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Introduction to Multiple Regression

Dale E. Berger

Claremont Graduate University http://wise.cgu.edu

Overview

Multiple regression is a flexible method of data analysis that may be appropriate whenever a quantitative variable (the dependent or criterion variable) is to be examined in relationship to any other factors (expressed as independent or predictor variables). Relationships may be nonlinear, independent variables may be quantitative or qualitative, and one can examine the effects of a single variable or multiple variables with or without the effects of other variables taken into account (Cohen, Cohen, West, & Aiken, 2003).

Multiple Regression Models and Significance Tests

Many practical questions involve the relationship between a dependent or criterion variable of interest (call it Y) and a set of k independent variables or potential predictor variables (call them X1, X2, X3,..., Xk), where the scores on all variables are measured for N cases. For example, you might be interested in predicting performance on a job (Y) using information on years of experience (X1), performance in a training program (X2), and performance on an aptitude test (X3). A multiple regression equation for predicting Y can be expressed a follows:

(1) [pic]

To apply the equation, each Xj score for an individual case is multiplied by the corresponding Bj value, the products are added together, and the constant A is added to the sum. The result is Y(, the predicted Y value for the case.

For a given set of data, the values for A and the Bjs are determined mathematically to minimize the sum of squared deviations between predicted Y( and the actual Y scores. Calculations are quite complex, and best performed with the help of a computer, although simple cases with only one or two predictors can be solved by hand with special formulas.

The correlation between Y( and the actual Y value is also called the multiple correlation coefficient, Ry.12...k, or simply R. Thus, R provides a measure of how well Y can be predicted from the set of X scores. The following formula can be used to test the null hypothesis that in the population there is no linear relationship between Y and prediction based on the set of k X variables from N cases:

(2) [pic].

For the statistical test to be accurate, a set of assumptions must be satisfied. The key assumptions are that cases are sampled randomly and independently from the population, and that the deviations of Y values from the predicted Y values are normally distributed with equal variance for all predicted values of Y.

Alternatively, the independent variables can be expressed in terms of standardized scores where Z1 is the z score of variable X1, etc. The regression equation then simplifies to:

(3) ZY( = ß1Z1 + ß2Z2 + ß3Z3 .

The value of the multiple correlation R and the test for statistical significance of R are the same for standardized and raw score formulations.

Test of R Squared Added

An especially useful application of multiple regression analysis is to determine whether a set of variables (Set B) contributes to the prediction of Y beyond the contribution of a prior set (Set A). The statistic of interest here, R squared added, is the difference between the R squared for both sets of variables (R2Y.AB) and the R squared for only the first set (R2Y.A). If we let kA be the number of variables in the first set and kB be the number in the second set, a formula to test the statistical significance of R squared added by Set B is:

(4) 1 [pic]

Each set may have any number of variables. Notice that Formula (2) is a special case of Formula (4) where kA=0. If kA=0 and kB=1, we have a test for a single predictor variable, and Formula (4) becomes equivalent to the square of the t test formula for testing a simple correlation.

Example: Prediction of Scores on a Final Examination

An instructor taught the same course several times and used the same examinations each time. The composition of the classes and performance on the examinations was very stable from term to term. Scores are available on a final examination (Y) and two midterm examinations (X1 and X2) from an earlier class of 28 students. The correlation between the final and the first midterm, rY1, is .60. Similarly, rY2=.50 and r12=.30. In the current class, scores are available from the two midterm examinations, but not from the final. The instructor poses several questions, which we will address after we develop the necessary tools:

a) What is the best formula for predicting performance on the final examination from performance on the two midterm examinations?

b) How well can performance on the final be predicted from performance on the two midterm examinations?

c) Does this prediction model perform significantly better than chance?

d) Does the second midterm add significantly to prediction of the final, beyond the prediction based on the first midterm alone?

Regression Coefficients: Standardized and Unstandardized

Standard statistical package programs such as SPSS REGRESSION can be used to calculate statistics to answer each of the questions in the example, and many other questions as well. Since there are only two predictors, special formulas can be used to conduct an analysis without the help of a computer.

With standardized scores, the regression coefficients are:

2(5) [pic]

Using the data from the example, we find:

[pic]

We can put these estimates of the beta weights into Formula (3) to produce a prediction equation for the standardized scores on the final examination. For a person whose standardized scores on the midterms are Z1 = .80 and Z2 = .60, our prediction of the standardized score on the final examination is:

ZY’ = (ß1)(Z1) + (ß2)(Z2) = (.49)(.80) + (.35)(.60) = .602.

Once we have the beta coefficients for standardized scores, it is easy to generate the Bj regression coefficients shown in Formula (1) for prediction using unstandardized or raw scores, because

(6) [pic]

It is important that Bj weights not be compared without proper consideration of the standard deviations of the corresponding Xj variables. If two variables, X1 and X2, are equally predictive of the criterion, but the SD for the first variable is 100 times larger than the SD for the second variable, B1 will be 100 times smaller than B2! However, the beta weights for the two variables would be equal.

To apply these formulas, we need to know the SD and mean for each test. Suppose the mean is 70 for the final, and 60 and 50 for the first and second midterms, respectively, and SD is 20 for the final, 15 for the first midterm, and 10 for the second midterm. We can calculate B1 = (.49)(20/15) = .653 and B2 = (.35)(20/10) = .700, and A = 70 – (.653)(60) – (.700)(50) = -4.18.

Thus, the best formula for predicting the score on the final in our example is

Y’ = -4.18 + .653 X1 + .700 X2

Multiple Correlation with Two Predictors

The strength of prediction from a multiple regression equation is nicely measured by the square of the multiple correlation coefficient, R2 . In the case of only two predictors, R2 can be found by using the formula

(7) [pic]

In our example, we find

[pic]

One interpretation of R2Y.12 is that it is the proportion of Y variance that can be explained by the two predictors. Here the two midterms can explain (predict) 47.3% of the variance in the final test scores.

Tests of Significance for R

It can be important to determine whether a multiple regression coefficient is statistically significant, because multiple correlations calculated from observed data will always be positive. When many predictors are used with a small sample, an observed multiple correlation can be quite large, even when all correlations in the population are actually zero. With a small sample, observed correlations can vary widely from their population values. The multiple regression procedure capitalizes on chance by assigning greatest weight to those variables which happen to have the strongest relationships with the criterion variables in the sample data. If there are many variables from which to choose, the inflation can be substantial. Lack of statistical significance indicates that an observed sample multiple correlation could well be due to chance.

In our example we observed R2=.473. We can apply Formula (2) to test for statistical significance to get

[pic]

1.

The tabled F(2, 25, .01) = 5.57, so our findings are highly significant (p

How to calculate the coefficient of determination? Steps to Find the Coefficient of Determination Find r, Correlation Coefficient Square ‘r’. Change the above value to a percentage.