Statistics for business decision making and analysis robert stine and foster chapter 05

Chapter Association between Categorical Variables Copyright © 2011 Pearson Education, Inc 5.1 Contingency Tables Which hosts send more buyers to Amazon.com?  To answer this question we must gather data on two categorical variables: Host and Purchase  Host identifies the originating site: MSN, RecipeSource, or Yahoo; Purchase indicates whether or not the visit results in a sale of 39 Copyright © 2011 Pearson Education, Inc 5.1 Contingency Tables Consider Two Categorical Variables Simultaneously  A table that shows counts of cases on one categorical variable contingent on the value of another (for every combination of both variables)  Cells in a contingency table are mutually exclusive of 39 Copyright © 2011 Pearson Education, Inc 5.1 Contingency Tables Contingency Table for Web Shopping of 39 Copyright © 2011 Pearson Education, Inc 5.1 Contingency Tables Marginal and Conditional Distributions • Marginal distributions appear in the “margins” of a contingency table and represent the totals (frequencies) for each categorical variable separately • Conditional distributions refer to counts within a row or column of a contingency table (restricted to cases satisfying a condition) of 39 Copyright © 2011 Pearson Education, Inc 5.1 Contingency Tables Conditional Distribution of Purchase for each Host (Column Counts and Percentages) of 39 Copyright © 2011 Pearson Education, Inc 5.1 Contingency Tables Conditional Distribution • Reveals the percentage of purchases among visitors from RecipeSource to be much less than for MSN and Yahoo • Host and Purchase are associated of 39 Copyright © 2011 Pearson Education, Inc 5.1 Contingency Tables Segmented Bar Charts • Used to display conditional distributions • Divides the bars in a bar chart into segments that are proportional to the percentage in each category of a second variable of 39 Copyright © 2011 Pearson Education, Inc 5.1 Contingency Tables Contingency Table of Purchase by Region 10 of 39 Copyright © 2011 Pearson Education, Inc 4M Example 5.2: AIRLINE ARRIVALS Mechanics – This is Simpson’s Paradox 25 of 39 Copyright © 2011 Pearson Education, Inc 4M Example 5.2: AIRLINE ARRIVALS Message The CEO should book on US Airways as it is more likely to arrive on time regardless of destination 26 of 39 Copyright © 2011 Pearson Education, Inc 5.3 Strength of Association Chi-Squared Statistic  A measure of association in a contingency table  Calculated based on a comparison of the observed contingency table to an artificial table with the same marginal totals but no association 27 of 39 Copyright © 2011 Pearson Education, Inc 5.3 Strength of Association Contingency Table 28 of 39 Copyright © 2011 Pearson Education, Inc 5.3 Strength of Association Calculating the Chi-Squared Statistic 29 of 39 Copyright © 2011 Pearson Education, Inc 5.3 Strength of Association Calculating the Chi-Squared Statistic x    30  40  40  10  40    70  60   10  60 60   10  40    50  40  40  10  (50  60)2  60 60  2.5  1.67  2.5  1.67  8.33 30 of 39 Copyright © 2011 Pearson Education, Inc 5.3 Strength of Association Cramer’s V  Derived from the Chi-Squared Statistic  Ranges in value from (variables are not associated) to 1(variables are perfectly associated) 31 of 39 Copyright © 2011 Pearson Education, Inc 5.3 Strength of Association Calculating Cramer’s V V x nmin  r  1, c  1 V = 0.20 for our example There is a weak association between group (students or staff) and attitude toward sharing copyrighted music 32 of 39 Copyright © 2011 Pearson Education, Inc 5.3 Strength of Association Checklist: Chi-Squared and Cramer’s V  Verify that variables are categorical  Verify that there are no obvious lurking variables 33 of 39 Copyright © 2011 Pearson Education, Inc 4M Example 5.3: REAL ESTATE Motivation Do people who heat their homes with gas prefer to cook with gas as well? What heating systems and appliances should a developer select for newly built homes? 34 of 39 Copyright © 2011 Pearson Education, Inc 4M Example 5.3: REAL ESTATE Method The developer contacts homeowners to obtain the data Two categorical variables: type of fuel used for home heating (gas or electric) and type of fuel used for cooking (gas or electric) 35 of 39 Copyright © 2011 Pearson Education, Inc 4M Example 5.3: REAL ESTATE Mechanics Chi-Squared = 98.62; Cramer’s V = 0.47 36 of 39 Copyright © 2011 Pearson Education, Inc 4M Example 5.3: REAL ESTATE Message Homeowners prefer gas to electric heat by about to The developer should build about two-thirds of new homes with gas heat Put electric appliances in all homes with electric heat and in half of the homes with gas heat (assuming that buyers for new homes have the same preferences) 37 of 39 Copyright © 2011 Pearson Education, Inc Best Practices  Use contingency tables to find and summarize association between two categorical variables  Be on the lookout for lurking variables  Use plots to show association  Exploit the absence of association 38 of 39 Copyright © 2011 Pearson Education, Inc Pitfalls  Don’t interpret association as causation  Don’t display too many numbers in a table 39 of 39 Copyright © 2011 Pearson Education, Inc ... Contingency Tables Marginal and Conditional Distributions • Marginal distributions appear in the “margins” of a contingency table and represent the totals (frequencies) for each categorical variable... Reveals the percentage of purchases among visitors from RecipeSource to be much less than for MSN and Yahoo • Host and Purchase are associated of 39 Copyright © 2011 Pearson Education, Inc 5.1 Contingency... that higher premiums for theft insurance should be charged for models that are more likely to be stolen 19 of 39 Copyright © 2011 Pearson Education, Inc 5.2 Lurking Variables and Simpson’s Paradox