22 Regression and Correlation
Objective
- Here you will learn about pairs of variables that are related in a linear fashion, including those with values occurring
in a slightly random manner. - Here you will learn to calculate the linear correlation coefficient, and how to use it to describe the relationship between an explanatory and response variable.
Linear Relationships
Imagine walking through the electronics section of your local department store. On the wall are examples of dozens of television sets, from little 1900 units made to sit on a kitchen counter to 72″+ monsters meant to be the centerpiece of a home theatre. Looking at the prices, you note without surprise that the 72″ model is more expensive than the 19″, and a 42″ model is priced in between. It seems rather clear that as the TV gets larger, the price goes up. Does that mean increased screen size causes increased price?
Look to the end of the section for the answer.
Watch This: Exploring Linear Relationships
When two quantities are compared, it is not uncommon to note a relationship between them that indicates both quantities increase and decrease at the same time, or that one increases as the other decreases. If both quantities are plotted on coordinate axes, the data points show a general or definite linear trend.
If the points actually form a clearly defined line, the variables may be an example of a deterministic relationship. A deterministic relationship indicates that the value of one variable can be reliably and accurately determined by the manipulation of the other variable. An example might be inches and centimeters: one inch is the same as 2.54 centimeters. If you know how many inches long something is, you can reliably and accurately calculate the number of centimeters long the same item is.
As you likely recall from Algebra, the slope describes the angle of the line created by plotting points from a linear relationship, and the point where the explanatory variable has a value of zero is called the y-intercept (commonly denoted b).
Often, particularly in research situations when one or both variables are measured, the plotted values are generally linear, but do not line up precisely. When two variables seem to show a linear relationship, but the values display some amount of randomness, we commonly visually describe the relationship with a scatter plot. As you will see throughout this chapter, the strength of the linear relationship of the variables can be described through mathematics.
Example 1
Given the equation y = 2.3x + 5:
- Create an x–y table to describe the values of at least four points
- What is the slope of the line?
- What is the y-intercept?
Solution
- Pick a value for x, substitute the chosen value for x in the equation, and calculate y:
Table 1 x calculation y 1 y = 2.3(1) + 5 7.3 2 y = 2.3(2) + 5 9.6 0 y = 2.3(0) + 5 5 −1 y = 2.3(−1) + 5 2.7 The equation in the problem is in y = mx + b form (also known as slope-intercept form), where b is the y-value w