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

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

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

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)

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

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

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

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