Contingency Tables cross tabs Generally used when variables

Contingency Tables (cross tabs) Ø Generally used when variables are nominal and/or ordinal l Even here, should have a limited number of variable attributes (categories) Inside the cells of the table are frequencies (number of cases that fit criteria) Ø To examine relationships within the sample, most use percentages to standardize the cells Ø

Example A survey of 2883 U. S. residents Ø Is one’s political ideology (liberal, moderate, conservative) related to their satisfaction with their financial situation? l Null = ideology is not related to satisfaction with financial status (the are independent) Ø Convention for bivariate tables Ø l l IV (Ideology) is on the top of the table (dictates columns) The DV ($ status)is on the side (dictates rows).

Are these variable related within the sample? Satisfaction With Current $ Situation Total Political Ideology Liberal Moderate Conservative Satisfied 242 300 334 876 More or Less 329 499 439 1267 Unsatisfied 213 294 233 740 Total 784 1093 1006 2883

The Test Statistic for Contingency Tables Ø Chi Square, or χ2 l Calculation • Observed frequencies (your sample data) • Expected frequencies (UNDER NULL) l l Intuitive: how different are the observed cell frequencies from the expected cell frequencies Degrees of Freedom: • 1 -way = K-1 • 2 -way = (# of Rows -1) (# of Columns -1)

Calculating Ø χ2 l = ∑ [(fo - fe)2 /fe] Where Fe= Row Marginal X Column Marginal N So, for each cell, calculate the difference between the actual frequencies (“observed”) and what frequencies would be expected if the null was true (“expected”). Square, and divide by the expected frequency. Ø Add the results from each cell. Ø

Satisfaction With Current $ Situation Satisfied Total Political Ideology Liberal Moderate Conservative 242 (238) 300 (332) 334 (305) 876 More or Less 329 499 439 1267 Unsatisfied 213 294 233 740 Total 784 1093 1006 2883 FIND EXPECTED FREQUENCIES UNDER NULL Example: 876(784)/2883 = 238

Satisfaction With Current $ Situation Political Ideology Liberal Moderate Total Conservative Satisfied 242 (238) 300 (332) 334 (305) 876 More or Less 329 (344) 499 (480) 439 (442) 1267 Unsatisfied 213 (201) 294 (280) 233 (258) 740 784 1093 1006 2883 Total FIND EXPECTED FREQUENCIES UNDER NULL Example: 876(784)/2883 = 238

Calculating χ2 Ø χ2 l l l = ∑ [(fo - fe)2 /fe] [(242 -238) 2 / 238 ] =. 067 [(300 -332) 2 / 332 ] = 3. 08 Do the same for the other seven cells… Calculate obtained χ2 Figure out appropriate df and then Critical χ2 (alpha =. 05) • Would decision change if alpha was. 01?

Interpreting Chi-Square Ø Chi-square has no intuitive meaning, it can range from zero to very large l As with other test statistics, the real interest is the “p value” associated with the calculated chi-square value • Conventional testing = find χ2 (critical) for stated “alpha” (. 05, . 01, etc. ) l Reject if χ2 (observed) is greater than χ2 (critical) • SPSS: find the exact probability of obtaining the χ2 under the null (reject if less than alpha)

SPSS Procedure Ø Analyze Descriptive Statistics Crosstabs Rows = DV Ø Columns = IV Ø Cells Ø l Column Percentages Ø Statistics l Chi square

Agenda Ø Today l Group based assignment (review for exam, review chi-square) Ø Monday l More “hands on” review for exam + final project time Ø Wednesday l Go over HW#4, review conceptual stuff, more practice