style="font-size:15px;"> 3 Smoke is a nominal qualitative variable.4 No. Cigarettes is a quantitative variable.
5 Age is a quantitative variable.
6 No. Physical Activity is a quantitative variable.
7 Health is an ordinal qualitative variable.
A quantitative variable can also be classified as either a discrete variable or a continuous variable. A quantitative variable is a discrete variable when it can take on a finite or a countable number of values; a quantitative variable is a continuous variable when it can take on any value in one or more intervals. Note that the values that a discrete variable can take on are distinct, isolated, and can be counted. In the previous example, the variables Age, No. Cigarettes, and No. Physical Activity are discrete variables. A counting variable is a specialized discrete variable that simply counts how many times a particular event has occurred. The values a counting variables can take on are the values 0,1,2,3,…,∞. For example, in the Framingham Heart Study the variables No. Cigarettes and No. Physical Activity are counting variables.
Example 2.5
The following variables are all counting variables:
1 The number of cancer patients in remission following treatment at a hospital.
2 The number of laboratory mice that survive in an experiment.
3 The number of white blood cells in a 10 ml blood sample.
Continuous variables are variables that can take on any value in one or more intervals. Examples of continuous variables are the exact weight of a subject, the exact dose of a drug, and the exact height of a subject. In most problems, there will be variables of interest that are continuous variables, but because the variable can only be measured to a specific accuracy, a discrete version of the variable is used. For example, the weight of a subject is a continuous variable, but when it is measured only in pounds or tenths of pounds, it is a discrete variable.
Example 2.6
The following variables are continuous variables that might be measured on a discrete measurement scale:
1 Body temperature since it is usually measured in tenths of degrees.
2 Lung capacity since it is a volume and is usually measured in cubic centimeters.
3 Tumor size since it is measured as a depth in tenths of centimeters.
It is important that a variable truly reflects the characteristic being studied. A variable is said to be a valid variable when the measurements on the variable truly represent the characteristic the variable is supposed to be measuring. The validity of a variable depends on the characteristic being measured and the measuring device being used to measure the characteristic. When a characteristic of a unit is subjective in nature, it will be difficult to measure the characteristic accurately, and in this case, the validity of any variables used to measure this subjective characteristic is usually questionable.
Example 2.7
The intelligence of an individual is a subjectively measured characteristic. There are many tests that have been developed to measure intelligence. For example, the Fagan test measures the amount of time an infant spends inspecting a new object and compares this time with the time spent inspecting a familiar object (Fagan and Detterman, 1992). The validity of the Fagan test as a measure of intelligence, however, has been questioned by several scientists who have studied the relationship between intelligence and the Fagan test scores.
The diagram in Figure 2.1 summarizes the different types of variables/data that can be observed.
Figure 2.1 Different types of classifications for variables.
2.1.3 Multivariate Data
In most research problems, there will be many variables that need to be measured. When the collection of variables measured on each unit consists of two or more variables, a data set is called a multivariate data set, and a multivariate data set consisting of only two variables is called a bivariate data set. In a multivariate data set, there is usually one variable that is of primary interest to a research question that is believed to be explained by some of the other variables measured in the study. The variable of primary interest is called a response variable and the variables believed to cause changes in the response are called explanatory variables or predictor variables. The explanatory variables are often referred to as the input variables and the response variable is often referred to as the output variable. Furthermore, in a statistical model, the response variable is the variable that is being modeled; the explanatory variables are the input variables in the model that are believed to cause or explain differences in the response variable. For example, in studying the survival of melanoma patients, the response variable might be Survival Time that is expected to be influenced by the explanatory variables Age, Gender, Clark’s Stage, and Tumor Size. In this case, a model relating Survival Time to the explanatory variables Age, Gender, Clark’s Stage, and Tumor Size might be investigated in the research study.
A multivariate data set often consists of a mixture of qualitative and quantitative variables. For example, in a biomedical study, several variables that are commonly measured are a subject’s age, race, gender, height, and weight. When data have been collected, the multivariate data set is generally stored in a spreadsheet with the columns containing the data on each variable and the rows of the spreadsheet containing the observations on each subject in the study.
In studying the response variable, it is often the case that there are subpopulations that are determined by a particular set of values of the explanatory variables that will be important in answering the research questions. In this case, it is critical that a variable be included in the data set that identifies which subpopulation each unit belongs to. For example, in the National Health and Nutrition Examination Survey (NHANES) study, the distribution of the weight of female children was studied. The response variable in this study was weight and some of the explanatory variables measured in this study were height, age, and gender. The result of this part of the NHANES study was a distribution of the weights of females over a certain range of age. The resulting distributions were summarized in the chart given in Figure 2.2 that shows the weight ranges for females for several different ages.
Figure 2.2 Weight-by-age chart for girls in the NHANES study.
Example 2.8
In the article “The validity of self-reported weight in US adults: a population based cross-sectional study” published in BMC Public Health (Villanueva, 2001), the author reported the results of a study on the validity of self-reported weight. The data set used in the study was a multivariate data set with response variable being the difference between the self-reported weight and the actual weight of an individual. The explanatory variables in this study were gender, age, race–ethnicity, highest educational attainment, level of activity, and perception of the individuals’ current weight.
2.2
