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UT Knoxville STAT 201 - Chapter 26

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1Chapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.Chapter 26 Comparing CountsChapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.2Test of Independence Contingency tables categorize counts on two categorical variables so that we can see whether the distribution of counts on one variable is contingent on the other. A test of independence examines whether there is a significant association between a pair of categorical variables.Chapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.3Test of Independence (Cont.) In Chapter 3, we saw an example of survival on board the Titanic. Was survival proportion associated with the passengers ticket class?Chapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.4Test of Independence (Cont.) In a test of independence of two categorical variables, the “generic” hypotheses are: H0: Row and column classifications are independent HA: Row and column classifications are notindependent (i.e., they are associated with eachother)Chapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.5Assumptions and Conditions Counted Data Condition: Check that the data are countsfor the categories of a categorical variable. Independence Assumption: The counts in the cells should be independent of each other. Randomization Condition: The counts should be a random sample from some population. 10% Condition: The sample should consist of less than 10% of the population of interest.Question: does the Titanic data meet the two condition above?Chapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.6Assumptions and Conditions (cont.) Sample Size Assumption: We must have enough data for the methods to work. Expected Cell Frequency Condition: We should “expect” to see at least 5 individuals in each cell.Chapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.7What Is Meant by “Expected” Cell Count? Given the row totals and column totals in the contingency table, what cell counts would we “expect” to see inside the table if the row and column classifications were independent of each other? What cell counts would represent perfect independence of row and column classifications (and maintain the row and column totals)?Chapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.8Calculating Expected Cell Counts If rows and columns were really independent, you should be able to take: the probability of being in a particular row TIMES the probability of being in a particular column TIMES the total number of observations to get the cell count for that corresponding cell. This represents the “expected” cell count. This is done for all cells in the contingency table. Mechanically, it is simpler (although equivalent) to calculate, for each cell: row total x column total / grand totalChapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.9Calculations (Cont.) How different are the actual (observed) cell counts from these “expected” cell counts? It is natural to look at the differences between the observed and expected counts in each cell: These differences are actually residuals, so adding up all of these differences will result in a sum of 0.)( ExpObsChapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.10Calculations (cont.) We’ll handle the residuals as we did in regression, by squaring them: To get an idea of the relative sizes of the differences, for each cell, we will divide these squared quantities by the expected cell count for that cell:2)( ExpObs ExpExpObs2)( Chapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.11Calculations (cont.) 2Obs  Exp2Expall cells We then add up all of these values. This is the test statistic, called the Chi-square (or Chi-squared) statistic:Chapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.12Calculations (cont.) Chi-square models are actually a family of distributions indexed by degrees of freedom (much like the t-distribution). DF = (#Rows-1)*(#Cols-1) in this case.Chapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.13Calculations and Hypotheses Recall, our null and alternative hypotheses: H0: Row and column classifications are independent HA: Row and column classifications are notindependent (i.e., they are associated with eachother) Use the Chi-square test statistic to find the P-value for this hypothesis test.Chapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.14Calculations and Hypotheses (Cont.) Large Chi-square values mean lots of deviation from the null hypothesis, so they give small P-values. A good Chi-square calculator for finding p-values can be found at:http://www.stat.tamu.edu/~west/applets/chisqdemo.htmlChapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.15Chi-Square P-Values The Chi-square statistic is used only for testing hypotheses, not for constructing confidence intervals. If the observed counts don’t match the expected, the Chi-square test statistic will be large—it can’t be “too small” or negative. If the calculated value of 2is large enough, we’ll reject the null hypothesis.Chapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.16The Chi-Square Statistic in UseMaintenance Experience Chevy Ford ToyotaRegular Maint. Only42 34 70Regular & Unexpected Maint.24 18 12Type of CarThe following data table represents the results of a survey of some randomly selected Chevy, Ford and Toyota owners and their maintenance experience during the first 5 years of owning their new car. Is maintenance experience associated with the type of car? Test with  = 0.05Ho: Maintenance experience is independent of the type of car.Ha : Maintenance experience is not independent of (or is associatedwith) the type of car.Chapter26 Presentation 1213Copyright © 2009 Pearson Education, Inc.17Find the Expected Cell CountsMaint. Experience ChevyFordToyotaReg. Maint. Only 42 34 70 Row 1 Total: (146)Reg. & Unexpect. 24 18 12 Row 2 Total: (54)Column 1 Total: (66) Column 2 Total: (52) Column 3 Total: (82) Number of Obs. (200)Type of CarExpected Cell Count = Eij = (ith row total)*(jth column total)/(number of obs.)Maint. Experience Chevy Ford ToyotaReg. Maint. Only (146*66)/200=48.18 37.96 59.86Reg. & Unexpect. 17.82 14.04 22.14Type of CarExpected


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UT Knoxville STAT 201 - Chapter 26

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