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Chapter 13 Categorical Data Analysis

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2 Categorical Data and the Multinomial Distribution Properties of the Multinomial Experiment 1.Experiment has n identical trials 2.There are k possible outcomes to each trial, called classes, categories or cells 3.Probabilities of the k outcomes remain constant from trial to trial 4.Trials are independent 5.Variables of interest are the cell counts, n 1, n 2 …n k, the number of observations that fall into each of the k classes

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3 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.

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4 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(n k )=np k, where n is the total number of trials, and p k is the hypothesized probability of being in class k according to H 0 The test statistic, 2, is calculated as and the rejection region is determined by the 2 distribution using k-1 df and the desired

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5 Testing Category Probabilities: One-Way Table Hypothesis Testing for a One-Way Table The null hypothesis is often formulated as a no difference, where H 0 : p 1 =p 2 =p 3 =…=p k =1/k, but can be formulated with non-equivalent probabilities Alternate hypothesis states that H a : at least one of the multinomial probabilities does not equal its hypothesized value

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6 Testing Category Probabilities: One-Way Table Hypothesis Testing for a One-Way Table The null hypothesis is often formulated as a no difference, where H 0 : p 1 =p 2 =p 3 =…=p k =1/k, but can be formulated with non-equivalent probabilities Alternate hypothesis states that H a : at least one of the multinomial probabilities does not equal its hypothesized value

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7 Testing Category Probabilities: One-Way Table One-Way Tables: an example H 0 : p Legal =.07, p decrim =.18, p existlaw =.65, p none =.10 H a : At least 2 proportions differ from proposed plan Rejection region with =.01, df = k-1 = 3 is Since the test statistic falls in the rejection region, we reject H 0

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

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9 Testing Category Probabilities: Two-Way (Contingency) Table Used when classifying with two qualitative variables H 0 : The two classifications are independent H a : The two classifications are dependent Test Statistic: Rejection region: 2 > 2 , where 2 has (r-1)(c-1) df

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10 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(n i ) at least 5 for every cell

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

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12 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 H 0 is not rejected, do not accept H 0 that the classifications are independent, due to the implications of a Type II error. Do not infer causality when H 0 is rejected. Contingency table analysis determines statistical dependence only.

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