Measures of association between variables are a way of summarizing the strength of the relationship between two or more variables. Two variables are said to be associated when information about one variable can help predict the value of the other. In addition to the strength, some measures of association can also reveal the direction of the relationship between variables.

The terms correlation and association are often used synonymously, but some authors distinguish between them. According to them, association is a broader concept and includes any type of relationship between variables, while correlation usually refers only to a linear relationship between variables. Many authors classify the analysis of the association of variables under descriptive statistics, rather than inferential statistics.

Types of association between variables

Measures of association can be linear and nonlinear. Linear association denotes a type of association between variables in which the value of one variable is expressed as another variable multiplied by its coefficient (slope), plus a constant that is added to this product. Mathematically, a linear relationship is represented as follows:

Y = a + b*X

where X and Y are the variables whose relationship is being observed, a is a constant, and b is the coefficient on X.

A linear relationship is graphically represented by a straight line. Nonlinear association denotes such a connection between variables in which the relationship between the variables cannot be reduced to a linear relationship, and when graphically represented, it can take the form of a curve of various shapes.

The intensity of the connection refers to the degree to which a change in one variable is associated with a change in the other variable. The general rule is that complete connection exists when a change in one variable follows a change in the other variable to the same extent, while complete independence (absence of association) exists when a change in the value of one variable does not follow the value of the other variable at all. Such extremes are very rare in practice. Perfect association (±1) is theoretically possible only if both variables share identical types of underlying distributions. It is common for the connection to be between these two extremes, so we speak of weak, medium, strong and very strong connections between variables. These categorizations are largely arbitrary and serve to facilitate the interpretation of the relationships that exist between variables.

The relationship between two variables can be positively or negatively associated (also called a positive or negative correlation, respectively). A positive correlation between two variables exists when an increase in the value of one variable on average leads to a higher value of the other variable. The opposite is true for a negative correlation  - an increase in the value of one variable on average causes lower values ​​of the other variable. This characteristic of the relationship is called the direction of the relationship.

A distinction should be made between symmetrical and asymmetrical types of relationship. In a symmetric type of relationship, a change in the value of the variable X follows a change in the value of the variable Y and vice versa, a change in the value of the variable Y follows a change in the value of the variable X. An asymmetric type of relationship denotes a relationship in which a change in the value of one variable affects a change in the value of the other variable, but the reverse is not true. Asymmetric relationships designate models where one variable is treated as a predictor and the other as an outcome, though this directional mapping does not inherently prove a causal mechanism.

When the relationship between two variables changes across different ranges of a variable, it is modeled using piecewise regression or non-linear functions, rather than a single relationship holding across the entire domain.

Methods for determining the association of variables

Measures of association can be determined by various types of analysis, all of which are built on the concept of proportional error reduction. Methods for detecting association can be based on a graphical representation of the relationship or some type of coefficient. The most commonly used in practice are a scatter plot and a correlation coefficient.

A scatter plot allows you to quickly see the association between variables, because it visually displays the strength and direction of the relationship, but does not provide precise numerical indicators of that association. On the other hand, coefficients are more precise in quantifying the relationship, but they express an average, and their interpretation is more complex. Therefore, these two methods are often combined, by first drawing a scatter diagram, and then proceeding to a more precise analysis using one of the coefficients. The choice of method for identifying the relationship also depends on the type of variables being examined.

It is possible to use other methods to determine the relationship between variables. When a study includes multiple variables, in order to analyze their association, one of the first steps in the study is to construct a correlation coefficient matrix that contains a coherent representation of all pairs of relationships between variables in the study, and includes information about the correlation coefficient and its statistical significance. Regression analysis allows for the calculation of multiple correlation coefficients and partial correlation coefficients that allow for the identification of associations between more than two variables. To determine the association can also be used χ2 (Chi-square) test of independence, relative risk-odds ratio, etc.