Three-Way Tables (continued):

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1 STAT5602 Categorical Data Analysis Mills 2015 page 110 Three-Way Tables (continued) Now let us look back over the br preference example. We have fitted the following loglinear models 1.MODELX,Y,Z logm ijk i X j Y k Z G ,df 4,p value MODELXY, Z logm ijk i X j Y k Z ij XY G ,df 3,pvalue MODELXY, XZ logm ijk i X j Y k Z ij XY ik XZ G ,df 2,pvalue MODEL XY, XZ, YZ logm ijk X i Y j Z k XY ij XZ YZ ik jk G ,df 1,p value MODEL XYZ logm ijk X i Y j Z k XY ij XZ ik YZ XYZ jk ijk G 2 0,df 0 When one model is nested within the other, we can conduct a Chi-square test to compare the fit of the two models. When the simpler model holds, the asymptotic distribution for the difference in the likelihood ratio statistics for the two models, G 2, is Chi-square with degrees of freedom equal to df,, the difference in the degrees of freedom associated with the two models (ref. Bishop, Fienberg & Holl 1975, Haberman1974, Rao 1973). We will use these ideas to find the simplest, best-fitting model.

2 STAT5602 Categorical Data Analysis Mills 2015 page 111 Consider the models XY,Z XY, XZ G with df The pvalue for testing the hypothesis that there is no difference in the fit of the two models is Therefore, we conclude that there is evidence to indicate that model XY,XZ results in a significantly better fit than model XY,Z. For models XY,XZ XY,XZ,YZ G with df The pvalue for testing the hypothesis that there is no difference in the fit of these two models is Therefore, we conclude here that there is insufficient evidence to indicate difference in the fits of these two models. These above models are only a subset of the possible models we could have fit to the br preference data. For example, we have not considered XZ, Y, YZ, X, XY, YZ XZ, YZ. The following table summarizes fitting the model of mutual independence, all possible models with one two-way interactions, all possible models with two two-way interactions, the model with three two-way interactions, the model with three-way interaction Note We are ignoring models that do not include some of the main effects. Model G 2 df p value X,Y,Z IJKIJK XY, Z 7.84 K 1IJ XZ, Y J 1IK YZ, X I 1JK XY, XZ 3.48 IJ 1K XY, YZ 6.59 JI 1K XZ, YZ KI 1J XY, XZ, YZ 2.79 I 1J 1K XYZ 0 0

3 STAT5602 Categorical Data Analysis Mills 2015 page 112 How to choose an appropriate model Stepwise Procedure In exploratory studies, we can use an algorithmic search method to help select a model. Goodman (1971) proposed methods analogous to forward selection backward elimination procedures in multiple regression analysis (Draper Smith, 1981, Chpt.6). Forward Selection Terms are added sequentially to the model until further additions do not improve the fit. At each stage, we add the term that gives the greatest improvement in fit. The maximum pvalue for the resulting model is a sensible criterion, since reduction in G 2 for different terms may have different degrees of freedom. To illustrate the forward selection process for the br preference example, we begin with the main effects model X,Y,Z logm ijk i X j Y k Z for which G ,df 4,pvalue Thus there is evidence that this model does not fit the data. If we are to add a term to this model, we have the choice of interaction terms represented by XY ij, XZ ik, or YZ jk. Thus we compare model X,Y,Z in turn to XY,Z logm ijk i X j Y k Z ij XY for which G ,df 3,pvalue XZ,Y logm ijk i X j Y k Z ik XZ for which G ,df 3,pvalue YZ, X logm ijk X i Y j Z k YZ jk for which G ,df 3,pvalue

4 STAT5602 Categorical Data Analysis Mills 2015 page 113 Comparison G 2 df p value X,Y,Z with XY, Z X, Y, Z with XZ, Y X, Y, Zwith YZ, X We look for the greatest improvement to the model X, Y, Z.The greatest improvement to this model is obtained by adding ij XY ; the corresponding pvalue of indicates that the improvement is significant. The resulting model is XY, Z ;it has an associated p value of , indicating that there is weak evidence that the model fits the data. However, using the forward selection process, it is possible we might be able to find a better fitting model yet. To do so, we would need to add an additional term to the model XY,Z. Possible choices are still two-way interaction terms represented by XZ ik, or YZ jk. Thus we compare model XY,Z in turn to XY,XZ logm ijk i X j Y k Z ij XY ik XZ for which G ,df 2,pvalue XY,YZ logm ijk X i Y j Z k XY ij YZ jk for which G ,df 2,pvalue Comparison G 2 df p value XY, Z with XY, XZ XY, Z with XY, YZ Here we see that the greatest improvement to the model XY, Z is obtained by adding XZ ik ; the pvalue associated with adding XZ ik is which indicates that the improvement is significant. The resulting model XY, XZ has an overall associated p value of , indicating that there is evidence that the model fits the data.

5 STAT5602 Categorical Data Analysis Mills 2015 page 114 Using the forward selection process, can we find an even better fitting model? To answer this question, we would need to add an additional term to the model XY,XZ. The possible option is to add the two-way interaction represented by YZ jk. Thus we compare model XY,XZ to XY,XZ,YZ logm ijk X i Y j Z k XY ij XZ YZ ik jk for which G , df 1, p value Comparing XY,XZ with XY,XZ,YZ yields G with df 1; this has associated YZ pvalue of This indicates that the model XY,XZ is not improved by adding jk (which was the only possibility). Thus the forward selection procedure terminates here, with XY, XZ as the final model of choice. Alternatively, backward elimination begins with a complex model sequentially removes terms. At each stage, we remove the term for which there is the least effect on the model (i.e. terms that are not significant). We stop when any further deletion leads to a significantly poorer-fitting model. To illustrate the backward elimination process for the br preference example, we begin with the saturated model XYZ logm ijk X i Y j Z k XY ij XZ ik YZ XYZ jk ijk for which G 2 0, df 0. This saturated model fits the data perfectly. If we are to delete just one term from this model, based on the hierarchical principle, the only possibility here would be XYZ ijk. Thus, we compare model XYZ toxy, XZ, YZ XY, XZ, YZ logm ijk X i Y j Z k XY ij XZ YZ ik jk for which G , df 1, pvalue With a pvalue of we have some evidence that the model XY, XZ, YZ fits the data. Comparing model XY, XZ, YZ with model XYZ yields G with df 1; the XYZ pvalue associated with dropping ijk from the model is This suggests there is no statistically significant difference in the fits of these two models hence suggests that we can eliminate XYZ ijk from the model. Using the backward elimination process, can we find an even more parsimonious model that fits the data? This would mean that we would need to remove a term from model XY, XZ, YZ. Our possible choices are two-way interactions represented by

6 STAT5602 Categorical Data Analysis Mills 2015 page 115 XY ij, XZ ik, or YZ jk.thus, we compare model XY, XZ, YZ in turn to XY, XZ logm ijk i X j Y k Z ij XY ik XZ for which G , df 2, pvalue XY, YZ logm ijk X i Y j Z k XY YZ ij ik for which G , df 2, pvalue XZ, YZ logm ijk X i Y j Z k XZ YZ ij ik for which G , df 2, pvalue Comparison G 2 df p value XY, XZ, YZ with XY, XZ XY, XZ, YZ with XY, YZ XY, XZ, YZ with XZ, YZ The smallest effect on the fit of the model XY, XZ, YZ occurs when removing YZ jk ; the p value associated with its removal is , indicating that the effect removed is not statistically significant. The resulting model is XY, XZ this model has an associated p value of , indicating that there is some evidence that this model fits the data. Can we find a still more parsimonious model that fits the data by removing a term from the model XY, XZ? We have two possible choices - ik XY or jk XZ.Thus, we compare model XY, XZ in turn to XY, Z logm ijk i X j Y k Z ij XY for which G , df 3, pvalue XZ, Y logm ijk i X j Y k Z ik XZ for which G , df 3, pvalue

7 STAT5602 Categorical Data Analysis Mills 2015 page 116 Comparison G 2 df p value XY, XZ with XY, Z XY, XZ with XZ, Y The smallest effect on the model XY, XZ results from removing jk XZ. The pvalue of associated with removing jk XZ indicates that the term being removed is statistically significant. Removing it results in a significantly poorer-fitting model. Thus, the backward elimination procedure would terminate, with XY, XZ as the model of choice. In this case, forward selection backward elimination result in the same model for the br preference example, namely XY, XZ. This is not generally the case. How should we interpret the relationships among the three variables? Recall that under our best model, estimated marginal partial odds ratios are given by ASSOCIATION Br-Prevuse Br-Temp Prevuse-Temp Marginal Partial Level Level If we treat br preference as a response variable, previous use of M temperature as explanatory variables, the relationships of interest would be between br preference previous use of br M we might wish to see if the relationship differs for individuals washing in high temperature vs low temperature These relationships could be described using the appropriate estimated partial odds ratios. For example, consider the relationship between br preference previous use of M. Given that an individual washes in high temperature, it is estimated the odds of preferring X for previous users of M is times the odds of preferring X for individuals who have never used M.

8 STAT5602 Categorical Data Analysis Mills 2015 page 117 We can make exactly the same statement for individuals who wash in low temperature. Thus, we estimate that regardless of temperature, the odds of preferring X for previous users of M is times the odds of preferring X for individuals who have never used M. Similarly, regardless of whether an individual has previously used M or not, we estimate that the odds of preferring X for individuals who wash in high temperature is times the odds of preferring X for individuals who wash in low temperature. NOTE Iit is possible to fit loglinear models to four- higher-way tables. Forward selection backward elimination procedures can be used to assist in identifying an appropriate model. However, the selection process becomes more onerous as the number of variables increases because of the rapid increase in possible associations interactions.