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## Chi-Square:

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**Chi-Square:**Introduction to Nonparametric Stats**Chi-square**• Parametric vs. nonparametric tests • Hypotheses about Frequencies • Two main uses: • Goodness of fit. 1 IV. • Test of independence. 2 or more IVs.**Goodness-of-fit test**Blind beer tasting study. Judges taste 4 beers and declare their favorite. 100 lucky judges. Results: If no difference in taste (or all the same beer) we expect about 25 people to choose each beer (null hypothesis). There are 100 people and 4 choices (100/4 = 25). We will test whether frequencies are equal across beers.**Goodness-of-fit (2)**Where O is an observed frequency and E is an expected frequency under the null.**Goodness-of-fit 3**Our test statistic was 30. The df for this test are k-1, where k is the number of cells. In our example k=4 and df = 3. Chi-square has a distribution found in tables. For alpha=.05 and 3 df, the critical value is 7.81, which is less than 30. We reject the null hypothesis. People can taste the difference among beers and have favorites.**Test of Independence (1)**Exit survey at polls. Voter preferences. Did you vote yes for: Use same formula. But now E is calculated by: E=(rowtotal*columntotal)/grandtotal or equivalently: E=pctr*pctc*N, where pct means percentage.**Test of Independence (2)**Find expected values: We use row and column totals to figure expected cell frequencies under the null hypothesis that all cell frequencies are proportional to their row and column frequencies in the population.**Test of Independence (3)**Find the value of chi-square: For the chi-square test of independence, the df are (rows-1) times (cols-1) or for this example, (2-1)*(3-1) = 2. From the chi-square table, we find the critical value is 5.99 for an alpha of .05, so we reject the null. Men and women have different voting preferences.**Effect Size**• Effect size – index of magnitude of relations • Statistical Significance – probability of outcome • Significant results when large magnitude or large sample size. Can have trivial magnitude but still significant results, so you want an effect size.**Effect Sizes for Contingencies - Phi**For 2x2 tables only Phi This is a strong relation. Anything larger than about .5 is unusual in psychology. Average is about .20. Data are hypothetical.**Contingency Coefficient**For 2-way tables other than 2x2, e.g., 3x2 or 4x3 This is a more typical result. There is a significant association, but the association is not very strong.