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A monotonic relationship is a type of relationship between two variables in which one variable increases or decreases as the other increases or decreases, without ever reversing direction. This means that the variables either have a positive or negative relationship, but never switch between the two. In a monotonic relationship, there is no change in the rate of the relationship.
In statistics, a monotonic relationship between two variables refers to a scenario where a change in one variable is generally associated with a change in a specific direction in another variable.
There are two types of monotonic relationships:
Positive Monotonic: When the value of one variable increases, the value of the other variable tends to increase as well.
Negative Monotonic: When the value of one variable increases, the value of the other variable tends to decrease.
If two variables don’t generally change in the same direction , then they are said to have a non-monotonic relationship.
Here’s one example of a non-monotonic relationship between two variables:
And here’s another example of a non-monotonic relationship between two variables:
As the value of x increases, the value of y sometimes increases, but sometimes the value of y decreases.
Strictly Monotonic vs. Non-Strictly Monotonic
Two variables are said to have a strictly monotonic relationship if changes in one variable are always associated with a change in the same direction in another variable.
For example, the following chart illustrates a strictly positive monotonic relationship between two variables:
When the value of x increases, the value of y always increases.
When the value of x increases, the value of y always decreases.
How to Quantify Monotonic Relationships
The most common way to quantify the relationship between two variables is to use the , which measures the linear association between two variables.
This coefficient always takes on a value between -1 and 1 where:
- -1 indicates a perfectly negative linear correlation between two variables
- 0 indicates no linear correlation between two variables
- 1 indicates a perfectly positive linear correlation between two variables
The closer the coefficient is to 1, the stronger the positive relationship between two variables. Conversely, the closer the coefficient is to -1, the stronger the negative relationship between two variables.
However, if the relationship between two variables is monotonic but non-linear (like an exponential relationship) then it’s a good idea to use the , which was designed to handle monotonic relationships well.
No matter which type of correlation coefficient you calculate, it’s always a good idea to create a scatterplot to visualize the relationship between the variables as well.