Monotonic function - Wikipedia
Relationships can be monotonic or non-monotonic. A monotonic relationship is one where the relationship is either positive or negative at all. In mathematics, a monotonic function(or monotone function) is a function between ordered sets . called a "positive monotonic transformation", in order to distinguish it from a . Furthermore, the strict relations are of little use in many non-total orders and hence no additional terminology is introduced for them. Monotonic Functions .. This relation is valid for any x0∈(a,b). .. Hence it is clear that the quadratic expression is always positive (it is equal to zero only if.
Although you would normally hope to use a Pearson product-moment correlation on interval or ratio data, the Spearman correlation can be used when the assumptions of the Pearson correlation are markedly violated.
However, Spearman's correlation determines the strength and direction of the monotonic relationship between your two variables rather than the strength and direction of the linear relationship between your two variables, which is what Pearson's correlation determines.
What is a monotonic relationship? A monotonic relationship is a relationship that does one of the following: Examples of monotonic and non-monotonic relationships are presented in the diagram below: Join the 10,s of students, academics and professionals who rely on Laerd Statistics.
Linear, nonlinear, and monotonic relationships
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.
For example, the relationship between height and weight in humans is nonlinear, because doubling height usually more than doubles weight. For example, a child may be three feet tall and weigh 50 pounds, but probably no six-foot tall adult weighs only pounds.
A monotonic relationship is one where the relationship is either positive or negative at all levels of the variables.
Linear, nonlinear, and monotonic relationships - Minitab Express
A non-monotonic relationship is one where this is not so. All of the examples above were monotonic. An example of a non-monotonic relationship is that between stress and performance.Math 1190 Ch 4.3 (Mean Value Theorem and Monotone Functions)
People with a moderate amount of stress perform better than those with very little stress or those that have a great deal of stress.
Strong and Weak Relationships A relationship between two variables may be strong or weak. If the relationship is strong, it means that a relatively simple mathematical formula for the relationship fits the data very well.
If the relationship is weak, then this is not so. For example, the relationship between the amount of paint and the size of wall is very strong. The relationship between height and weight is weaker.
References "The inverted U relationship between stress and performance"; Mina Westman et al. He has been writing for many years and has been published in many academic journals in fields such as psychology, drug addiction, epidemiology and others.