# Monotonic relationship between two variables examples

The strength and direction of a monotonic relationship between two variables can be measured by the Spearman Rank-Order Correlation. Correlation: when two variables move together in some fashion. Correlations measure with a straight line. Common monotonic relationships, but not linear: .. Example: Suppose a hotel manager surveys guest who indicate they will not. Now if you want to distinguish a weak non-monotonic relationship from a strong non-monotonic relationship, A small example for illustration.

Part 5 By Neep Hazarika August 16, Review of Previous Parts In Part 1we described how two datasets can have strong, weak or no correlation depending on how the points in the resultant scatter plot are located relative to one another along a straight line. We looked at how positive or negative correlations occur when an increase in the value of one variable X leads to a corresponding increase or decrease respectively in the value in the other variable Y.

In Part 4we showed how the Pearson correlation coefficient is calculated, and how it is used to identify and test the strength of the linear relationship between two sets of data.

All of the above examples however, assume a linear relationship between the two variables. In this part, we will focus on datasets that may exhibit a nonlinear association between the two variables. Correlation It is important to note that, although the terms association and correlation are frequently used interchangeably, this is not, strictly speaking, an accurate description.

An association exists between two variables if knowledge of the value of one variable provides us with an assessment about the probable value of the other.

### Linear, nonlinear, and monotonic relationships - Minitab Express

There is correlation between the variables if the association is linear; this can be represented by a straight line on a scatter plot. This situation is illustrated in Figure 1. Correlation between two variables X and Y Now let us consider the case shown in Figure 2.

In the linear case, the correlation coefficient gives a measure of the strength of the association; the closer the points lie along a straight line, the stronger is the correlation. Nonlinear Association between two variables X and Y Monotonic and Non-Monotonic Associations A linear association between variables implies that the relationship can be modelled best by a straight line. This is illustrated in Figure 3.

## Spearman's Rank-Order Correlation using SPSS Statistics

Different Types of Association The datasets of Figure 3 a and 3 bwhere the points are distributed more closely around a straight line, will in general exhibit a higher degree of correlation than the dataset shown in Figure 3 cwhere the points are more widely scattered. The dataset in Figure 3 a is said to be monotonically increasing, i.

Figure 3 b shows a monotonically decreasing association between X and Y, i. Lastly, the association shown in Figure 3 c is non-monotonic — as X increases, Y sometimes increases and sometimes decreases.

A monotonic relationship is thus one in which Y moves in only one direction either up or down as X increases, but the relationship is not necessarily but can be linear.

A monotonic association is more general than correlation, since correlation implies an underlying linear trend, while, for example, the exponential and logarithmic functions are also monotonically increasing functions but are nonlinear in nature.

Nevertheless, it is still possible to obtain positive correlations from them. Figures 1 and 2 are also examples of monotonic relationships. We will recall from Part 1 that perfect correlations occur when all the points on the scatter plot lie on a straight line. We designate this association perfect because an increase in X always leads to a corresponding increase Figure 4 a or decrease Figure 4 b in variable Y.

It is therefore natural to ask if there is a simple way of assessing whether a strong non-linear association between two variables can be transformed into a linear relationship, so that the correlation coefficient can be calculated. Fortunately, a statistician named Charles Spearman developed a clever method to achieve this.

Spearman Rank Correlation Spearman used the ranks of the items rather than their actual values. There are many scenarios when the only data available are in the form of ranks, as for example when judges rank competitors in order, 1st, 2nd, 3rdetc. Join the 10,s of students, academics and professionals who rely on Laerd Statistics. Spearman's correlation measures the strength and direction of monotonic association between two variables.

Monotonicity is "less restrictive" than that of a linear relationship. For example, the middle image above shows a relationship that is monotonic, but not linear.

A monotonic relationship is not strictly an assumption of Spearman's correlation. That is, you can run a Spearman's correlation on a non-monotonic relationship to determine if there is a monotonic component to the association.

- Spearman's Rank-Order Correlation
- Majestic Beginners Guide to Correlation: Part 5
- Linear, nonlinear, and monotonic relationships

However, you would normally pick a measure of association, such as Spearman's correlation, that fits the pattern of the observed data. That is, if a scatterplot shows that the relationship between your two variables looks monotonic you would run a Spearman's correlation because this will then measure the strength and direction of this monotonic relationship.

On the other hand if, for example, the relationship appears linear assessed via scatterplot you would run a Pearson's correlation because this will measure the strength and direction of any linear relationship.

You will not always be able to visually check whether you have a monotonic relationship, so in this case, you might run a Spearman's correlation anyway.

How to rank data?