**MATHEMATICS IN MEDICINE**, and kindly contributed to R-bloggers)

# Clarifying difference between Ratio and Interval Scale of Measurement

## Introduction

Recently while preparing lecture on scales of measurements and types of statistical data, I came across two scales of measurement when numbers are used to denote a quantitative variable. I took some time to clarify the difference between “Interval and "Ratio” scales of measurements. I am writing down what I understand of the above mentioned scales.

## Process of measuring a variable

First step in variable measurement is to understand the concept we want to measure, i.e., we would like to define the variable on a conceptual level. Then we need to make an operational definition of the variable, which includes the following steps:

Setting up of a domain of all the possible values the variable can assume.

Understanding the meaning of different values the variable can assume.

- Different values can just mean different classes (categories) - Nominal scale
- Different values can mean different classes (categories) with some ordering (direction of difference) between the classes - Ordinal scale
- Different values can mean different classes (categories) with ordering and specified distance between them (direction and magnitude/distance of difference) - Interval and Ratio scale

Checking if a real origin (“0”) exists for the variable in the particular scale. Origin (“0”) should mean absolute absence of the variable.

Designing a device which will measure the variable.

Validating the measurement from the device.

## Prerequisites for Ratio Scale

There are two prerequisites for a measurement scale to be a Ratio Scale:

- Presence of “Real Origin”.
- Scale is uniformly spaced across its full domain.

## What happens when the above prerequisites are met?

Let us assume that we have made numerical observations \( A_{ratio} \) and \( B_{ratio} \) for a variable in ratio scale and that \( B_{ratio} > A_{ratio} \). There are **two valid ways** to denote the difference between A and B:

**Arithmetic difference between \( A_{ratio} \) and \( B_{ratio} \):**It is denoted by \( B_{ratio} - A_{ratio} \). It is a valid measure of difference because of the fact that the scale is uniformly spaced across the domain.**Ratio difference between \( A_{ratio} \) and \( B_{ratio} \):**It is denoted by \( B_{ratio}/A_{ratio} \). It indicates that \( B_{ratio} \) is \( B_{ratio}/A_{ratio} \) times larger than \( A_{ratio} \). We say this as a valid measure of difference because the origin is an absolute one and is same for both observations. Note that there is no unit as the result is a ratio. It is also equivalent to arithmetic difference of log transformation of observations, \( log(B_{ratio}) - log(A_{ratio}) \).**Location transformation:**If we shift the observations by \( x \) units, we get \( Ax_{ratio} = A_{ratio} + x \) and \( Bx_{ratio} = B_{ratio} + x \). Arithmetic difference between the two transformed observations, \( Bx_{ratio} - Ax_{ratio} = B_{ratio} - A_{ratio} \), which is the same as original observations.**Scale transformation:**If we multiply each of the observations by \( x \) units, we get \( Ax_{ratio} = A_{ratio} \cdot x \) and \( Bx_{ratio} = B_{ratio} \cdot x \). Ratio difference between the two transformed observations, \( Bx_{ratio}/Ax_{ratio} = B_{ratio}/A_{ratio} \), which is the same as original observations.

So, for ratio scale, *both arithmetic and ratio difference are valid measures of difference between observations and the difference remain same after both location and scale transformations.*

## General transformation of measuring scale

Any transformation (\( X_{trans} \)) of the original ratio scale, say \( X_{ratio} \) can be depicted as follows

\[ X_{ratio} = f(X_{trans},S(X_{trans}),L(X_{trans})) \]

where, \( S(X_{trans}) \) denotes **scale transformation parameter** as a function wrt location in transformed scale and \( L(X_{trans}) \) denotes **location transformation parameter** as a function wrt location in transformed scale.

If we assume constant \( S \) and \( L \) wrt location in transformed scale, one of the simplest scale transformation will be:

\[ X_{ratio} = (X_{trans} + L) \cdot S \]

where, \( S \neq 0 \)

and interval scale of measurement (\( X_{int} \)) will be the one with \( L \neq 0 \) in addition to the above constraints.

## What happens in Interval Scale?

In interval scale, the zero doesnot mean absolute nothingness, but it is an arbitrarily chosen one and corresponds to a distance of \( L \) from the real origin in ratio scale.

We continue our example from the above section:

Let us say that we make two observations in interval scale, \( A_{int} \) and \( B_{int} \), and want to assess difference between both the observations as done earlier.

Observation \( A_{int} \) will be mapped as \( (A_{int} + L) \cdot S \) and observation \( B_{int} \) will be mapped as \( (B_{int} + L) \cdot S \) in ratio scale. We will have to use values in ratio scale for comparision, as it has got “real origin”.

**Arithmetic difference between \( A_{int} \) and \( B_{int} \):**It shows that the arithmetic difference measured in interval scale is linearly related to the difference measured in ratio scale and that the arithmetic difference in ratio scale is independent of absolute values of \( A_{int} \) and \( B_{int} \). Moreover, interval scale is uniformly spaced across the full domain. Because of the above reasons, arithmetic difference measured in interval scale is a valid way of representing difference between observations \( A_{int} \) and \( B_{int} \). \[ [B_{int} - A_{int}]_{int\ scale} = [(B_{int} + L) \cdot S - (A_{int} + L) \cdot S]_{ratio\ scale} = [(B_{int} - A_{int}) \cdot S]_{ratio\ scale} \]**Ratio difference between \( A_{int} \) and \( B_{int} \):**Unlike the arithmetic difference, ratio difference in interval scale is dependent on the absolute values of \( A_{int} \) and \( B_{int} \), with ratio approaching \( B_{int}/A_{int} \) with \( B_{int}, A_{int} >> L \). So, ratio difference is not a valid measure of difference between two observations in interval scale. \[ [B_{int}/A_{int}]_{int\ scale} = [(B_{int} + L) \cdot S/(A_{int} + L) \cdot S]_{ratio\ scale} = [(B_{int} + L)/(A_{int} + L)]_{ratio\ scale} \]**Location transformation:**If we shift the observations by x units on interval scale, we get \( Ax_{int} = ((A_{int} + x) + L) \cdot S \) and \( Bx_{int} = ((B_{int} + x) + L) \times S \). Arithmetic difference between the two transformed observations, \( [Bx_{int} - Ax_{int}]_{int\ scale} = [(B_{int} - A_{int}) \cdot S]_{ratio\ scale} = [B_{int} - A_{int}]_{int\ scale} \), remains the same as original observations on interval scale. So, arithmetic difference remains same with location transformation.**Scale transformation:**If we multiply each of the observations by x units on original scale, we get \( Ax_{int} = (A_{int} \cdot x + L) \cdot S \) and \( Bx_{int} = (B_{int} \cdot x + L) \cdot S \). Ratio difference between the two transformaed observations, \( [Bx_{int}/Ax_{int}]_{int\ scale} = [(B_{int} \cdot x + L)/(A_{int} \cdot x + L)]_{ratio\ scale} \neq [B_{int}/A_{int}]_{int\ scale} \). With scale transformation, the ratio difference becomes different from the the ratio difference of the original observations in interval scale.

So, for interval scale, *only arithmetic difference is a valid measure of difference between observations.*

## Conclusion

The aim of this post is to express what I understand of the interval and ratio scale of measurements. Comments, suggestions and criticisms are welcome.

Bye.

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