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Concept Version 11
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Making Inferences About the Slope

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

Learning Objective

  • Infer how variables are related based on the slope of a regression line


Key Points

    • It is important to interpret the slope of the line in the context of the situation represented by the data.
    • A fitted linear regression model can be used to identify the relationship between a single predictor variable xxx and the response variable yyy when all the other predictor variables in the model are "held fixed".
    • The interpretation of mmm (slope) is the expected change in yyy for a one-unit change in xxx when the other covariates are held fixed.

Terms

  • covariate

    a variable that is possibly predictive of the outcome under study

  • intercept

    the coordinate of the point at which a curve intersects an axis

  • slope

    the ratio of the vertical and horizontal distances between two points on a line; zero if the line is horizontal, undefined if it is vertical.


Full Text

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 yyy changes for every one unit increase in the independent variable xxx, on average.

Remember the equation for a line is:

y=mx+by = mx+by=mx+b

where yyy is the dependent variable, xxx is the independent variable, mmm is the slope, and bbb is the intercept.

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

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 xxx, so that once that variable is in the model, there is no contribution of xxx to the variation in yyy. Conversely, the unique effect of xxx 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 yyy, but they mainly explain said variation in a way that is complementary to what is captured by xxx. In this case, including the other variables in the model reduces the part of the variability of yyy that is unrelated to xxx, thereby strengthening the apparent relationship with xxx.

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.

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