Transcript PRINCIPLES OF MULTIPLE REGRESSION

PRINCIPLES OF
MULTIPLE REGRESSION
ON RESEARCH PROJECTS
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Papers due at Week 10 section meeting
Hard copy only (4-6 pages + tables, graphs)
See “handout” on course website
Lateness policy:
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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
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
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)
OUTLINE
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Purposes of Multiple Regression
The Basic Model
Key Concepts
An Illustration
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
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
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)
Visualizing a Plane of Least Squares
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]
An Illustration of the Principles
Problem: Effects of public health expenditures
Y = infant mortality rate
X1 = health expenditures
X2 = % nonwhite population
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.