Slope

The equation for a straight line is Y = A + BX, where Y is the dependent variable, A is the y-intercept, B is the slope, and X is the independent variable. The slope is calculated by taking any two points on the line and dividing the rise by the run, where the run is the increase in X between the two points and the rise is the increase (or decrease) in Y between the two points (see Figure 1). A positive relationship between the two variables is indicated by a positive slope; a negative relationship is indicated by a negative slope.

When regression analysis is used in the social sciences, the data never fit perfectly along a straight line, meaning that we have error in prediction. The usual simple regression equation is Y = A + BX + E, where E represents the error associated with predicting Y given X. The slope is now interpreted as the average rate by which the dependent variable changes with a 1-unit change in the independent variable. In simple regression (i.e., where there is only one independent variable), the slope can tell us the marginal relationship between income and education. By way of an example, assume that we regress income (measured in dollars) on education (measured in years of schooling), obtaining a slope of 5000. We would argue, then, that an increase of one year of education is associated with, on average, an increase in income of $5,000.

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In multiple regression analysis (i.e., when there are several independent variables), the individual slope estimates for each independent variable pertain to partial relationships. In other words, each slope refers to the effect of a single independent variable on the dependent variable controlling for all other independent variables in the model. For example, assume that we regress income (again measured in dollars) on years of education and gender, obtaining a slope for education of 4000. The interpretation of the slope would then be the following: Holding gender constant, with every additional year of education, income increases, on average, by $4,000.

Reporting the slope coefficients is usually sufficient when the functional form between two quantitative variables is linear because it is easily interpreted. If the functional form of the relationship between two variables is not linear (i.e., the slope of the regression line is not constant through the range of the independent variable), however, a single slope coefficient cannot summarize the relationship. In such cases, several slope estimates are needed, and graphs are usually required in order to fully comprehend the relationship. For example, in polynomial regression (see polynomial equation), the independent variable has one slope coefficient more than the number of changes in the direction of the line. In nonparametric regression (e.g., local regression), there are no slope coefficients, and the relationship can be displayed only in a graph.

The concept of slope can also be generalized to extensions of the linear model, such as generalized linear models and multilevel analysis. In nonlinear models such as logistic regression, slopes are usually best interpreted by calculating fitted probabilities.

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