# What is a non monotonic relationship

### Non-monotonic logic - Wikipedia Now if you want to distinguish a weak non-monotonic relationship from a strong non-monotonic relationship, we need to get a little bit creative: One option is to. A non-monotonic logic is a formal logic whose consequence relation is not monotonic. In other words, non-monotonic logics are devised to capture and. Request PDF on ResearchGate | Identifying Monotonic and Non-Monotonic Relationships | The basic assumption in regression analysis is that.

Spearman's correlation measures the strength and direction of monotonic association between two variables. Monotonicity is "less restrictive" than that of a linear relationship.

## Detect Non-Linear and Non- Monotonic Relationship between Variables

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. 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.

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• When should you use the Spearman's rank-order correlation?

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.

Using ranks rather than data values yields two new attributes the ranks. Let us illustrate this concept with an example: Consider two datasets X and Y shown in the table in Figure 5, and graphically displayed in Figure 6 a.

## Spearman's Rank-Order Correlation

Plots of a Original values and b Ranks Note that X and Y have a strictly monotonically increasing relationship. The first step is to calculate the rank of each element of X, i. This is shown in the fourth column labelled Rank X in Figure 5. Once all the elements of X have been ranked, the exact procedure is applied to the elements in dataset Y, and the values are shown in the fifth column labelled Rank Y. Observe that although the association between the data points of X and Y is nonlinear, the points on the plot of Rank Y versus Rank X all lie on a straight line as shown in Figure 6 bindicating a perfect correlation.

In fact, if we calculate the Pearson correlation coefficient between the original values of the datasets X and Y using the method described in Part 4we obtain a value of 0.

### Detect Non-Linear and Non- Monotonic Relationship between Variables

But what about the case when the two datasets are not strictly monotonic or have duplicate repeated or tied ranks? Nonlinear relationship The data points in Plot 3 appear to be randomly distributed. They do not fall close to the line indicating a very weak relationship if one exists.

If a relationship between two variables is not linear, the rate of increase or decrease can change as one variable changes, causing a "curved pattern" in the data.

This curved trend might be better modeled by a nonlinear function, such as a quadratic or cubic function, or be transformed to make it linear. Plot 4 shows a strong relationship between two variables. This relationship illustrates why it is important to plot the data in order to explore any relationships that might exist. Monotonic relationship In a monotonic relationship, the variables tend to move in the same relative direction, but not necessarily at a constant rate. In a linear relationship, the variables move in the same direction at a constant rate.