Review2 Test2 Test for IndependenceTrue proportions and estimated proportionsIs there a relationship between the row and the column variableFormal 2 test2 TestAnalysis Result of 2 TestExample: Link between Diabetes and Heart DiseaseExample: Relationship between Exposure to R-rated Movies and Adolescent SmokingChapter 9 - Analysis of Two-Way TablesAnh DaoAugust 3rd, 2009Chapter 9 - Analysis of Two-Way TablesReviewDepending on the situation, one of the variables is the explanatoryvariable (can be called grouping variable) and the other is the responsevariable. In this case, we look at the percentages of one variable foreach level of the other variable.Examples:Gender and Soda PreferenceCountry of Origin and Marital StatusSmoking Habits and Socioeconomic StatusChapter 9 - Analysis of Two-Way TablesReviewTwo-way tables come about when we are interested in the relationshipbetween two categorical variables.Two-way tables are also called contingency tables or crosstabulation.One of the variables is the row variable.The other is the column variable.The combination of a row variable and a column variable is a cell.Chapter 9 - Analysis of Two-Way TablesReviewExampleChapter 9 - Analysis of Two-Way Tablesχ2Testχ2Test for IndependenceTo test whether or not there is a relationship between the row variableand the column variable, we use the chi-square statistic (X2), which canbe calculated in the computer.The null hypothesis (H0) is no relationship among the two variables,i.e. the variables are independent.The alternative hypothesis (Ha) is that there is a relationship, i.e.the variables are not independent.Assumptions needed for χ2-test:For 2×2 tables, we require that all four expected cell counts be 5or more.For tables larger than 2×2, we will use this approximationwhenever the average of the expected counts is 5 or more and thesmallest expected count is 1 or more.Chapter 9 - Analysis of Two-Way Tablesχ2Testχ2Test for IndependenceA comparison of the proportion of “successes” in two populationsleads to a 2×2 table.We can compare two population proportions either by the χ2test or bythe two-sample z-test from section 8.2.z =p1− p2rˆp (1 −ˆp )1n1+1n2, whereˆp =X1+ X2n1+ n2These tests always give exactly the same result when we compare 2populations.χ21(chi-square with 1 degree of freedom) critical values are equal to thesquares of the corresponding N(0, 1) critical values. In fact:χ21,α=zα/22.Chapter 9 - Analysis of Two-Way Tablesχ2Testχ2Test for IndependenceGraphical summary of categorical dataRecall in Chapter 2 we learned how to describe the relationship between 2categorical variables using graphs or tables. For this purpose we can use:Side-by-side bar graphsSide-by-side pie chartsNow, we use χ2test to numerical summary for categorical data.Chapter 9 - Analysis of Two-Way Tablesχ2Testχ2Test for IndependenceAdvantage of the z-test: We can test either one-sided or two-sidedalternatives, while χ2-test always tests the two-sided alternative.Advantage of χ2-test: We can compare more than two populations,while z-test compares only two populations.Chapter 9 - Analysis of Two-Way Tablesχ2Testχ2Test for IndependenceExample 9.8 Textbook: Market researchers know that background music caninfluence the mood and purchasing behavior of customers. One study in asupermarket in Northern Ireland compared three treatments: no music,French accordion music, and Italian string music. Under each condition, theresearchers recorded the numbers of bottles of French, Italian, and otherwine purchased.Chapter 9 - Analysis of Two-Way Tablesχ2Testχ2Test for IndependenceThere is a convention that the column variable is explanatory and therow variable is a response.In this example, the researchers expected that music would influencesales, so music type is the explanatory variable and the type of winepurchased is the response variable.Denote r = # of row categories, c = # of column categories. In thisexample, r = 3 and c = 3.Chapter 9 - Analysis of Two-Way Tablesχ2Testχ2Test for IndependenceHow do we interpret a 2-way table?There are as many populations as the levels of explanatory variable (cpopulations).In the wines and music example there are 3 populations:Population 1: Customers in the store when no music isplayed.Population 2: Customers in the store when French music isplayed.Population 3: Customers in the store when Italian music isplayed.Chapter 9 - Analysis of Two-Way Tablesχ2Testχ2Test for IndependenceHow do we interpret a 2-way table?There are r × c cells in the 2-way table. Each cell contains a count ofsubjects that fall into the corresponding category.In the wines and music example there 3 × 3 = 9 cells. Forinstance, the cell with Italian music and French wine contains 30people.Chapter 9 - Analysis of Two-Way Tablesχ2TestTrue proportions and estimated proportionsFor each population we may compute the proportions or percents ofpeople who fall into each category of the response variable.Column percents for the music and wine example are shownbelow:Chapter 9 - Analysis of Two-Way Tablesχ2TestTrue proportions and estimated proportionsThere are r true proportions corresponding to each population. Theproportions p1,1, . . . , p3,3computed from the data displayed in theprevious table estimate the true proportions.No Music French Music Italian MusicFrench wine π11π12π13Italian wine π21π22π23Other wine π31π32π33Proportions in the column add up to 1. In this example:π11+ π21+ π31= 1, π12+ π22+ π23= 1, π13+ π23+ π33= 1.Chapter 9 - Analysis of Two-Way Tablesχ2TestTrue proportions and estimated proportionsIn Chapter 8, we had 2 populations and therefore 4 populationproportions. However, note that we need only π1and π2in order tospecify 4 true proportions. We have the following correspondence withthe new notation:π11= π1, π12= 1 − π1= π11π21= π2, π22= 1 − π2= π21Due to the condition that the proportions in the column add up to 1, weneed to know only r × (c − 1) in order to specify r × c true proportions.In reality, we just need to estimate r × (c − 1) proportions.Chapter 9 - Analysis of Two-Way Tablesχ2TestIs there a relationship between the row and the column variableNo relationship (independence) between the row and column variableswill mean that the distributions of the row (response) variable will be thesame across the levels of the column (explanatory) variable.Mathematically it will mean that the row proportions are thesame,
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