Application of multiple regression to explaining the vote for Reagan by state in 1984:

• The setting of the 1984 election:
  • Republican President Ronald Reagan ran against the Democratic candidate, Walter Mondale
    • How well would Reagan do in 1984 comapred with his election in 1980?
  • Walter Mondale had won the nomination handily over Jesse Jackson, who promised to mobilize the black vote in 1984
    • What impact would the "Jackson factor" have on turning out the Democratic vote in 1984?
  • Mondale's running-mate for Vice President was Geraldine Ferraro
    • What impact would the "Ferraro factor" have on turning out women to vote for the ticket in 1984?
  • Let's include all three variables in a model to explain Reagan's vote in 1984.
• Matrix of correlations:
  • 
• Use of Multiple Regression in SPSS 10 to execute this model:
  • Under Analyze in the Menu select Linear Regression
    • In the "Dependent" box, enter the name of the variable you want to explain: Reagan84
    • In the "Independents" box enter the names of the explanatory variables: Reagan80, PctBlack, PctWomen
  • for this analysis, select STEPWISE [enters the variables one at a time] in the "Method" box.
  • Click on the "Statistics" button and check
    • "Estimates" for Regression Coefficients and
    • "Model Fit"
    • Press "Continue"
  • Click on "OK"

• Interpretation:
  • Although three independent variables were offered for inclusion using the stepwise procedure, only Reagan84 was included in Model 1
  • The box for the "Excluded Variables" shows that % Black and % Women were not significant at the .05 level and therefore were not selected for inclusion.

• In 1980, Reagan ran against President Jimmy Carter, a southerner
  • Southern states have the highest proportions of black voters
  • Perhaps Carter in 1980 ran relatively better in the south against Reagan than Minnesotan Mondale did in 1984
  • So perhaps we need to control for the south before assessing the impact of the Jackson factor
• Let's redo the analysis entering the variable, South, scored 1 if the state is one of the 11 in the deep south, or 0 otherwise.
  • Here's the new correlation matrix
  • Note that the correlation between "south" and "Reagan80" was negative (as expected) but very small.

  CORRELATION:	REAGAN80	PCTBLACK	PCTWOMEN	SOUTH
  REAGAN84	0.900	-0.565	-0.513	0.123
  REAGAN80		-0.595	-0.537	-0.082
  PCTBLACK			0.522	0.511
  PCTWOMEN				0.211

• Still, let's carry out the new stepwise analysis using four independent variables:

• Adding south as a variable completely alters the analysis.
• In Model 2, South is added to Reagan as a second explanatory variable, raising the R² from .81 to .85.
• In Model 3, once South is controlled for, %Black enters the equation as a significant variable (.05 level), raising the R² to ,88.
• However, %Women is not added into the equation to further increase the explanation:
  • As shown in the Excluded Variables table above, it is only significant at .612 -- which is well short of .05 and so not added.
  • In truth, there is relatively little variance among states in % women, making that a weak explanatory variable. 

•  REGRESSION coefficients
  • Regression coefficients are embedded within a regression EQUATION
    • "b" coefficients are "unstandardized" coefficients and can be interpreted using the scale of measurement for the raw data.
    • "beta" coefficients are "standardized" and must be interpreted in terms of "standard deviation" units.
    • Because of the standardization, "beta" coefficients can be compared WITHIN equations; "b" coefficients cannot.
  • The regression coefficients in a multiple regression equation are "partial" coefficients because they state the "effect" of a given independent variable on the dependent variable while "controlling" for effects of other variables in the equation.
  • Regression coefficients measure CHANGE in Y, not percent of variance explained.
•   CORRELATION coefficients:
  • Measure the STRENGTH of a relationship, i.e., the fit of observed Y values around the line representing predicted Y values according to a regression equation.
  • Simple correlations pertain to bivariate relationships involving only one independent variable and are expressed by r.
  • Multiple correlations pertain to multivariate relationships using multiple independent variables and are expressed by R.
  • The regression procedure refers to the "partial" correlations of variables not in the equation to choose which variables (if any) are to be added to the equation next.
    • Partial correlations computed by regression pertain to the correlation between a dependent variable, Y, and an independent variable, Xi, out of the equation while "controlling for" the effects of the other independent variables already in the equation.
    • In essence, partial correlations express the correlation between the residuals (i.e., deviations) of Y regressed on X_i and the residuals around the line of the regression equation built from the other variables in the equation.