8 Estimating and Making Inferences About the Slope

The purpose of a multiple regression is to find an equation that best predicts the YY variable as a linear function of the XX variables.

Learning Objectives

After completion of this session, you will be able to:

  • Discuss how partial regression coefficients (slopes) allow us to predict the value of YY given measured XX values.

Key Takeaways

Key Points

  • Partial regression coefficients (the slopes ) and the intercept are found when creating an equation of regression so that they minimize the squared deviations between the expected and observed values of YY.
  • If you had the partial regression coefficients and measured the XXvariables, you could plug them into the equation and predict the corresponding value of YY.
  • The standard partial regression coefficient is the number of standard deviations that YYwould change for every one standard deviation change in X1X1, if all the other XX variables could be kept constant.

You use multiple regression when you have three or more measurement variables. One of the measurement variables is the dependent (YY) variable. The rest of the variables are the independent (XX) variables. The purpose of a multiple regression is to find an equation that best predicts the YY variable as a linear function of the XXvariables.

How It Works

The basic idea is that an equation is found like this:

Yexp=a+b1X1+b2X2+b3X3+⋯Yexp=a+b1X1+b2X2+b3X3+⋯

The YexpYexp is the expected value of YY for a given set of XX values. b1b1 is the estimated slope of a regression of YY on X1X1, if all of the other XX variables could be kept constant. This concept applies similarly for b2b2, b3b3, et cetera. aa is the intercept. Values of b1b1, et cetera, (the “partial regression coefficients”) and the intercept are found so that they minimize the squared deviations between the expected and observed values of YY.

How well the equation fits the data is expressed by R2R2, the “coefficient of multiple determination. ” This can range from 0 (for no relationship between the XX and YY variables) to 1 (for a perfect fit, i.e. no difference between the observed and expected YY values). The pp-value is a function of the R2R2, the number of observations, and the number of XX variables.

Importance of Slope ()

When the purpose of multiple regression is prediction, the important result is an equation containing partial regression coefficients (slopes). If you had the partial regression coefficients and measured the XX variables, you could plug them into the equation and predict the corresponding value of YY. The magnitude of the partial regression coefficient depends on the unit used for each variable. It does not tell you anything about the relative importance of each variable.

When the purpose of multiple regression is understanding functional relationships, the important result is an equation containing standard partial regression coefficients, like this:

y′exp=a+b′1x′1+b′2x′2+b′3x′3+⋯yexp′=a+b1′x1′+b2′x2′+b3′x3′+⋯

Where b′1b1′ is the standard partial regression coefficient of yy on X1X1. It is the number of standard deviations that YY would change for every one standard deviation change in X1X1, if all the other XX variables could be kept constant. The magnitude of the standard partial regression coefficients tells you something about the relative importance of different variables; XX variables with bigger standard partial regression coefficients have a stronger relationship with the YY variable.

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