I. IntroductionPrior Use of MNo Prior Use of MTotalsBrand XBrand MPrior Use of MNo Prior Use of MTotalsBrand XBrand MPrior Use of MNo Prior Use of MTotalsBrand XBrand MPrior Use of MNo Prior Use of MTotalsBrand XBrand MIII. Two-Way Log-Linear Model: Preference Between Two Brands Vs. Prior use of OneA. The ModelPrior Use of Brand MNo Prior Use of Brand MV. Quantitative Analysis of the Three-Way Contingency TableNov. 12, 2009 LEC #14 ECON 240A-1 L. PhillipsGoodman Log-Linear Model for Qualitative DataI. IntroductionGoodman’s log-linear model can be used to extend bivariate analysis for qualitative variables from 2x2 Chi-Square tables (or more generally mxn tables) to trivariate and multivariate analyses. This tool is the analog to multivariate linear regression, exploring the relationship between a dependent variable and an explanatory variable, conditional on other independent variables. Analysis of the relationship between two variables is called two-way analysis, between three variables 3-way analysis, etc. However, this technique is seldom used beyond 4-way or 5-way analysis in contrast to multivariate regression. Another differenceis that if there are 3 or more categories for a qualitative variable, non-linear aspects of a relationship can be revealed.We will use an example of brand preference for detergent, dependent on water hardness (soft, medium, hard) and prior use of brand M (yes or no). This example is from Yvonne Bishop, Stephen Fienberg, and Paul Holland, Discrete Multivariate Analysis, Theory and Practice. We begin by developing the log-linear model for 2-way analysis and then extend it to three-way analysis for application to detergent preference. If one of the variables in the analysis is clearly the dependent variable, then the log-linear model can be simplified and expressed in terms of the logarithm of the odds. In the case of this example, the log odds of preferring brand X to brand M.II. Survey of Detergent PreferenceNov. 12, 2009 LEC #14 ECON 240A-2 L. PhillipsGoodman Log-Linear Model for Qualitative DataThe survey of detergent preference included 1008 people. They were asked a number of questions. One question is whether they preferred brand X or brand M. Another question was the hardness of their water. A third question was whether they had previously used brand M or not. The results of the survey are described in the tables below, depending on whether they had previously used brand M or not.Table 1A: Previous use of Brand M: YesWater Hardness: Prefer Brand X Prefer Brand M TotalsSoft 76 77 153Medium 70 102 172Hard 61 95 156Totals 207 274 481 Table 1B: Previous Use of Band M: NoWater Hardness: Prefer Brand X Prefer Brand M TotalsSoft 92 80 172Medium 99 73 172Hard 110 72 182Totals 301 225 526As a preview of coming attractions, we will see how revealing looking at the odds can be.In the next table, we list the odds of preferring brand X to brand M, as it varies with water softness and prior use of brand M. The odds are calculated by dividing column 2 bycolumn 3 in Table 1A, and proceeding in a similar fashion for Table 1B.Table 2: Odds of Preferring Brand X Over Brand M Vs.Water Hardness and Prior UseWater Hardness Prior Use of Brand M No Prior Use of Brand MSoft 0.987 1.15Medium 0.686 1.36Nov. 12, 2009 LEC #14 ECON 240A-3 L. PhillipsGoodman Log-Linear Model for Qualitative DataHard 0.642 1.53By comparing columns 2 and 3 of Table 2, it is apparent that the odds for preferring brand X are higher for those with no prior use of brand M. Note that for these consumers, the odds of preferring brand X increase with water hardness. In contrast, for those consumers who had used brand M previously, the odds for brand X decrease with water hardness. It would appear that preference for detergent brand X not only depends on water hardness, but that the nature of that dependence is conditional on whether or notthere has been prior use of brand M.From this exploratory analysis using the odds approach, it would appear that three-way analysis is appropriate. To begin at a simple starting point, we collapse Tables 1A and 1B into a 2x2 two-way analysis by summing over water hardness. The survey results are reported in Table 3. Using the row sums and column sums from table 3, as well as the grand total of 1007, we calculate the marginal probabilities reported in Table 4. Using the marginal probabilities from Table 4 and the grand total of 1007, we calculatethe expected cell counts reported in Table 5.------------------------------------------------------------------Table 3: 1007 People Given Two Brands of Detergent , Observed CountsPrior Use of M No Prior Use of M TotalsBrand X 207 301 508Brand M 274 225 499481 526 1007Table 4: 1007 People Given Two Brands of Detergent, Marginal Probabilities Prior Use of M No Prior Use of M TotalsNov. 12, 2009 LEC #14 ECON 240A-4 L. PhillipsGoodman Log-Linear Model for Qualitative DataBrand X 0.5045Brand M 0.49550.4777 0.5223 1Table 5: 1007 People Given Two Brands of Detergent, Expected Cell CountsPrior Use of M No Prior Use of M TotalsBrand X 242.7 265.3Brand M 238.4 260.61Lastly, Using the observed cell counts in Table 3, and the expected cell counts in Table 5, we calculate the contribution to Chi-Square for each cell, as reported in Table 6.Table 6: 1007 People Given Two Brands of Detergent, Contribution to 2Prior Use of M No Prior Use of M TotalsBrand X 5.3 4.8Brand M 5.3 4.912 = 5.3 + 4.8 + 5.3 + 4.9 = 20.3, where the critical value at the 5% level is 3.84, so we reject the null hypothesis of no association between brand preference for these two detergents and prior use of brand M. III. Two-Way Log-Linear Model: Preference Between Two Brands Vs. Prior use of OneA. The ModelThe probability of each of the cells, for example, in Table 3, is Pij. For example, the observed probability for the first row and the first column is P11 = 207/1007 = 0.206. The probabilities for each cell are postulated to depend on the exponential of a linear function of a number of parameters, where the superscripts B and U refer to brand X orNov. 12, 2009 LEC #14 ECON 240A-5 L. PhillipsGoodman Log-Linear Model for Qualitative DataM and yes or no for prior use of brand M. The subscripts i and j refer to the row and column in the 2x2 table.PijBU = exp[u + uiB + ujU+ uijB,U] (1)Taking natural logarithms, lnPijBU = u + uiB + ujU+ uijB,U (2)hence the name log-linear model. The parameter u is an overall effect. There are two parameters uiB, one for each row