rows, you must use the byrow option, like so,

matrix(1:6, nrow = 3, byrow = TRUE)
##      [,1] [,2]
## [1,]    1    2
## [2,]    3    4
## [3,]    5    6

If you want to access a particular element of a matrix, you can use the [] brackets, similar to accessing parts of a vector. For example x[3] is the third element of the vector x. Since a matrix has both rows and columns, we need to provide R two indices - one row number and one column number. So, M[3,4] is the element in the 3rd row, 4th column of the matrix M.

mat = matrix(1:6, nrow = 3, byrow = TRUE)
mat
##      [,1] [,2]
## [1,]    1    2
## [2,]    3    4
## [3,]    5    6
mat[3,2]
## [1] 6

You can also access an entire row or column by leaving a blank. Leaving the column number blank accesses all columns, while leaving the row number blank accesses all rows.

mat[2,]
## [1] 3 4
mat[,1]
## [1] 1 3 5

You can flip, or transpose, a matrix using the t() function. This switches the rows and columns of the matrix.

t(mat)
##      [,1] [,2] [,3]
## [1,]    1    3    5
## [2,]    2    4    6

3.1.2 Example Dataset

Let’s say a drug company is interested in evaluating the performance of two new drugs in development, New Drug 1 (D1) and New Drug 2 (D2), in alleviating Disease Y symptoms. They want to test it against the current standard drug (ST). They enroll 1000 people in a large clinical trial, and found that:

  1. out of the 400 people put on D1, 200 found their health status improve,
  2. out of the 200 people put on D2, 150 found their health status improve, and
  3. out of the 400 people put on ST, 240 found their health status improve.

You can create the table directly in R:

drug <- matrix(c(200, 200, 150, 50, 240, 160), ncol = 2, byrow = TRUE)
colnames(drug) <- c("Improved", "NotImproved")
rownames(drug) <- c("D1", "D2", "ST")
#drug <- as.table(drug)
drug
##    Improved NotImproved
## D1      200         200
## D2      150          50
## ST      240         160

3.1.3 Working with Two-Way Tables

Here we look at some of the commands available to help look at the information in a table in different ways. First, there are a couple of ways to get graphical views of the data:

barplot(drug, legend=TRUE, beside=TRUE,
        main='Health Improvement by Drug Treatment (bar plot)')

barplot(t(drug), legend=TRUE, beside=FALSE, 
        main='Drug Treatment by Health Improvement (bar plot)',
        args.legend = list(x="top"))

mosaicplot(drug, main="Health Improvement by Drug Treatment (mosaic plot)")

Discussion

  • What questions might you be able to answer with the first graph? What questions might you be able to answer with the second graph?
  • Which display is most informative? Why?

3.1.4 Distributions in Two-Way Tables

There are a number of ways to get marginal distributions using the margin.table() function. If you pass just the table (the first argument) to the command it calculates the total number of observations.

margin.table(drug)
## [1] 1000

If you pass it the optional second argument margin, it will calculate the marginal frequencies. We use margin = 1 for the row margins (row totals),

margin.table(drug, margin = 1)
##  D1  D2  ST 
## 400 200 400

And we use margin = 2 for the column margins (column totals),

margin.table(drug, margin = 2)
##    Improved NotImproved 
##         590         410

You can get a table of the joint proportions with prop.table(),

prop.table(drug)
##    Improved NotImproved
## D1     0.20        0.20
## D2     0.15        0.05
## ST     0.24        0.16

To calculate conditional probabilities, margin can be specified similarly to margin.table(). We can calculate P(Health Improvement|Drug) by using margin = 1.

prop.table(drug, margin = 1)
##    Improved NotImproved
## D1     0.50        0.50
## D2     0.75        0.25
## ST     0.60        0.40

We can calculate P(Drug|Health Improvement) by using margin = 2.

prop.table(drug, margin = 2)
##     Improved NotImproved
## D1 0.3389831   0.4878049
## D2 0.2542373   0.1219512
## ST 0.4067797   0.3902439

So, how well do you think the new treatment drug candidates perform relative to the standard treatment?

3.1.5 Exercises

Using your new R capabilities, check the answers you found above using R:

  • What is the marginal distribution of subjects who improved/did not improve?
  • What is the marginal distribution of drug assignments?
  • What is the conditional distribution of subjects who improved given that they received drug D2?
  • Among subjects who improved, what proportion received D1?
  • What is the joint proportion of subjects who improved and received the standard drug?
  • What proportion of all subjects received drug D2 and did not improve?
  • If there is no association between improvement and drug choice, how many of the subjects which took drug D1 would you expect to improve?
  • What is the conditional distribution of improvement given drug received? What drug seems to be working the best? worst?