Download Education 793 Class Notes

Education 793 Class Notes
Multiple Regression
19 November 2003
Today’s Agenda
• Class and lab announcements
– Draft of final paper to reviewer on the 15th
– Reviews and final version to instructors on
the 19th
• Your questions?
• Multiple Regression
Multiple Regression
• Purpose: To help researcher predict
some dependent variable from a set of
predictors (X1, X2, X3…Xn)
• Based on prior research and theory,
researchers are able to build
comparative models
Review: Basic Equation for One
Independent Variable
yˆ  a  bx
where
a  y  bx
and
b  rxy
sy
sx
Regression equation tells us that for every one
unit increase in x, there is a b increase in y
In analogous fashion, with k independent
predictors:
yˆ  a  b1 X 1  b2 X 2  ...  bk X k
Regression Coefficients
yˆ  a  b1 X 1  b2 X 2  ...  bk X k
• a is the y intercept
• b’s are partial regression coefficients
– A partial regression coefficient shows the
relationship between the dependent variable and
one independent variable controlling for the other
independent variables in the model
Multiple Correlation Coefficient
• We can estimate the magnitude of the
relationship between the dependent variable
and the best linear combination of
independent variables.
• R-multiple correlation coefficient ranges from
0 to 1. It is the correlation between the
dependent variable and the predicted values
from the regression equation
R2
• The square of the multiple correlation
coefficient is the proportion of variation
in Y accounted for by the set of
independent variables
Tests of Significance
• Ho: Is there a systematic relationship
between the dependent variable and the set
of predictors?
• The formal F-test compares the proportion of
variance predictable by the X’s to the
proportion that is unpredictable by the X’s.
• It is an omnibus test, it does not test the
predictors individually.
Tests of Significance
• Besides the omnibus test, we have ttests for each independent predictor
individually.
• The t-test is the test of Ho: b=0
– If the p-value < .05, then the independent
variable is a significant predictor of the
dependent variable
Design Requirements
• There is one dependent variable and
two or more independent variables that
are correlated to the dependent
• Minimum sample size is approximately
50 and a general rule is that there
should be at least 10 cases for every
independent variable in the model
Assumptions
• Subjects are independent
• Dependent variable is normally
distributed
• Constant variance across the range of
predictor values
• The relationship between X and Y is
linear
Example: Gender Effect on
Predicting SAT
• Sample from 1998 University of
Michigan CIRP
• Dependent Variable: SAT Verbal
• Independent Variables
– HSGPA
– Sex
– Academic Rating Ability
Descriptive Statistics
HSPGA is an 8 category ordinal variable that we treated as
interval
Sex is a 2 category variable 1=male, 2=female
Correlation Matrix
• What do we notice here?
ANOVA Table
• What do we notice here?
Test of the Coefficients
• Is Sex a significant predictor of SAT Verbal?
• Is HSGPA a significant predictor of SAT Verbal?
• What is the effect of HSGPA on SAT Verbal
Laptop Exercise
• Using the Cirp98, Run a multiple
regression with a continuous dependent
variable.
– Chose 3-4 independent variables
– Be ready to interpret your results
Next Week
• Enjoy Your Thanksgiving Break
Please let the great bird live, choose a
vegetarian alternative 