PRINCIPLES OF MULTIPLE REGRESSION

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

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 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.