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University of Waterloo STAT 431 gehan_88s

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STAT 431 ASSIGNMENT 2 DUE: Monday, February 13, 2012, AT 10:30 AM QUESTION 1 (a) We start our investigation by fitting the main effects model, that is, the model with only TNF (denoted by x1) and I... FN (denoted by x2) doses included, and the model that includes TNF and IFN doses and their interaction (denoted by x3 = x1 × x2). Table 1 gives the name of each model as in the R-output, the covariates included in the model, and the corresponding residual deviances and degrees of freedom. Table 1: Logistic Models for Cell Differentiation Bioassay Data Model Covariates D df model1 log (1, x1, x2) 233.25 13 model2 log (1, x1, x2, x3 = x1 × x2) 192.36 12 The regression equations for these two models are: model1 log: log( πi 1−πi ) = β0 + β1 × xi1 + β2 × xi2. model2 log: log( πi 1−πi ) = β0 + β1 × xi1 + β2 × xi2 + β3 × xi3, where i = 1, . . . , 16. Referring to the R-output, model2 log - the model with TNF and IFN doses and their interaction - is deemed to be the most appropriate. Compare ∆D = 233.25−192.36 = 40.89 to the χ 2 (1) distribution to obtain a significance level p < 0.001. As we found out that the model with interaction is the most appropriate, we use this model for inference. Normally, we would first check the fit of the model by looking at residual plots before making inferences. However, for this assignment, we first get some relevant estimates and then we check the fit of the model in part (b). The parameter estimates for model2 log (with associated standard errors in brackets) are given below: Coefficient Estimate (s.e.) β0 -1.730 (0.0705) β1 0.0252 (0.00135) β2 0.0107 (0.00123) β3 0.000384 (0.000085) We conclude that larger doses of TNF and IFN help to induce cell differentiation when considered either separately or together (coefficients corresponding to the main effects and their interaction are positive). Furthermore, the significant positive interaction suggests that 2 the probability of cell differentiation is highest at large doses of both agents (e.g. higher than one would anticipate if the two agents were acting independently of one another). A useful interpretation in terms of odds ratios is possible with the logistic model. For example, the log of odds ratio of cell differentiation at TNF and IFN doses of x1 + 1 and x2 + 1, respectively, as compared to x1 and x2 is given by: [−1.73 + 0.0252(x1 + 1) + 0.0107(x2 + 1) + 0.000384(x1 + 1)(x2 + 1)] − [−1.73 + 0.0252x1 + 0.0107x2 + 0.000384x1x2] = 0.0252 + 0.0107 + 0.000384 + 0.000384(x1 + x2) = 0.03626659 + 0.000384(x1 + x2) [Show More]

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