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Distributions of Nominal Variables

This guide explores the analysis of nominal data, focusing on hypothesis testing with chi-square tests. It explains the concepts of distributions of nominal variables, the binomial test, and how to extend it to multinomial tests. Key ideas include evaluating frequencies, calculating z-scores, and understanding independence between variables. Learn to formulate null hypotheses and compute expected frequencies, observed frequencies, and chi-square statistics for various categories, illustrated through examples, including dog breed distributions in shelters. ###

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Distributions of Nominal Variables

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  1. Distributions of Nominal Variables 12/03

  2. Nominal Data • Some measurements are just types or categories • Favorite color, college major, political affiliation, how you get to school, where you’re from • Minimal mathematical structure, but we can still do hypothesis testing • Hypotheses about frequencies or probabilities • Are all categories equally likely? • Do two groups differ in their distributions? • Are two nominal variables related or independent?

  3. Extending the Binomial Test • Binomial test • Frequency of observations in yes/true category • Compare to prediction of null hypothesis • Normal approximation • Treat binomial distribution as Normal • Convert frequency to z-score mfreq f

  4. Extending the Binomial Test • Multinomial test • Count observations in every category • Observed frequencies, fobs • Convert each to z-score • H0 predicts each z should be near 0 • Chi-square statistic • Sum of squared z-scores • Measures deviation from null hypothesis • p-value • Probability of result greater than c2 • Uses chi-square distribution • df = k – 1 (k is number of categories) • Counts are not independent; last constrained by rest

  5. Details of z-score • Expected frequencies: • Frequency of each category predicted by H0 • Expected value or mean of sampling distribution • Category probability times number of observations (n) • If all categories equally likely: • Standard error • Denominator of z formula • Standard deviation of sampling distribution (adjusted for degrees of freedom) • Equals square root of expected frequency:

  6. Example: Favorite Colors • Choices: Red, Yellow, Green, Blue, Purple • Are they all equally popular? • Null hypothesis: For each color, • Deviances: • Squared z-scores: • Chi-square statistic: • Critical value (df = 4, a = 5%): 9.49

  7. Independence of Nominal Variables • Are two nominal variables related? • Same question as correlation, but need different approach • Do probabilities for one variable differ between categories of another? • Experimental condition vs. success of learning;sex vs. political affiliation; origin vs. major • Independent nominal variables • Probabilities for each variable unaffected by other • Example: 80% from CO, 10% psych majors • 80% of psych majors are from CO • 80%10% = 8% both psych and from CO • p(x & y) = p(x)p(y)

  8. Chi-square Test of Independence • Null hypothesis: Variables are independent • Use H0 to calculate expected frequencies • Find observed marginal frequencies for each variable • Total count for each category, ignoring levels of other variable • Multiply marginal frequencies to get expected frequency for combination • Same formula as before:

  9. General Principles of Chi-square Tests • Can use any prediction about data as null hypothesis • Very general approach • Measure goodness of fit • Actually badness of fit • Deviation of data from prediction • Nominal data • Calculate z-score for each frequency within its sampling distribution • Observed minus expected frequency, divided by sqrt(f exp) • Square zs and sum, to get c2 • Distribution of one variable; dependence between two variables • Compare GoF to chi-square distribution to get p-value • p = p(c2df > c2) • df comes from number of parameters constrained by H0 • k – 1 for multinomial test; (kX – 1)(kY – 1) for independence test

  10. Review Dogs in the U.S. are 30% Labrador 20% Chihuahua 15% German Shepherd 35% Other The local shelter has 200 dogs. How many of each would you expect if sheter dogs had the same distribution as the general population? • 50 Lab, 50 Chihuahua, 50 GSD, 50 other • 30 Lab, 20 Chihuahua, 15 GSD, 35 other • 60 Lab, 40 Chihuahua, 30 GSD, 70 other • 25 Lab, 25 Chihuahua, 25 GSD, 25 other

  11. Review The actual breed counts are as follows: Labrador Chihuahua GSD Other Observed 65 50 20 65 Expected 60 40 30 70 Calculate a c2 statistic for testing whether this shelter reliably deviates from the breed distribution in the general population. • 0.74 • 6.61 • 7.77 • 250 p = .085

  12. Review Next we count how many get adopted in the next month: Labrador Chihuahua GSD Other Total Adopted 20 35 4 21 80 Not 45 15 16 44 120 Total 65 50 20 65 200 If being adopted is independent of breed, how many German Shepherds would you have expected to be adopted? • 8 • 10 • 20 • 25

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