The book features interactive demos, simulations and case studies.
Introduction to Simple Linear Regression
Prerequisite
Measures of Variability, Describing Bivariate Data
Learning Objectives
1. Define linear regression
2. Identify errors of prediction in a scatterplot with a regression line
In simple linear regression, we predict scores on one variable from the scores on a second variable. The variable we are predicting is called the criterion variable and is referred to as Y. The variable we are basing our predictions on is called the predictor variable and is referred to as X. When there is only one predictor variable, the prediction method is called simple regression. In simple linear regression, the topic of this section, the predictions of Y when plotted as a function of X form a straight line.
The example data in Table 1 are plotted in Figure 1. You can see that there is a positive relationship between X and Y. If you were going to predict Y from X, the higher the value of X, the higher your prediction of Y.
Table 1. Example data.
| X | Y |
|---|---|
| 1.00 | 1.00 |
| 2.00 | 2.00 |
| 3.00 | 1.30 |
| 4 | 3.75 |
| 5 | 2.25 |
Figure 1. A scatterplot of the example data.
Linear regression consists of finding the best-fitting straight line through the points. The best-fitting line is called a regression line. The black diagonal line in Figure 2 is the regression line and consists of the predicted score on Y for each possible value of X. The vertical lines from the points to the regression line represent the errors of prediction. As you can see, the red point is very near the regression line; its error of prediction is small. By contrast, the yellow point is much higher than the regression line and therefore its error of prediction is large.
Figure 2. A scatterplot of the example data. The black line consists of the predictions, the points are the actual data, and the vertical lines between the points and the black line represent errors of prediction.
The error of prediction for a point is the value of the point minus the predicted value (the value on the line). Table 2 shows the predicted values (Y’) and the errors of prediction (Y-Y’). For example, the first point has a Y of 1.00 and a predicted Y of 1.21. Therefore its error of prediction is -0.21.
Table 2. Example data.
| X | Y | Y’ | Y-Y’ | (Y-Y’)2 |
|---|---|---|---|---|
| 1.00 | 1.00 | 1.210 | -0.210 | 0.044 |
| 2.00 | 2.00 | 1.635 | 0.365 | 0.133 |
| 3.00 | 1.30 | 2.060 | -0.760 | 0.578 |
| 4.00 | 3.75 | 2.485 | 1.265 | 1.600 |
| 5.00 | 2.25 | 2.910 | -0.660 | 0.436 |
You may have noticed that we did not specify what is meant by “best fitting line.” By far the most commonly used criterion for the best fitting line is the line that minimizes the sum of the squared errors of prediction. That is the criterion that was used to find the line in Figure 2. The last column in Table 2 shows the squared errors of prediction. The sum of the squared errors of prediction shown in Table 2 is lower than it would be for any other regression line.
The formula for a regression line is
Y’ = bX + A
where Y’ is the predicted score, b is the slope of the line, and A is the Y intercept. The equation for the line in Figure 2 is
Y’ = 0.425X + 0.785
For X = 1,
Y’ = (0.425)(1) + 0.785 = 1.21.
For X = 2,
Y’ = (0.425)(2) + 0.785 = 1.64.
Computing the Regression Line
In the age of computers, the regression line is typically computed with statistical software. However, the calculations are relatively easy are given here for anyone who is interested. The calculations are based on the statistics shown in Table 3. MX is the mean of X, MY is the mean of Y, sX is the standard deviation of X, sY is the standard deviation of Y, and r is the correlation between X and Y.
Formula for standard deviation
Formula for correlation
Table 3. Statistics for computing regression line.
| MX | MY | sX | sY | r |
|---|---|---|---|---|
| 3 | 2.06 | 1.581 | 1.072 | 0.627 |
The slope (b) can be calculated as follows:
b = r sY/sX
and the intercept (A) can be calculated as
MY – bMX.
For these data,
b = (0.627)(1.072)/1.581 = 0.425
A = 2.06 – (0.425)(3)=0.785
Note that the calculations have all been shown in terms of sample statistics rather than population parameters. The formulas are the same; simply use the parameter values for means, standard deviations, and the correlation.
Standardized Variables
The regression equation is simpler if variables are standardized so that their means are equal to 0 and standard deviations are equal to 1 for then b = r and A = 0. This makes the regression line:
ZY’ = (r)(ZX)
where ZY’ is the predicted standard score for Y, r is the correlation, and ZX is the standardized score for X. Note that the slope of the regression equation for standardized variables is r.
Figure 3 shows a scatterplot with the regression line predicting the standardized Verbal SAT from the standardized Math SAT.
A Real Example
The case study, Predicting GPA contains high school and university grades for 105 computer science majors at a local state school. We now consider how we could predict a student’s university GPA if we knew his or her high school GPA.
Figure 3 shows a scatterplot of University GPA as a function of high school GPA. You can see from the figure that there is a strong positive relationship. The correlation is 0.78. The regression equation is
University GPA’ = (0.675)(High School GPA) + 1.097
Therefore a student with a high school GPA of 3 would be predicted to have a university GPA of
University GPA’ = (0.675)(3) + 1.097 = 3.12.
Figure 3. University GPA as a function of High School GPA.
Assumptions
It may surprise you, but the calculations shown in this section are assumption free. Of course, if the relationship between X and Y is not linear, a different shaped function could fit the data better. Inferential statistics in regression are based on several assumptions, and these assumptions are in a later section of this chapter.