1. Bivariate Linear Regression July 14, 2008
Ivan Katchanovski, Ph.D.
POL 242Y-Y

2. Bivariate Linear Regression
Linear Regression: Statistical technique of finding association between variables and testing research hypotheses
Widely used in political science research
Originally concerned regression of height toward the population mean (Galton and eugenics)
Assumes linear relationship
Bivariate Regression: Association between two variables at interval-ratio level
Can be used for ordinal variables with certain assumptions

3. Bivariate Linear Regression Formula
Regression Formula: Y = a + bX
Y = the value of the dependent variable
a = the Y intercept, or the value of Y when X = 0
b = the regression coefficient, the slope of the regression line, or the amount of change produced in Y by a unit change in X
Positive sign of regression coefficient: positive direction of association
Negative sign of regression coefficient: negative direction of association
X = the value of the independent variable

4. R Square: Coefficient of Determination
R Square: Proportion of the variation in the dependent variable (Y ) that is explained by the independent variable (X)
R square=Explained variation/Total variation
R square=Correlation coefficient (r) squared
Ranges between 0 (no association) and 1 (perfect association)

5. Regression Line
Regression line: The best-fitting straight line that summarizes the relationship between two variables

6. Statistical Significance
Statistical Significance of Regression Coefficient (b):
Statistically significant if in SPSS p(obtained)<p(critical)=.05 or .01 or .001
Statistically nonsignificant if SPSS p(obtained)>p(critical)=.05
Direction of association should be reported only for statistically significant regression coefficients

7. Example
Research hypothesis: The level of economic development has a positive effect on civil liberties in countries of the world
Dependent variable: civil liberties
Interval-ratio
Independent variable: GDP per capita ($1000)
Measure of the level of the economic development

8. Example Regression Coefficient=.257
Increase of $1000 in the level of GDP per capita increases the civil liberties score by .257
Statistical significance of the regression coefficient:
SPSS: p(obtained)=.000 <p(critical)=.001=.1%
Statistically significant at the .001 or .1% level
R square=.525
GDP per capita explains 52.5% of variation in civil liberties
Research hypothesis: supported by bivariate regression analysis
The level of economic development has a positive and statistically significant effect on civil liberties