Categorical Data Analysis Chapter 13 Categorical Data Analysis Slides for Optional Sections No Optional Sections

Categorical Data and the Multinomial Distribution Properties of the Multinomial Experiment Experiment has n identical trials There are k possible outcomes to each trial, called classes, categories or cells Probabilities of the k outcomes remain constant from trial to trial Trials are independent Variables of interest are the cell counts, n1, n2…nk, the number of observations that fall into each of the k classes

Testing Category Probabilities: One-Way Table In a multinomial experiment with categorical data from a single qualitative variable, we summarize data in a one-way table.

Testing Category Probabilities: One-Way Table Hypothesis Testing for a One-Way Table Based on the 2 statistic, which allows comparison between the observed distribution of counts and an expected distribution of counts across the k classes Expected distribution = E(nk)=npk, where n is the total number of trials, and pk is the hypothesized probability of being in class k according to H0 The test statistic, 2, is calculated as and the rejection region is determined by the 2 distribution using k-1 df and the desired 

Testing Category Probabilities: One-Way Table Hypothesis Testing for a One-Way Table The null hypothesis is often formulated as a no difference, where H0: p1=p2=p3=…=pk=1/k, but can be formulated with non-equivalent probabilities Alternate hypothesis states that Ha: at least one of the multinomial probabilities does not equal its hypothesized value

Testing Category Probabilities: One-Way Table Hypothesis Testing for a One-Way Table The null hypothesis is often formulated as a no difference, where H0: p1=p2=p3=…=pk=1/k, but can be formulated with non-equivalent probabilities Alternate hypothesis states that Ha: at least one of the multinomial probabilities does not equal its hypothesized value

Testing Category Probabilities: One-Way Table One-Way Tables: an example H0: pLegal=.07, pdecrim=.18, pexistlaw=.65, pnone=.10 Ha: At least 2 proportions differ from proposed plan Rejection region with =.01, df = k-1 = 3 is 11.3449 Since the test statistic falls in the rejection region, we reject H0

Testing Category Probabilities: One-Way Table Conditions Required for a valid 2 Test Multinomial experiment has been conducted Sample size is large, with E(ni) at least 5 for every cell

Testing Category Probabilities: Two-Way (Contingency) Table Used when classifying with two qualitative variables H0: The two classifications are independent Ha: The two classifications are dependent Test Statistic: Rejection region:2>2, where 2 has (r-1)(c-1) df

Testing Category Probabilities: Two-Way (Contingency) Table Conditions Required for a valid 2 Test N observed counts are a random sample from the population of interest Sample size is large, with E(ni) at least 5 for every cell

Testing Category Probabilities: Two-Way (Contingency) Table Sample Statistical package output

A Word of Caution about Chi-Square Tests When an expected cell count is less than 5, 2 probability distribution should not be used If H0 is not rejected, do not accept H0 that the classifications are independent, due to the implications of a Type II error. Do not infer causality when H0 is rejected. Contingency table analysis determines statistical dependence only.