Measures of Association

When examining relationships (or the lack thereof) between nominal- and ordinal-level variables, Crosstabs are our instruments of analysis. The Chi-square (x 2 ) tells us if the variability (or dispersal) of a dependent variable differs significantly from what we expect if the null hypothesis (no difference, no relationship between the ind. variable and the dep. variable) is the reality. If the variability (or dispersal) is something we would find only 5% of the time or less due to chance (random sampling error), then we reject the null hypothesis and conclude that there is a statistically significant relationship between the ind. and dep. variables. p-value [ for measuring statistical significance ].05 = 5 chances out of 100 that the relationship you found does not truly exist in the population from which the sample was taken (95% Confidence Interval).01 = 1 chance out of 100 that the relationship you found does not truly exist in the population from which the sample was taken (99% Confidence Interval) Statistical Significance

Measures of Association Now we need statistical measures to tell us how much (if any) knowing an independent variables tells us about (or predicts) the variability of a dependent variable. In other words, “how strong is the relationship between an independent variable and an dependent variable.” Proportional Reduction in Error (PRE) “measures of association” help us estimate how much predictability a given independent variable provides for estimating the variability of a dependent variable.

Measures of Association Vote on Conservation Funding Bill by Political Party Affiliation DemocratsRepublicansTotal No202 Yes212 Total414 Proportional Reduction in Error (PRE) = (E 1 – E 2 ) E 1 Without knowledge of the independent variable (party id), you could only guess how legislators voted based on the total number of “no” and “yes” votes. In this case, you would have to guess “yes” every time (because it’s the largest or “modal” option), which would leave you 212 correct predictions, but 202 incorrect predictions (E 1 = 202).

Measures of Association Vote on Conservation Funding Bill by Political Party Affiliation DemocratsRepublicansTotal No Yes Total Proportional Reduction in Error (PRE) = (E 1 – E 2 ) E 1 With knowledge of the independent variable (party id), you would guess “yes” every time if it was a Democrat and “no” every time if it was a Republican. Now you could increase your number of “correct” predictions to 314 and you would reduce your number of “incorrect” predictions (E 2 = = 100). PRE = (202 – 100) = 102 =.50 (or 50%) = Lambda (a PRE measure when one or both of the variables is nominal)

Measure of Association: the higher the value of Cramer’s V, the stronger the relationship (.304 is evidence of a strong relationship, in this case, between race and the ’96 presidential vote) PRE: Cramer’s V is not a PRE measure (not a measure of the reduction in error)

Measures of Association If BOTH variables are ordinal-level, then use either Kendall’s tau-b or Kendall’s tau-c. You use Kendall’s tau-b when your crosstab table is SQUARE (3 rows and 3 columns). You use Kendall’s tau-c when your crosstab table is NOT SQUARE (3 rows, 2 columns). With Kendall’s tau-b and Kendall’s tau-c, you get the same PRE “measure of association” or gauge of the strength of the relationship between two variables (0-1) as Lambda for nominal- level variables, PLUS a measure of the direction of the relationship between the two variables (positive or negative).

Direction: positive (the older one is, the more likely they are to be interested in campaigns) PRE: by knowing the ind. variable (age cohort), we can reduce by 20% the number of errors in predicting the dep. variable (attention paid to political campaigns)

Direction: negative (the more one prays, the less willing they are to permit legal abortion) PRE: by knowing the ind. variable (frequency of prayer), we can reduce by 30% the number of errors in predicting the dep. variable (support for legal abortions)

Measures of Association PRE Measures (Lambda, Kendall’s tau-b, tau-c): less than 0.1 = “weak” relationship 0.1 to 0.19 = “moderate” relationship 0.2 to.29 = “moderately strong” relationship > than 0.3 = “strong” relationship