Categorical (qualitative) data

Categorical data is derived from populations that consist of some number of subpopulations and we record only the subpopulation membership of selected individuals. In such cases the basic data summary is a frequency table that counts the number of individuals within each category. If there is more than one set of categories, then we can summarize the data using a multi-dimensional frequency table. For example, here is part of a dataset that records the hair color, eye color, and sex of a group of 592 students.

Hair   Eye    Sex
--------------------
Black  Brown  Female
Red    Green  Male
Blond  Blue   Male
Brown  Hazel  Female
...

Sometimes numerical codes are used in place of names, but it is important to remember that these codes are not quantitative values, just labels. The frequency table for hair color in this dataset is:

Black Brown   Red Blond 
  108   286    71   127

The basic graphical tool for categorical data is the barplot. This plots bars for each category, the height of which is the frequency or relative frequency of that category.

Barplots are more effective than pie charts because we can more readily make a visual comparison of rectangles that have the same base width than a visual comparison of wedges in a pie.

If a second categorical variable also is observed, for example hair color and gender, a barplot with side-by-side bars for each level of the first variable plotted contiguously, and each such group plotted with space between groups, is most effective to compare each level of the first variable across levels of the second. For example, the following plot shows how hair color is distributed for a sample of males and females. A comparison of the relative frequencies for males and females shows that a relatively higher proportion of females have blond hair and somewhat lower proportion of females have black or brown hair.

We can also display the relationship between hair and eye color using a 2-dimensional frequency table and barplot. The areas of the rectangles in this plot represent the relative frequency of the corresponding category combination.

       Eye
Hair    Brown Blue Hazel Green
  Black    68   20    15     5
  Brown   119   84    54    29
  Red      26   17    14    14
  Blond     7   94    10    16

Are hair color and eye color related? Although we will consider this question in detail later, we can think about how to interpret this question here. First note that there are a total of 108 people who have black hair, 68 of whom have brown eyes. That is, about 63% (68/108) of those with black hair have brown eyes. First note the correspondence between the structure of this sentence and the arithmetic that goes with it. The reference group for this percentage is defined by the prepositional phrase, of those with black hair, and this count is the denominator. The verb plus object in this sentence is have brown eyes. The count of people who have brown eyes within the reference group is the numerator of this percentage. Now the total counts for eye color are:

Brown  Blue Hazel Green 
  220   215    93    64

so 220 of the 592 people in this data have brown eyes. That is, 37% of all people in this data set have brown eyes, but brown eyes occurs much more frequently among people with black hair, 63%. So it appears that the percentage of people who have brown eyes depends on whether or not they have black hair. If the two percentages were equal, that is, if 37% of people with black hair also had brown eyes, then we would say that having brown eyes does not depend on whether or not a person has black hair. Therefore, for those two outcomes to be independent, there should have been 40 people (37% of 108) with black hair and brown eyes. This is the expected count under the assumption of independence between brown eyes and black hair. We can do the same for each combination of categories in this table to give the expected frequencies:

       Brown   Blue Hazel Green
Black  40.14  39.22 16.97 11.68
Brown 106.28 103.87 44.93 30.92
Red    26.39  25.79 11.15  7.68
Blond  47.20  46.12 19.95 13.73

If all of the observed counts had been equal to these expected counts, then hair and eye color would be completely independent. Obviously that is not the case. We can define a measure of distance between the observed counts and these expected counts under the assumption of independence by

$$D = \sum \frac{(O-E)^2}{E},$$

where the sum is over all combinations of hair and eye categories. Note that the expected count for a cell can be expressed as $RC/N$, where $R$ denotes the row total, $C$ denotes the column total, and $N$ denotes the grand total. For this data, $D = 138.3$. Later in the course we will discuss how to interpret this distance and determine whether or not this is large. The contribution to this distance from each cell is:

       Eye
Hair    Brown  Blue Hazel Green
  Black 19.35  9.42  0.23  3.82
  Brown  1.52  3.80  1.83  0.12
  Red    0.01  2.99  0.73  5.21
  Blond 34.23 49.70  4.96  0.38

Note that blond hair with brown or blue eyes are the greatest contributors to the distance of these counts from independence. This can be visualized by the R function assocplot:

data(HairEyeColor) #load HairEyeColor data set
HairEyeColor #this is a 3-d array
HairEye = apply(HairEyeColor,c(1,2),sum) #sum over gender, keep dims 1,2
Hair = apply(HairEye,1,sum) #get totals for hair color
Eye = apply(HairEye,2,sum) #get totals for eye color
# there are several ways to compute R*C. One way is to replicate Hair in each
# column of a matrix that is the same size as HairEye
# and replicate Eye in each row of a matrix with same size.
N = sum(Hair) #grand total
HairMat = matrix(Hair,nrow=length(Hair),ncol=length(Eye))
#note that matrix fills by column by default
EyeMat = matrix(Eye,nrow=length(Hair),ncol=length(Eye),byrow=TRUE)
# byrow arg makes matrix fill by row
ExpHairEye = HairMat*EyeMat/N # expected counts
round(ExpHairEye,2) # print rounded to 2 decimal places
#an easier way is to use the function outer
ExpHairEye = outer(Hair,Eye)/N
round(ExpHairEye,2) #note that outer preserves names of hair and eye
# now get distance from independence
D = ((HairEye - ExpHairEye)^2)/ExpHairEye
round(D,2) # gives contribution from each cell
sum(D) # print total distance
# now use R function paste to combine text and value
paste("Distance of Hair-Eye data from Independence:",round(sum(D),2))
# if round is not used then lots of decimal places will be printed!
# now visualize cell contributions using assocplot()
assocplot(HairEye,main="Relationship Between Hair and Eye Color: Males")
# first dim of HairEye is plotted on x-axis. Use transpose to put Hair on y-axis
assocplot(t(HairEye),main="Relationship Between Hair and Eye Color: Males")

We will see later that this data is very far from independence!