Measures of Association June 25, 2008 Ivan Katchanovski, Ph.D. POL 242Y-Y

2 Measures of Association Association refers to the relationship between two (or more) variables – Example: Relationship between preference for freedom and national culture Measures of Association provide information about: – Strength of relationship – Direction of association (ordinal or interval-ratio variables) Helpful in tests of research hypotheses

Proportionate Reduction of Error (PRE) A logical model for assessing the strength of a relationship by asking how much knowing values on one variable would reduce our errors in guessing values on the other – Example: If we know how much education people have, we can improve our ability to estimate their income, thus indicating there is a relationship between the two variables

PRE-Based Measures of Association For nominal variables: If at least one variable is nominal – Lambda: based on ability to guess values on one of the variables For ordinal variable s: – Gamma: based on guessing the ordinal arrangement of values – Kendall’s tau-b If the dependent and independent variables have the same number of categories – Kendall’s tau-c If the dependent and independent variables do not have the same number of categories

5 Cramer’s V: Chi-square Based Measure of Association Non-PRE measure Cramer’s V formula: _______________________________ – Cramer’s V= √χ 2 /N(Minimum of rows-1 or columns-1) Varies between 0 (no association) and 1 (perfect association) Most appropriate for nominal variables

Criteria of Strength of Association Lambda and Cramer’s V Vary between 0 (no association) and 1 (perfect association) Gamma, Kendall’s tau-b, Kendall’s tau-c – Vary between -1 (perfect negative association) and 1 (perfect positive association) 0: no association 0-0.1: weak association : moderate association : strong association 6

Direction of Association Direction of the association (ordinal or interval-ratio variables): – Positive association: relationship where the variables vary in the same direction Example: Positive association between income and education level – Negative association: relationship where the variables vary in opposite directions 7

Example Table 1. Preference for freedom and equality in the US and Canada, percent Source: 1996 Lipset/Meltz survey Nominal independent variable: Country – Cramer’s V= (moderate association) – Lambda=0.092 (weak) 8 United StatesCanada Freedom6756 Equality3344 Total, %100 N

Example Less than universityUniversity A great deal63 Quite a lot3527 Not very much4655 None at all1315 Total, %100 N Table 1. Confidence in television in Canada by education level, 2000 World Values Survey, % Ordinal variables: Confidence in television and education Gamma=0.183 (moderate level of association) The dependent and independent variables do not have the same number of categories Kendall’s tau-c=0.093 (weak level of association) What is direction of association between these variables?

SPSS Commands SPSS Commands for Measures of Association: – Analyze=Descriptive Statistics-Crosstabs – “Row” box: select dependent variable – “Column” box: select independent variable – “Cells” Option: Column percentages – “Statistics” Option: Chi-square and measure of association For nominal variables: Cramer’s V or Lambda For ordinal variables: Gamma or Kendall’s tau 10