Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 11 Analyzing the Association Between Categorical Variables Section 11.5 Small Sample Sizes: Fisher’s Exact Test

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 3 Fisher’s Exact Test  The chi-squared test of independence is a large- sample test.  When the expected frequencies are small, any of them being less than about 5, small-sample tests are more appropriate.  Fisher’s exact test is a small-sample test of independence.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 4  The calculations for Fisher’s exact test are complex.  Statistical software can be used to obtain the P-value for the test that the two variables are independent.  The smaller the P-value, the stronger the evidence that the variables are associated. Fisher’s Exact Test

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 5 Example: A Tea-Tasting Experiment This is an experiment conducted by Sir Ronald Fisher.  His colleague, Dr. Muriel Bristol, claimed that when drinking tea she could tell whether the milk or the tea had been added to the cup first.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 6 Experiment:  Fisher asked her to taste eight cups of tea:  Four had the milk added first.  Four had the tea added first.  She was asked to indicate which four had the milk added first.  The order of presenting the cups was randomized. Example: A Tea-Tasting Experiment

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 7 Results: Table Result of Tea-Tasting Experiment. The table cross-tabulates what was actually poured first (milk or tea) by what Dr. Bristol predicted was poured first. She had to indicate which four of the eight cups had the milk poured first. Example: A Tea-Tasting Experiment

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 8 Analysis: Table Result of Fisher’s Exact Test for Tea-Tasting Experiment. The chi-squared P- value is listed under Asymp. Sig. and the Fisher’s exact test P-values are listed under Exact Sig. “Sig” is short for significance and “asymp.” is short for asymptotic. Example: A Tea-Tasting Experiment

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 9 The one-sided version of the test pertains to the alternative that her predictions are better than random guessing. Does the P-value suggest that she had the ability to predict better than random guessing? Example: A Tea-Tasting Experiment

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 10 The P-value of does not give much evidence against the null hypothesis. The data did not support Dr. Bristol’s claim that she could tell whether the milk or the tea had been added to the cup first. If she had predicted all four cups correctly, the one-sided P-value would have been We might then believe her claim. Example: A Tea-Tasting Experiment

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 11 Summary of Fisher’s Exact Test of Independence for 2x2 Tables 1.Assumptions:  Two binary categorical variables  Data are random 2.Hypotheses:  the two variables are independent  the two variables are associated

Copyright © 2013, 2009, and 2007, Pearson Education, Inc Test Statistic:  First cell count (this determines the others given the margin totals). 4.P-value:  Probability that the first cell count equals the observed value or a value even more extreme as predicted by. 5.Conclusion:  Report the P-value and interpret in context. If a decision is required, reject when P-value significance level. Summary of Fisher’s Exact Test of Independence for 2x2 Tables