Making Inferences About the Slope

Making Inferences About the Slope

The slope of the regression line describes how changes in the variables are related. It is important to interpret the slope of the line in the context of the situation represented by the data. You should be able to write a sentence interpreting the slope in plain English.

The slope of the best fit line tells us how the dependent variable $y$ changes for every one unit increase in the independent variable $x$, on average.

Remember the equation for a line is:

$y = mx+b$

where $y$ is the dependent variable, $x$ is the independent variable, $m$ is the slope, and $b$ is the intercept.

A fitted linear regression model can be used to identify the relationship between a single predictor variable, $x$, and the response variable, $y$, when all the other predictor variables in the model are "held fixed". Specifically, the interpretation of $m$ is the expected change in $y$ for a one-unit change in $x$ when the other covariates are held fixed—that is, the expected value of the partial derivative of $y$ with respect to $x$. This is sometimes called the unique effect of $x$ on $y$. In contrast, the marginal effect of $x$ on $y$ can be assessed using a correlation coefficient or simple linear regression model relating $x$ to $y$; this effect is the total derivative of $y$ with respect to $x$.

Care must be taken when interpreting regression results, as some of the regressors may not allow for marginal changes (such as dummy variables, or the intercept term), while others cannot be held fixed.

It is possible that the unique effect can be nearly zero even when the marginal effect is large. This may imply that some other covariate captures all the information in $x$, so that once that variable is in the model, there is no contribution of $x$ to the variation in $y$. Conversely, the unique effect of $x$ can be large while its marginal effect is nearly zero. This would happen if the other covariates explained a great deal of the variation of $y$, but they mainly explain said variation in a way that is complementary to what is captured by $x$. In this case, including the other variables in the model reduces the part of the variability of $y$ that is unrelated to $x$, thereby strengthening the apparent relationship with $x$.

The meaning of the expression "held fixed" may depend on how the values of the predictor variables arise. If the experimenter directly sets the values of the predictor variables according to a study design, the comparisons of interest may literally correspond to comparisons among units whose predictor variables have been "held fixed" by the experimenter. Alternatively, the expression "held fixed" can refer to a selection that takes place in the context of data analysis. In this case, we "hold a variable fixed" by restricting our attention to the subsets of the data that happen to have a common value for the given predictor variable. This is the only interpretation of "held fixed" that can be used in an observational study.

The notion of a "unique effect" is appealing when studying a complex system where multiple interrelated components influence the response variable. In some cases, it can literally be interpreted as the causal effect of an intervention that is linked to the value of a predictor variable. However, it has been argued that in many cases multiple regression analysis fails to clarify the relationships between the predictor variables and the response variables when the predictors are correlated with each other and are not assigned following a study design.