2.2.1 Nonmetric or Qualitative Variables

As we will study in Chapter 3, the representation of the characteristics of nonmetric or qualitative variables can be done through frequency distribution tables or in a graphical way, without having to calculate the measures of position, dispersion, and shape. The only exception is the mode, a measure that provides the variable's most frequent value, and it can also be applied to nonmetric variables.

Imagine that a questionnaire will be used to collect data on family income from a sample of consumers, based on certain salary ranges. Table 2.1 shows the variable categories.

Note that both variables are qualitative, since the data are represented by ranges. However, it is very common for researchers to classify them incorrectly, mainly when the variable has numerical values in the data. In this case, it is only possible to calculate the frequencies, and not the summary measures, such as, the mean and standard deviation.

The frequencies obtained for each income range can be seen in Table 2.2.

A common error found in papers that use qualitative variables represented by numbers is the calculation of the sample mean, or any other summary measure. First of all, the researcher calculates the mean of the limits of each range, assuming that this value corresponds to the real mean of the consumers found in that range. However, since the data distribution is not necessarily linear or symmetrical around the mean, this hypothesis is often violated.

In order for us to be able to calculate summary measures, such as, the mean and standard deviation, the variable being studied must necessarily be quantitative.

2.2.2 Metric or Quantitative Variables

Quantitative variables can be represented in a graphical way (line charts, scatter plots, histograms, stem-and-leaves, and boxplots), through measures of position or location (mean, median, mode, quartiles, deciles, and percentiles), measures of dispersion or variability (range, average deviation, variance, standard deviation, standard error, and coefficient of variation), or through measures of shape, such as, skewness and kurtosis, as we will study in Chapter 3.

These variables can be discrete or continuous. Discrete variables can take on a finite set of values that frequently come from a count, such as, for example, the number of children in a family (0, 1, 2…). Conversely, continuous variables take on values that are in an interval with real numbers, such as, for example, an individual's weight or income.

Imagine a dataset with 20 people's names, age, weight and height, as shown in Table 2.3.

The data are available in the file VarQuanti.sav. To classify the variables on SPSS (Fig. 2.2), let’s click on Variable View. Note that the variable Name is qualitative (a string), and it is measured on a nominal scale (column Measure). On the other hand, variables Age, Weight, and Height are quantitative (Numeric), and they are measured in scale (Scale). The variable scales of measurement will be studied in more detail in Section 2.3.

2.3.1 Nonmetric Variables—Nominal Scale

The nominal scale classifies the units into classes or categories regarding the characteristic represented, not establishing any magnitude or order relationship. It is called nominal because the categories are only differentiated by their names.

We can assign numerical labels to the variable categories, but arithmetic operations, such as, addition, subtraction, multiplication, and division over these numbers are not allowed. The nominal scale only allows some basic arithmetic operations. For instance, we can count the number of elements in each class or apply hypotheses tests regarding the distribution of the population units in the classes. Thus, most of the usual statistics, such as, the mean and standard deviation, do not make any sense for nominal scale qualitative variables.

As examples of nonmetric variables on nominal scales, we can mention professions, religion, color, marital status, geographic location, or country of origin.

Imagine a nonmetric variable related to the country of origin of 10 large multinational companies. To represent the categories of the variable Country of origin, we can use numbers, assigning value 1 to the United States, 2 to the Netherlands, 3 to China, 4 to the United Kingdom, and 5 to Brazil, as shown in Table 2.4. In this case, the numbers are only labels or tags to help identify and classify objects.

This scale of measurement is known as a nominal scale, that is, the numbers are randomly assigned to the object categories, without any kind of order. To represent the behavior of nominal data, we can use descriptive statistics, such as, frequency distribution tables, bar or pie charts, or the calculation of the mode (Chapter 3).

Now, we will discuss how to define labels for qualitative variables on a nominal scale by using the SPSS software (Statistical Package for the Social Sciences). After that, we will be able to construct absolute and relative frequencies tables and charts.

Before generating the dataset, let’s define the characteristics of the variables being studied in Variable View (visualization of variables). In order to do that, click on the respective spreadsheet that is available in the lower left side of the Data Editor, or click twice on the column var.

The first variable, called Company, is a string, that is, its data are inserted as characters or letters. It was established that the maximum number of characters of the respective variable would be 18. In the column Measure, the scale of measurement of the variable Company is defined, which is nominal.

The second variable, called Country, is numerical, since its data are inserted as numbers. However, the numbers are only used to categorize or label the objects, so, the scale of measurement of the respective variable is also nominal (Fig. 2.4).

To insert the data from Table 2.4, we are going to go back to Data View. The information must be typed as shown in Fig. 2.5 (the columns represent the variables and the rows represent the observations or individuals).

Since the variable Country is represented by numbers, it is necessary to assign labels to each variable category, as shown in Table 2.5.

In order to do that, we must click on Data → Define Variable Properties… and select the variable Country, according to Figs. 2.6 and 2.7.

Since the nominal scale of measurement of the variable Country has already been defined in the column Measure in Variable View, we can see that it already appears correctly in Fig. 2.8. Defining the labels for each category must be done at this moment, and it can also be seen in the same figure.

The database starts being seen with the label names assigned, as shown in Fig. 2.9. By clicking on Value Labels, located on the toolbar, it is possible to alternate from the numerical values of the nominal or ordinal variable and their respective labels.

Having structured the dataset, it is possible to construct absolute and relative frequencies tables and charts on SPSS.

The descriptive statistics to represent the behavior of a single qualitative variable and of two qualitative variables will be studied in Chapters 3 and 4, respectively.

2.3.2 Nonmetric Variables—Ordinal Scale

A nonmetric variable on an ordinal scale classifies the units into classes or categories regarding the characteristic being represented, establishing an order between the units of the different categories. An ordinal scale is a scale on which data is shown in order, determining a relative position of the classes according to one direction. Any set of values can be assigned to the variable categories, as long as the order between them is respected.

As in the nominal scale, arithmetic operations (sums, subtractions, multiplications, and divisions) between these values do not make any sense. Thus, the application of the usual descriptive statistics is also limited to nominal variables. Since the scale numbers are only meant to classify them, the descriptive statistics that can be used for ordinal data are frequency distribution tables, charts (including bar and pie charts), and the mode, as will study in Chapter 3.

Examples of ordinal variables include consumers' opinion and satisfaction scales, educational level, social class, age, etc.

Imagine a nonmetric variable called Classification that measures a group of consumers' preference regarding a certain wine brand. The definition of labels for each ordinal variable category can be found in Table 2.6. Value 1 is assigned to the worst classification, value 2 to the second worst, and so on, until value 5, which is the best classification, as shown in this table.

Instead of using scales from 1 to 5, we could have assigned any other numerical scale, as long as the order of classification had been respected. Thus, the numerical values do not represent a score of the product's quality, they are only meant to classify it. So, the difference between these values does not represent the difference of the attribute analyzed. These scales of measurement are known as ordinal scales.

Fig. 2.10 shows the characteristics of the variables being studied in Variable View on SPSS. The variable Customer is a string (its data are inserted as characters or letters) with a nominal scale of measurement. On the other hand, the variable Classification is numerical (numerical values were assigned to represent the variable categories) with an ordinal scale of measurement.

The procedure for defining labels for qualitative variables on an ordinal scale is the same as the one already presented for nominal variables.

2.3.3 Quantitative Variable—Interval Scale

According to Stevens classification (1946), metric or quantitative variables have data in an interval or ratio scale.

Besides ordering the units based on the characteristic being measured, the interval scale has a constant unit of measure. The origin or point zero of this scale of measurement is random, and it does not express an absence of quantity.

A classic example of an interval scale is temperature measured in Celsius (°C) or in Fahrenheit (°F). Choosing temperature zero is random and differences of equal temperatures are determined by the identification of equal expansion volumes in the liquid inside the thermometer. Hence, the interval scale allows us to infer differences between the units to be measured. However, we cannot state that a value in a specific interval of the scale is a multiple of another one. For instance, assume that two objects are measured at 15°C and 30°C, respectively. Measuring the temperature allows us to determine how much one object is hotter than the other. However, we cannot state that the object with 30°C is twice as hot as the other with 15°C.

The interval scale does not vary under positive linear transformations. So, an interval scale can be transformed into another through a positive linear transformation. Transforming degrees Celsius into degrees Fahrenheit is an example of a linear transformation.

Most descriptive statistics can be applied to variable data with an interval scale, except statistics based on the ratio scale, such as, the variation coefficient.

2.3.4 Quantitative Variable—Ratio Scale

Analogous to the interval scale, the ratio scale orders the units based on the characteristic measured and has a constant unit of measure. On the other hand, the origin (or point zero) is unique and value zero expresses the absence of quantity. Therefore, it is possible to know if a value in a specific interval of the scale is a multiple of another.

Equal ratios between values of the scale correspond to equal ratios between the units measured. Thus, ratio scales do not vary under positive proportion transformations. For example, if a unit is 1 m high and the other 3 m, we can say that the latter is three times higher than the former.

Among the scales of measurement, the ratio scale is the most complete, because it allows us to use all arithmetic operations. In addition to this, all the descriptive statistics can be applied to the data of a variable expressed on a ratio scale.

Examples of variables whose data can be on the ratio scale include income, age, how many units of a certain product were manufactured, and distance traveled.

2.4.1 Dichotomous or Binary Variable (Dummy)

A dichotomous or binary variable (dummy) can only take on two categories, and the values 0 or 1 are assigned to these categories. Value 1 is assigned when the characteristic of interest is present in the variable and value 0 if otherwise. As examples, we have: smokers (1) and nonsmokers (0), a developed country (1) and an underdeveloped country (0), vaccinated patients (1) and nonvaccinated patients (0).

Multivariate dependence techniques have as their main objective to specify a model that can explain and predict the behavior of one or more dependent variables through one or more explanatory variables. Many of these techniques, including the simple and multiple regression analysis, binary and multinomial logistic regression, regression for count data, and multilevel modeling, among others, can easily and coherently be applied with the use of nonmetric explanatory variables, as long as they are transformed into binary variables that represent the categories of the original qualitative variable. In this regard, a qualitative variable with n categories, for example, can be represented by (n − 1) binary variables.

For instance, imagine a variable called Evaluation, expressed by the categories good, average, or bad. Thus, two binary variables may be necessary to represent the original variable, depending on the researcher’s objectives, as shown in Table 2.7.

Further details about the definition of dummy variables in confirmatory models will be discussed in Chapter 13, including the presentation of the operations necessary to generate them on software such as Stata.

2.4.2 Polychotomous Variable

A qualitative variable can take on more than two categories and, in this case, it is called polychotomous. As examples, we can mention social classes (lower, middle, and upper) and educational levels (elementary school, high school, college, and graduate school).

2.4.3 Discrete Quantitative Variable

As described in Section 2.2.2, discrete quantitative variables can take on a finite set of values that frequently come from a count, such as, for example, the number of children in a family (0, 1, 2…), the number of senators elected, or the number of cars manufactured in a certain factory.

2.4.4 Continuous Quantitative Variable

Continuous quantitative variables, on the other hand, are those whose possible values are in an interval with real numbers and result from a metric measurement, as, for example, weight, height, or an individual’s salary (Bussab and Morettin, 2011).