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Lecture 8

PSYC 3430 Lecture Notes - Lecture 8: Xm Satellite Radio, Partial Correlation, Pearson Product-Moment Correlation Coefficient


Department
Psychology
Course Code
PSYC 3430
Professor
Peter K Papadogiannis
Lecture
8

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Instructor Notes - Chapter 16 - page 233
Chapter 16: Introduction to Regression
Chapter Outline
16.1 Introduction to Linear Equations and Regression
Linear Equations
Regression
The Least-Squared Error Solution
Using the Regression Equation for Prediction
Standardized Form of the Regression Equation
The Standard Error of Estimate
Relationship Between the Standard Error and the Correlation
16.2 Analysis of Regression : Testing the Significance of the Regression Equation
Significance of Regression and Significance of the Correlation
16.3 Introduction to Multiple Regression with Two Predictor Variables
Regression Equations with Two Predictors
R2 and Residual Variance
Computing R2 and 1 R2 from the Residuals
The Standard Error of Estimate
Testing the Significance of the Multiple-Regression Equation: Analysis of Regression
16.4 Evaluating the Contribution of Each Predictor Variable
Multiple Regression and Partial Correlations
Learning Objectives and Chapter Summary
1. Students should understand the concept of a linear equation including the slope and
Y-intercept.
All linear equations have the form Y = bX + a where a and b are constants. The value of
b is the slope of the line and measures how much Y changes when X is increased by one
point. The value of a is the Y-intercept and identified the point where the line crosses the
Y axis.
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Instructor Notes - Chapter 16 - page 234
2. Students should understand the concept of a least-squared-error solution.
The regression equation is obtained by first finding the distance (or error) between the Y
values in the data and the Y values predicted by the regression equation. Each error is
then squared to make the values consistently positive. The goal of regression is to find
the equation that produces the smallest total amount of squared error. Thus, the
regression equation produces the “best fitting” line for the data points.
3. Students should understand and be able to compute the linear regression equation for
predicting Y values from the X values in a set of correlational data
The regression equation is determined by the slope constant, b = SP/SSX and the Y-
intercept, a = MY bMX, producing a linear equation of the form Y = bX + a. The
equation can be used to compute a predicted Y for each X value in the data.
4. Students should understand the concept of multiple regression. Increasing the number of
predictor variables usually produces more accurate predictions. Although the task of computing
a multiple-regression equation is tedious, students should be able to follow the formulas and
produce a two-predictor formula for a relatively simple set of data
The simple concept is that each variable provides more information and allows for
more accurate predictions. Having two predictors in the equation will produce more
accurate predictions (less error and smaller residuals) than can be obtained using either
predictor by itself
5. Students should understand the concept and process of evaluating the significance of a
regression equation by comparing the predicted variance with the unpredicted (residual) variance
The total variability of the Y scores is measured by SSY with df = n 1. Part of this
variability is predicted by the regression equation and part is not. For linear regression,
the predicted portion is determined by r2 and has df = 1. The unpredicted portion is the
remaining or residual part and is determined by 1 r2 and has df = n 2. For multiple
regression, the predicted portion is determined by R2 and has df = 2. The unpredicted
portion is determined by 1 R2 and has df = n 3. In each case, SS/df can be used to
compute a variance for the predicted portion and a corresponding variance or MSresidual
for the unpredicted portion. Finally, the significance of the equation is determined by an
F-ratio
MSregression
F = ─────
MSresidual
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Instructor Notes - Chapter 16 - page 235
Other Lecture Suggestions
1. With linear regression, the calculations are simple enough to allow computation of the actual
regression equation, especially if you use a simple set of scores such as the data in problems 7
and 10 at the end of the chapter. However, it is probably best to focus on the concepts of
predicted versus unpredicted (residual) variance, and using the regression equation to find a
predicted Y value for each X.
a. For each X, you can compute the predicted Y, then find the residual (Y Ŷ) and
squared residual. The sum of the squared residuals is equal to (1 r2)SSY.
b. An alternative is to convert the original Y values into deviations before finding the
regression equation. Plugging X values into the resulting equation will now produce
predicted deviation scores. If the predicted deviations are squared, the sum of the
squared values will be equal to r2SSY. = SSregression.
2. For multiple-regression is probably is best to start with the regression equation or use a
computer in class to generate the regression equation. Then, focus on the concepts of predicted
versus unpredicted variance and use the equation to find the predicted Y value for each pair of
predictors.
3. The following values provide another example of relatively easy numbers for a linear
regression equation.
X Y
3 3 For these data, SP = 10, SSX = 10, and SSY = 40, producing
4 9 a correlation of r = 0.50 and a regression equation of
0 5 Ŷ = X + 2
2 2
1 1
4. The concepts of partial correlation and multiple regression are probably easier for students to
understand if a third variable is added to an existing X-Y correlation. The third variable is the
second predictor for multiple regression and is the variable to be controlled by the partial
correlation. In this context, the role of the third variable is easier for students to understand if it
is presented as a variable that separates the sample into distinct groups. For example, within a
specific age group, there probably is no consistent relationship between a student’s weight (X)
and his/her performance on a mathematics skill test (Y). Thus, the underlying correlation
between X and Y is r = 0. However, if age is added as a third variable so we now have three
groups consisting of 6-year-old, 8-year-old, and 10-year-old children, the combined sample will
show a clear, positive correlation between weight and math performance. The correlation is due
to the fact that weight and math skill both increase as age goes up. The partial correlation
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