Note that a variable is a quantitative or qualitative variable based on the possible values the variable can take on.
Example 2.1
In a study of obesity in the population of children aged 10 or less in the United States, some possible quantitative variables that might be measured include age, height, weight, heart rate, body mass index, and percent body fat; some qualitative variables that might be measured on this population include gender, eye color, race, and blood type. A likely choice for the response variable in this study would be the qualitative variable Obese defined by
2.1.1 Qualitative Variables
Qualitative variables take on nonnumeric values and are usually used to represent a distinct quality of a population unit. When the possible values of a qualitative variable have no intrinsic ordering, the variable is called a nominal variable; when there is a natural ordering of the possible values of the variable, then the variable is called an ordinal variable. An example of a nominal variable is Blood Type where the standard values for blood type are A, B, AB, and O. Clearly, there is no intrinsic ordering of these blood types, and hence, Blood Type is a nominal variable. An example of an ordinal variable is the variable Pain where a subject is asked to describe their pain verbally as
No pain,
Mild pain,
Discomforting pain,
Distressing pain,
Intense pain,
Excruciating pain.
In this case, since the verbal descriptions describe increasing levels of pain, there is a clear ordering of the possible values of the variable Pain levels, and therefore, Pain is an ordinal qualitative variable.
Example 2.2
In the Framingham Heart Study of coronary heart disease, the following two nominal qualitative variables were recorded:
and
Example 2.3
An example of an ordinal variable is the variable Baldness when measured on the Norwood–Hamilton scale for male-pattern baldness. The variable Baldness is measured according to the seven categories listed below:
1 I Full head of hair without any hair loss.
2 II Minor recession at the front of the hairline.
3 III Further loss at the front of the hairline, which is considered “cosmetically significant.”
4 IV Progressively more loss along the front hairline and at the crown.
5 V Hair loss extends toward the vertex.
6 VI Frontal and vertex balding areas merge into one and increase in size.
7 VII All hair is lost along the front hairline and crown.
Clearly, the values of the variable Baldness indicate an increasing degree of hair loss, and thus, Baldness as measured on the Norwood–Hamilton scale is an ordinal variable. This variable is also measured on the Offspring Cohort in the Framingham Heart Study.
2.1.2 Quantitative Variables
A quantitative variable is a variable that takes only numeric values. The values of a quantitative variable are said to be measured on an interval scale when the difference between two values is meaningful; the values of a quantitative variable are said to be measured on a ratio scale when the ratio of two values is meaningful. The key difference between a variable measured on an interval scale and a ratio scale is that on a ratio scale there is a “natural zero” representing absence of the attribute being measured, while there is no natural zero for variables measured on only an interval scale. Some scales of measurement will have natural zero and some will not. When a measurement scale has a natural zero, then the ratio of two measurements is a meaningful measure of how many times larger one value is than the other. For example, the variable Fat that represents the grams of fat in a food product is measured on a ratio scale because the value Fat = 0 indicates that the unit contained absolutely no fat. When a scale of measurement does not have a natural zero, then only the difference between two measurements is a meaningful comparison of the values of the two measurements. For example, the variable Body Temperature is measured on a scale that has no natural zero since Body Temperature = 0 does not indicate that the body has no temperature.
Since interval scales are ordered, the difference between two values measures how much larger one value is than another. A ratio scale is also an interval scale but has the additional property that the ratio of two values is meaningful. Thus, for a variable measured on an interval scale the difference of two values is the meaningful way to compare the values, and for a variable measured on a ratio scale both the difference and the ratio of two values are meaningful ways to compare difference values of the variable. For example, body temperature in degrees Fahrenheit is a variable that is measured on an interval scale so that it is meaningful to say that a body temperature of 98.6 and a body temperature of 102.3 differ by 3.7 degrees; however, it would not be meaningful to say that a temperature of 102.3 is 1.04 times as much as a temperature of 98.6. On the other hand, the variable weight in pounds is measured on a ratio scale, and therefore, it would be proper to say that a weight of 210 lb is 1.4 times a weight of 150 lb; it would also be meaningful to say that a weight of 210 lb is 60 lb more than a weight of 150 lb.
Example 2.4
The following questions were asked in the Framingham Heart Study on the Offspring Cohort and the corresponding variables recorded. The variables are listed in parentheses after each question. Determine which of these variables are qualitative and which quantitative. For the qualitative variables determine whether they are nominal or ordinal variables.
1 What is your gender? (Gender)
2 Systolic blood pressure (Systolic Blood Pressure)
3 Do you smoke? (Smoke)
4 How many cigarettes do you smoke per day? (No. Cigarettes)
5 What is your age? (Age)
6 How many times per week do you engage in intense physical activity? (No. Physical Activity)
7 How is your health now? (Health)
Solutions
1 Gender is a nominal qualitative variable.
2 Systolic Blood Pressure
