The basic idea behind regression is to find the equation of the straight line that comes as close as possible to as many of the points as possible. Collect a sample of ( x, y) pairs, and plot them on a set of x, y axes. You believe that y, the dependent variable, is a linear function of x, the independent variable - y depends on x. Regression also allows you to test to see if there is a functional relationship between the variables, by testing the hypothesis that each of the slopes has a value of zero.įirst, let us consider the simple case of a two-variable function. Regression allows you to estimate directly the parameters in linear functions only, though there are tricks that allow many non-linear functional forms to be estimated indirectly. There can also be non-linear functions, where the value of the dependent variable ( y in all of the examples we have used so far) depends on the values of one or more other variables, but the values of the other variables are squared, or taken to some other power or root or multiplied together, before the value of the dependent variable is determined. There can be functions where one variable depends on the values values of two or more other variables where x 1 and x 2 together determine the value of y. If the values of α and β are known, you can find the y that goes with any x by putting the x into the equation and solving. Α and β are parameters, remaining constant as x and y change. The simplest functional form is the linear function where: The most general is y = f( x), which simply says that the value of y depends on the value of x in some regular fashion, though the form of the relationship is not specified. Functions are written in a number of forms. The definition of a function can be formal, like the one in my freshman calculus text: “A function is a set of ordered pairs of numbers ( x, y) such that to each value of the first variable ( x) there corresponds a unique value of the second variable ( y)” (Thomas, 1960). More intuitively, if there is a regular relationship between two variables, there is usually a function that describes the relationship. What is regression?īefore starting to learn about regression, go back to algebra and review what a function is.
Once again, the t-distribution and F-distribution will be used to test hypotheses. In this chapter, the basics will be discussed. Though the simplest regression techniques seem limited in their applications, statisticians have developed a number of variations on regression that greatly expand the usefulness of the technique. With that estimated function, you will be able to infer or forecast things like unit costs, interest rates, or sales over a wide range of conditions.
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Once you learn how to use regression, you will be able to estimate the parameters - the slope and intercept - of the function that links two or more variables. Regression analysis is one of the most used and most powerful multivariate statistical techniques for it infers the existence and form of a functional relationship in a population. If there is a relationship between studying and grades, the location of that distribution of grades will change in an orderly manner as you move from lower to higher levels of studying. For each level of amount studied, there will be a distribution of grades. Some students are taking harder courses, like chemistry or statistics some are smarter some study effectively and some get lucky and find that the professor has asked them exactly what they understood best.
Notice that even if students who study more make better grades, the relationship in the population would not be perfect the same amount of studying will not result in the same grades for every student (or for one student every time). You could then complete your inference and test your hypothesis by gathering a sample of (amount studied, grades) data from some students and use regression to see if the relationship in the sample is strong enough to safely infer that there is a relationship in the population. If you say that students who study more make better grades, you are really hypothesizing that there is a positive relationship between one variable, studying, and another variable, grades. These relationships are seldom exact because there is variation caused by many variables, not just the variables being studied. Regression analysis, like most multivariate statistics, allows you to infer that there is a relationship between two or more variables.
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