methods require that everything is measured. It is of great importance to select and identify the scale used for the measurement of each study variable, or attribute, because the scale determines the statistical methods that will be used for the analysis. There are only four scales of measurement.
The simplest scale is the binary scale, which has only two values. Patient sex (female, male) is an example of an attribute measured in a binary scale. Everything that has a yes/no answer (e.g. obesity, previous myocardial infarction, family history of hypertension, etc.) was measured in a binary scale. Very often the values of a binary scale are not numbers but terms, and this is why the binary scale is also a nominal scale. However, the values of any binary attribute can readily be converted to 0 and 1. For example, the attribute sex, with values female and male, can be converted to the attribute female sex with values 0 meaning no and 1 meaning yes.
Next in complexity is the categorical scale. This is simply a nominal scale with more than two values. In common with the binary scale, the values in the categorical scale are usually terms, not numbers, and the order of those terms is arbitrary: the first term in the list of values is not necessarily smaller than the second. Arithmetic operations with categorical scales are meaningless, even if the values are numeric. Examples of attributes measured on a categorical scale are profession, ethnicity, and blood type.
It is important to note that in a given person an attribute can have only a single value. However, sometimes we see categorical attributes that seem to take several values for the same person. Consider, for example, an attribute called cardiovascular risk factors with values arterial hypertension, hypercholesterolemia, diabetes mellitus, obesity, and tabagism. Obviously, a person can have more than one risk factor and this attribute is called a multi‐valued attribute. This attribute, however, is just a compact presentation of a set of related attributes grouped under a heading, which is commonly used in data forms. For analysis, these attributes must be converted into binary attributes. In the example, cardiovascular risk factors is the heading, while arterial hypertension, hypercholesterolemia, diabetes mellitus, obesity, and tabagism are binary variables that take the values 0 and 1.
When values can be ordered, we have an ordinal scale. An ordinal scale may have any number of values, the values may be terms or numbers, and the values must have a natural order. An example of an ordinal scale is the staging of a tumor (stage I, II, III, IV). There is a natural order of the values, since stage II is more invasive than stage I and less than stage III. However, one cannot say that the difference, either biological or clinical, between stage I and stage II is larger or smaller than the difference between stage II and stage III. In ordinal scales, arithmetic operations are meaningless.
Attributes measured in ordinal scales are often found in clinical research. Figure 1.2 shows three examples of ordinal scales: the item list, where the subjects select the item that more closely corresponds to their opinion, the Likert scale, where the subjects read a statement and indicate their degree of agreement, and the visual analog scale, where the subjects mark on a 100 mm line the point that they feel corresponds to their assessment of their current state. Psychometric, behavioral, quality of life, and, in general, many questionnaires commonly used in clinical research have an ordinal scale of measurement.
When the values are numeric, ordered, and the difference between consecutive values is the same all along the scale, that is an interval scale. Interval scales are very common in research and in everyday life. Examples of attributes measured in interval scales are age, height, blood pressure, and most clinical laboratory results. Some interval‐measured attributes are continuous, for example, height, while others are not continuous and they are called discrete. Examples of discrete attributes are counts, like the leukocyte count. Because all values in the scale are at the same distance from each other, we can perform arithmetic operations on them. We can say that the difference between, say, 17 and 25 is of the same size as that between, say, 136 and 144. In practical data analysis, however, we often make no distinction between continuous and discrete variables.
Figure 1.2 Examples of commonly used ordinal scales.
Figure 1.3 Difference between an ordinal and an interval scale.
Figure 1.3 illustrates the difference between ordinal and interval scales: in ordinal scales, consecutive values are not necessarily equidistant all over the range of values of the scale, while that is always true in interval scales. One easy way to differentiate ordinal from interval scales is that interval scales almost always have units, like cm, kg, l, mg/dl, mmol/l, etc. while ordinal scales do not.
If an interval scale has a meaningful zero, it is called a ratio scale. Examples of ratio scales are height and weight. An example of an interval scale that is not a ratio scale is the Celsius scale, where zero does not represent the absence of temperature, but rather the value that was by convention given to the temperature of thawing ice. In ratio scales, not only are sums and subtractions possible, but multiplications and divisions as well. The latter two operations are meaningless in non‐ratio scales. For example, we can say that a weight of 21 g is half of 42 g, and a height of 81 cm is three times 27 cm, but we cannot say that a temperature of 40°C is twice as warm as 20°C. With very rare exceptions, all interval‐scaled attributes that are found in research are measured in ratio scales.
1.3 Central Tendency Measures
We said above that one important purpose of biostatistics is to determine the characteristics of a population in order to be able to make predictions on any subject belonging to that population. In other words, what we want to know about the population is the expected value of the various attributes present in the elements of the population, because this is the value we will use to predict the value of each of those attributes for any member of that population. Alternatively, depending on the primary aim of the research, we may consider that biological attributes have a certain, unknown value that we attempt to measure in order to discover what it is. However, an attribute may, and usually does, express variability because of the influence of a number of factors, including measurement error. Therefore, the mean value of the attribute may be seen as the true value of an attribute, and its variability as a sign of the presence of factors of variation influencing that attribute.
There are several possibilities for expressing the expected value of an attribute, which are collectively called central tendency measures, and the ones most used are the mean, the median, and the mode.
The mean is a very common measure of central tendency. We use the notion of mean extensively in everyday life, so it is not surprising that the mean plays an extremely important role in statistics. Furthermore, being a sum of values, the mean is a mathematical quantity and therefore amenable to mathematical processing. This is the other reason why it is such a popular measure in statistics.
As a measure of central tendency, however, the mean is adequate only when the values are symmetrically distributed about its value. This is not the case with a number of attributes we study in biology and medicine – they often have a large number of small values and few very large values. In this
