1. PRINCIPLES OF MULTIPLE REGRESSION

2. ON RESEARCH PROJECTS Papers due at Week 10 section meeting
Hard copy only (4-6 pages + tables, graphs)
See “handout” on course website
Lateness policy:
5% off for 24 hours lateness
10% off for 48 hours lateness
20% off for 72 hours lateness
NOT ACCEPTED after 72 hours
If completed over weekend, send electronic copy to TA and submit a hard copy on Monday, June 2

3. Postscripts Calculating intercept a:
a = Y – b X (note b = positive or negative)
Defining t-ratio or “t” statistic:
t = (b – ß)/SE, where b is sample slope and ß is population parameter
In null hypothesis, ß = 0, thus
t = b/SE, and
If t > 2, can reject the null hypothesis

4. READINGS Pollock, Essentials, ch. 7 (pp. 165-176)
Pollock, SPSS Companion, ch. 9
Course Reader, Selections 5-6 (Smith & Ziegler, Governmental Performance, and Inglehart, Mass Support for Democracy)

5. OUTLINE Purposes of Multiple Regression The Basic Model Key Concepts
An Illustration

6. Purposes of Multiple Regression
Incorporating more than one independent variable into
the explanation of a dependent variable
Measuring the cumulative impact of independent variables
on a dependent variable
Determining the relative importance of independent
variables

7. The Basic Model Ŷ = a + b1X1 + b2X2 + b3X3 …. bkXk
Note: Signs can be positive or negative!
PRE = R2
Standardized regression coefficient (beta):
= bi (st.dev.Xi/st.dev Y)
Partial correlation coefficient:
= rYX2.X1, or r13.2

8. Key Concepts
Measuring the cumulative impact on Y of X1 and X2 (via PRE or R2)
Examining relationship between Y and X2, controlling for the effects of X1 (via partial correlation coefficient)
Detecting the identifiable impact of independent
variables (Xs) on Y (via beta weights)
Assessing significance of overall relationship and of
individual regression coefficients (via significance tests,
including standard errors)

9. Visualizing a Plane of Least Squares

10. Detecting Relationships
Spurious = relationship between Y and X1 vanishes (i.e., approaches zero) with X2 in equation [check correlation between X1 and X2]
Enhancement = cumulative strength of relationship (R2) much higher with X1 and X2 in equation than with just X1
Specification = see use of dummy variables [next time]

11. An Illustration of the Principles
Problem: Effects of public health expenditures
Y = infant mortality rate
X1 = health expenditures
X2 = % nonwhite population

15. Since a = Y – bX
Y = 0 (as mean value of residuals)
X = 0 (as mean value of residuals)
the value of a for this equation = 0
so there is no intercept.