This chapter examines the relationship between two variables using linear regression and correlation. Linear regression estimates a linear equation that describes the relationship, whereas correlation measures the strength of that linear relationship.
Simple Linear Regression
When you plot two variables against each other in a scatter plot, the values usually don’t fall exactly in a perfectly straight line. When you perform a linear regression analysis, you attempt to fi nd the line that best estimates the relationship between two variables (the y, or dependent, variable, and the x, or independent, variable). The line you fi nd is called the fi tted regression line, and the equation that specifi es the line is called the regression equation.
The Regression Equation
If the data in a scatter plot fall approximately in a straight line, you can use linear regression to fi nd an equation for the regression line drawn over the data. Usually, you will not be able to fi t the data perfectly, so some points will lie above and some below the fi tted regression line. The regression line that Excel fits will have an equation of the form y 5 a 1 bx. Here y is the dependent variable, the one you are trying to predict, and x is the independent, or predictor, variable, the one that is doing the predicting
Checking the Regression Model
As in any statistical procedure, for statistical inference on a regression, you are making some important assumptions. There are four:
- The straight-line model is correct.
- The error term e is normally distributed with mean 0.
- The errors have constant variance.
- The errors are independent of each other.
Whenever you use regression to fi t a line to data, you should consider these assumptions. Fortunately, regression is somewhat robust, so the assumptions do not need to be perfectly satisfi ed. One point that cannot be emphasized too strongly is that a signifi cant regression is not proof that these assumptions haven’t been violated. To verify that your data do not violate these assumptions is to go through a series of tests, called diagnostics.