1. Analyzing categorical data S-012

2. Categorical data Non-continuous (discrete values) Categories such as: – “high” or “low” – Yes / no – Graduate / non-graduate – College vs. non-college Non-continuous (discrete values) Categories such as: – “high” or “low” – Yes / no – Graduate / non-graduate – College vs. non-college These are binary variables Or multiple categories High/medium/low Agree/neutral/disagree Elementary/middle/high school Public / private school Or multiple categories High/medium/low Agree/neutral/disagree Elementary/middle/high school Public / private school Sometimes the categorical variables are just categories (sometimes call “nominal”). Other times they may represent some order (ordinal). Sometimes the categorical variables are just categories (sometimes call “nominal”). Other times they may represent some order (ordinal).

3. Approaches for comparing groups When we have categorical data Parallel to our strategies for comparing means 1.Construct separate confidence intervals – Do they overlap? 2.Construct a confidence interval on the difference – Does the CI include zero? 3.A z-test on the difference between the proportions – Very similar to a t-test for the means 4.The chi-square test (NHST) – Is the probability value less than.05 (our alpha level or significance level)? Parallel to our strategies for comparing means 1.Construct separate confidence intervals – Do they overlap? 2.Construct a confidence interval on the difference – Does the CI include zero? 3.A z-test on the difference between the proportions – Very similar to a t-test for the means 4.The chi-square test (NHST) – Is the probability value less than.05 (our alpha level or significance level)? Options 1,2 and 3 are parallels to our approaches for comparing means. These all work fine for two groups. Option 4 is something new. It works for two groups, or more than two groups. So it is very versatile. Options 1,2 and 3 are parallels to our approaches for comparing means. These all work fine for two groups. Option 4 is something new. It works for two groups, or more than two groups. So it is very versatile.

4. CI approach Do they overlap? Example 1: Group AGroup B n = 100n = 200 sample sizes p =.75p =.65 sample proportions.75±.08.65±.07.67 ≤ P a ≤.83.58 ≤ P b ≤.72 95% CI: Null hypothesis: The proportions in the populations are equal. H0: P a = P b or H0:  a =  b Null hypothesis: The proportions in the populations are equal. H0: P a = P b or H0:  a =  b Conclusion: The CIs overlap. We cannot reject Ho. We cannot conclude that the population proportions are different.

5. CI for the difference 95% CI:.10 ±.11 Or: [ -0.01,.21 ] 95% CI:.10 ±.11 Or: [ -0.01,.21 ] Conclusion: The CI includes zero. We cannot reject Ho. We cannot conclude that the population proportions are different.

6. A z-test for the difference between the sample proportions = This is very parallel to a t-test on the means. In this example, we get z = 1.75 p =.04 (1 tailed) p =.08 (2-tailed) In this example, we get z = 1.75 p =.04 (1 tailed) p =.08 (2-tailed) Conclusion: The probability value is not less than 0.05. We cannot reject Ho. We cannot conclude that the population proportions are different.

7. Example 2 Three approaches School ASchool B n = 60n = 40 p =.50p =.75 CI overlap.50 +/-.127.75 +/-.134 [.313,.627][.616..884] CI for the difference.50 -.75 = -0.25 +/- 0.196 [-0.446, -0.054] Z –testZ = -2.50Prob =.006 (1-tailed) Prob =.012 (2-tailed) Next: Let’s use a chi-square test

8. Examples: