Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter 13 Categorical Data Analysis Categorical Data and the Multinomial Distribution Properties of the Multinomial Experiment 1. 2. 3. 4. 5. 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 2 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. Schema for one-way table for an experiment with k outcomes Outcomes Counts k1 n1 k2 n2 … … k nk 3 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 2 k ni E (ni ) 2 2 •The test statistic, , is calculated as E ni i 1 and the rejection region is determined by the 2 distribution using k-1 df and the desired 4 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 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 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 6 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 7 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 8 Testing Category Probabilities: Two-Way (Contingency) Table Used when classifying with two qualitative variables Row General r x c Contingency Table Column 1 2 … 1 n11 n12 … 2 n21 n22 … … … … r nr1 nr2 … Column Totals C1 C2 … c n1c n2c … nrc Cc Row Totals R1 R2 … Rr n H0: The two classifications are independent Ha: The two classifications are dependent 2 n E Test Statistic: 2 ij ij where Eij Ri C j Eij n Rejection region:2>2, where 2 has (r-1)(c-1) df 9 Testing Category Probabilities: TwoWay (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 10 Testing Category Probabilities: TwoWay (Contingency) Table Sample Statistical package output 11 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. 12