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Lecture10 - Two-way tables review Inference for Two-Way...

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Inference for Two-Way Tables Analysis of Two-Way Tables Section 9.1 © 2009 W.H. Freeman and Company Two-way tables - review ! An experiment has a two-way design if two categorical factors are studied with several levels of each factor. ! Two-way tables organize data about two categorical variables obtained from a two-way, or block, design. ! Example : We call Education the row variable and Age group the column variable . " Each combination of values for these two variables is called a cell . Marginal distributions (review) We can look at each categorical variable separately in a two-way table by studying the row totals and the column totals. They represent the marginal distributions , expressed in counts or percentages (They are written as if in a margin.) 2000 U.S. census Conditional distributions (review) # In the table below, the 25 to 34 age group occupies the first column. To find the complete distribution of education in this age group, look only at that column. Compute each count as a percent of the column total. # These percents should add up to 100% because all persons in this age group fall in one of the education categories. These four percents together are the conditional distribution of education, given the 25 to 34 age group. 2000 U.S. census
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The percents within the table represent the conditional distributions . Comparing the conditional distributions allows you to describe the “relationship” between both categorical variables. We can also compute each count as a percent of the column total. These percents should add up to 100% and together are the conditional distribution. (Here the percents are calculated by age range (columns)). Conditional distributions (review) 29.30% = 11071 37785 = cell total . column total Hypothesis: no association " The question of interest is whether there is a relation between the row variable and the column variable " We want to know if the differences in sample proportions are likely to have occurred just by chance, because of the random sampling. " We use the chi-square ( ! 2 ) test to assess the null hypothesis of no relationship between the two categorical variables of a two-way table. Specific Hypotheses " The hypotheses for the chi-square ( ! 2 ) test are… H 0 : There is no association between the row and column variables H a : There is an association between the row and column variables " These hypotheses are neither on or two tailed " For an r x c two way table, in general " H 0 means that the c distributions of the row variable are identical " H a means that the distributions are not the same " The question being posed is “Are the observed cell counts different from the expected counts?” Expected counts in two-way tables " Two-way tables sort the data according to two categorical variables. We want to test the hypothesis that there is no relationship between these two categorical variables ( H 0 ).
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