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## 1.2 Scales of information

Whenever we work with information, we need to understand how to measure it properly. If we cannot do that, then we cannot make proper decisions, supported by some evidence. For example, if a person feels ill but we cannot say what the temperature of their body is, then we cannot decide, whether anything needs to be done to reduce it. If we can measure something then we can model it and produce forecasts. Continuing our example, if the temperature is 39°C, then we can conclude that the person is sick and needs to take Paracetamol or some other pills that would reduce the temperature. So, whenever we collect some sort of information about a system’s behaviour or about a process, we will inevitably deal with scales of information and it is important to understand what we are dealing with in order to process that information correctly. There are four fundamental scales:

- Nominal,
- Ordinal,
- Interval,
- Ratio.

The first two form the so called “categorical” scale, while the latter two are typically united in the “numerical” one. Each one of these scales can have one of the following characteristics:

- Description,
- Order,
- Distance,
- Natural zero,
- Natural unit.

The last characteristics is typically ignored analytics and forecasting as it does not provide any useful information. But as for the other four, they provide important properties to the scales of information, giving them more flexibility. Here we discuss the scales in detail.

### 1.2.1 Nominal scale

This is the scale that only has “description” characteristics. It does not have an order, a distance or a natural zero. There is only one operation that can be done in this scale, and it is comparison, whether the value is “equal” or “not equal” to something. An example of data measured in such scale is the following question in a survey:

What is your nationality?

- Russian,
- English,
- Greek,
- Swiss,
- Belgian,
- Lebanese,
- Indonesian,
- Other.

In this case after collecting the data we can only say whether each respondent is Russian or not, English or not etc. So, the only thing that can be done with the data measured in this scale is to produce a basic summary, showing how many people selected one option or another. Among the statistical instruments, *only the mode* is useful, as it shows which of the options was selected the most. If there are several variables measured in nominal scale, we can calculate some measures of association to see if there are any patterns in respondents behaviour (e.g. those who select “Russian” would prefer Vodka, while those who selected “Belgian” will tend to drink “Beer”).

When it comes to constructing models, the nominal scale is typically transformed in a set of dummy variables, which will be discussed later in regression analysis of this textbook.

If you are not sure, whether your data is measured in nominal or another scale, you can do a simple test: if changing the places of two values does not break the scale, then this is the nominal one. For example, in the question above, moving “Greek” to the first place will not change anything, so this is indeed the nominal scale. Another example of nominal scale is the number on the T-shirt of football players. They are only descriptive, and if two players change numbers, this will not change anything (although it might confuse football fans).

### 1.2.2 Ordinal scale

In addition to description, the ordinal scale has the “order.” It is possible to say that one value can be placed higher or lower than the other on a scale (thus, permitting operations “greater” and “smaller” in addition to the “equal” and “not equal”) However, it is not possible to say how far the elements are placed from each other, so the number of operations in the scale is still limited. Here is an example of a survey question with such scale:

How old are you?

- Too young,
- Young,
- Not too young,
- Not too old
- Old,
- Too old.

In this scale above we have a natural order, and when collecting the data in this scale we can conclude, whether a respondent is older than another one or not. Sometimes ordinal scales look confusing and seem to be of a higher level than they are, here is an example:

How old are you?

- Younger than 16,
- 16 - 25,
- 26 - 40,
- 41 - 60
- Older than 60.

This is still an ordinal scale, because it has the natural order, and because we cannot measure the distance between the value: if, for example we subtract “16 - 25” from “26 - 40,” we will not get anything meaningful.

The ordinal scale, being more complex than the nominal one, allows using some additional statistical instruments (besides the mode), such as quantiles of distribution, including median. Unfortunately, the *arithmetic mean is not applicable to the data in ordinal scale*, because of the absence of distance. Even if you encode every answer in numbers, the resulting average will not be meaningful. Indeed, if in the question above with the five options, we use the numbers (“1” for the first option, “2” for the second one, etc.) and take average, the resulting number of, for example, 3.75 will not mean anything, as there is no element in the scale that would correspond to that number.

When it comes to measuring relations between two variables in ordinal scale, we can use Kendall’s \(\tau\) correlation coefficient, Goodman-Kruskal’s \(\gamma\) and Yule’s Q. These are discussed in detail in Section 2.6. As for using the variables in ordinal scale in modelling, the typical thing to do would be to create a set of dummy variables, similarly to how it is done for variables in nominals scale.

As for the identification of scale, if in doubt, you can do any transformation of elements of scale without the loss of its meaning. For example, if we assign numbers from 1 to 5 to the responses above, we can square each one of them and get 1, 4, 9, 16 and 25, which would not change the original scale, but only encode the answers differently (select “16” for the option “41 - 60”).

### 1.2.3 Interval scale

This scale is even more complex than the previous two, as in addition to description and order it also has a distance. This permits doing addition and subtraction to the elements of scale, which are meaningful operations in this case. Arithmetic mean and standard deviation become available in this scale in addition to all those used in lower level scales discussed above. The classical example of a variable measured in this scale is the temperature. Indeed, we can not only say if the temperature of one person is higher than the temperature of the other one, but also by how much: 39°C - 37°C = 2°C, which is a meaningful number in the scale. The only limitation in this scale is that there is no natural zero. 0°C does not mean the absence of temperature, but rather means the point at which water starts freezing. If we switch to Fahrenheit (although why would anyone do that?!), then the 0°F would correspond to the point, where the mixture of ice, water, and ammonium chloride used to stabilise back in 1724, when Fahrenheit proposed the scale.

The relation between two variables in interval scale can be measured by Pearson’s correlation coefficient. The scale can be used in the model as is, although some error metrics cannot be used for the measurement of accuracy of models for this scale (for example, MAPE cannot be used as it assumes meaningful zero).

Finally, when it comes to the identification of scale, only linear transformations are permitted for the variables without the loss of its properties. This means that if we measure temperature of two respondents and then do their linear transformations via \(y=a+bx\), then the characteristics of scale will not be broken: it will still have description, order and distance with the same meaning as prior to the transformation. In the example of temperature, this is how you switch, for example, from Celsius to Fahrenheit (\(y=a+bx\)).

### 1.2.4 Ratio scale

The most complex of the four, this ratio has a natural zero (in addition to all the other characteristics). It permits any operations to the values of scale, including product, division, and non-linear transformations. Coefficient of variation can be used together in addition to all the previous instruments. An example of the information measured in this scale is the height of respondents in meters. You can compare two respondents via their height and not only say that one is higher than the other, but also by how much and how many times. All these operations will be meaningful in this scale.

If you need to check, whether the variables is indeed in ratio scale, note that only the transformation via multiplication would maintain the meaning of the scale. For example, height measured in meters can be transformed into height in feet via the multiplication by approximately 3.28. If you add a constant to the values of scale, it will break it.

Being the most complex, this scale permits usage of all correlation coefficients and all error metrics.

Finally, the variables measured in this scale can be either integer or continuous. This might cause some confusions, because the integer numbers sometimes look suspiciously similar to the values of ordinal scale, but the tools of identification discussed above might help. If a company needs to buy 7 planes, then this is an integer variable measured in ratio scale: 7 planes is more than 6 planes by one plane, and zero planes means that there are no planes (all the characteristics of ratio scale). Furthermore, squaring the number of planes breaks the distance between them (\(7^2 - 6^2 \neq 1^2\)), while linear transformation breaks the scale (\(7\times 2 + 3\) has a completely different meaning in the scale than just 7).