Logistic Regression July 28, 2008 Ivan Katchanovski, Ph.D. POL 242Y-Y

Binary Logistic Regression Appropriate when the dependent variable is a dummy variable – Dummy variable: a variable that includes two categories which assume values 1 and 0 – Example: “Conservative party supporter”: Yes=1; No=0 – Binary: two values One or many independent variables Assumes non-linear relationship 2

Regression Coefficients and Odds Ratio Regression coefficients: – Interpretation is similar to interpretation of unstandardized regression coefficients in linear regression Effect of a change of one unit of an independent variable on the logged odds of the dependent variable Logged odds are not very easy to grasp Odds Ratio: – Effect of a change of one unit of an independent variable on the change in the odds of the dependent variable Better to grasp If odds ratio more than 1: positive relationship If odds ratio less than 1: negative relationship If odds ratio equal to1: no relationship 3

Statistical Significance Statistical significance of a regression coefficient: – Statistically significant if p(obtained)<p(critical)=.05 or.01 or.001 – Statistically nonsignificant if p(obtained)>p(critical)=.05 Direction of association should be reported only for statistically significant regression coefficients 4

Pseudo R Square R Square analogs in logistic regression – Power of independent variables in predicting the dependent variable Cox & Snell R square – Ranges between 0 (no association) and less than 1 (perfect association) Nagelkerke R square – Adjusts Cox & Snell R square so that its maximum value can equal 1 – Ranges between 0 (no association) and 1 (perfect association) 5

Example: Multiple Research Hypotheses First : The level of economic development has a positive effect on the odds that countries are democratic Second: Former British colonies are more likely to be democratic compared to other countries Third : Protestant countries are more likely to be democratic compared to other countries Fourth: Ethnic and linguistic homogeneity has a positive effect on the odds of countries being democratic 6

Example: Variables Dataset: World Dependent Variable: – Democracy (Is country democratic?) Dummy variable Independent Variables: – GDP per capita ($1000) Interval-ratio – Ethno-linguistic heterogeneity Ordinal treated as interval-ratio – Colony variable Transformed into dummy variables – Religious culture variable Transformed into dummy variables 7

Binary Logistic Regression: SPSS Commands SPSS Command: Analyze-Regression-Binary Logistic “Dependent” box: Select the dependent variable “Covariates” box: Select independent variables Method: “Enter” 8

Table: Determinants of democracy Regression coefficients B (Standard error) Odds ratio Exp(B) GDP per cap ($1000).336*** (.105) French colony (1.148).198 Spanish colony.433 (.823) Other country (1.035) Catholic.389 (1.218) Muslim (1.253).336 Other religion (1.149).824 Ethno-linguistic heterogeneity (.434).705 Constant (1.557).531 Nagelkerke R square.645 N92 9 *** Statistically significant at the.01 level, ** statistically significant at the.05 level, * statistically significant at the.1 level

Example: Statistical Significance Number of cases: N=92.1 or 10% significance level can be used Regression coefficient of the GDP variable: SPSS: p(obtained)=.001 <p(critical)=.01=1% Statistically significant at the.01 or 1% level Regression coefficients of the other independent variables: SPSS: p(obtained)=from.159 to.866 >p(critical)=.1 Statistically insignificant 10

Example: Regression Coefficients and Odds Ratio Regression Coefficient of GDP per capita variable=.336 Increase of $1000 in the level of GDP per capita increases the logged odds of country being democratic by.336 Odds ratio of GDP per capita: Increase of $1000 in the level of GDP per capita increases the odds of country being democratic by about 1.4 times 11

Example: Interpretation Nagelkerke R square=.645 The logistic regression model has a strong predictive power The first research hypothesis is supported by logistic regression analysis The level of economic development has a positive and statistically significant effect on the odds of countries being democracies All other research hypotheses are not supported by logistic regression analysis 12