The file Auto Accident. dat contains data on severity of drivers' injuries in auto ac- cidents...

  

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The file Auto Accident. dat contains data on severity of drivers' injuries in auto ac- cidents (Injury: 1-Not Severe, 2-Severe), along with the type of accident (Type: 1=Collision, 2 Rollover), the type of automobile (Car: 1=Small, 2-Standard), and whether or not the driver was ejected from the vehicle during the accident (Ejected: 1=No, 2-Yes). The data are also available with qualitative descriptors at Auto Accident.csv. Type Collision Rollover Car Ejected Injury Small No Not Severe 350 60 Severe 150 112 Yes Not Severe 26 19 Severe 23 80 Standard No Not Severe 1878 148 Severe 1022 404 Yes Not Severe 111 22 Severe 161 265 Analyze the data treating severity of injury as the response variable. Include the following pieces in your analysis: (a) Fit a model with main effects and a model with all two way interactions. [10 pts] (b) Report the goodness-of-fit test for both models. [10 pts] (c) Use both the likelihood ratio test and Wald test to compare the model with just main effects against the model with all two-way interactions. What do you conclude? [10 pts] (d) Report the estimates, standard errors, and Wald tests associated with each parameter in the main effects model. Give a table with the conditional odds ratios for each variable, along with 95% confidence intervals (be sure to clearly state what the baseline group is for each odds ratio). [Hint: the connection between conditional odds ratio and the value of parameter] [20 pts] 2. The file leukemia. dat contains data on the survival of 33 leukemia patients as a function of their white blood cell (wbc) count and the existence of a certain mor- phological characteristic in the cells, referred to as either AG positive or negative (AG: 1-positive, 0-negative). The binary response is survival of at least 52 weeks beyond the time of diagnosis (Survival: 1=Yes, 0=No). (a) Fit a model with separate slopes and intercepts for AG positive and negative patients (i.e., a model with AG:wbc interaction and main effects). What is the statistical significance of the interaction term? [10 pts] (b) Fit the model in part a) again using a log-transformation of wbc. Compare this model with the model in part a), which fit is better and why? [10 pts] (c) For the model in part b), calculate and plot predicted probabilities of survival as a function of AG and wbc (HINT: There should be two curves, one for AG positive and one for AG negative). By how much do the log of odds increase for each unit increase in log wbc counts for the AG positive group, and for the AG negative group? Taking all of the above together, interpret the nature of the interaction between AG and wbc. [15 pts] (d) Continuing with the model in part b), fit a model without the interaction term, and test for statistical significance of the interaction using a likelihood ratio test. Test the significance of AG and log wbc in the main effects model simultaneously by using the likelihood ratio test and Wald test. [15 pts] The file Auto Accident. dat contains data on severity of drivers' injuries in auto ac- cidents (Injury: 1-Not Severe, 2-Severe), along with the type of accident (Type: 1=Collision, 2 Rollover), the type of automobile (Car: 1=Small, 2-Standard), and whether or not the driver was ejected from the vehicle during the accident (Ejected: 1=No, 2-Yes). The data are also available with qualitative descriptors at Auto Accident.csv. Type Collision Rollover Car Ejected Injury Small No Not Severe 350 60 Severe 150 112 Yes Not Severe 26 19 Severe 23 80 Standard No Not Severe 1878 148 Severe 1022 404 Yes Not Severe 111 22 Severe 161 265 Analyze the data treating severity of injury as the response variable. Include the following pieces in your analysis: (a) Fit a model with main effects and a model with all two way interactions. [10 pts] (b) Report the goodness-of-fit test for both models. [10 pts] (c) Use both the likelihood ratio test and Wald test to compare the model with just main effects against the model with all two-way interactions. What do you conclude? [10 pts] (d) Report the estimates, standard errors, and Wald tests associated with each parameter in the main effects model. Give a table with the conditional odds ratios for each variable, along with 95% confidence intervals (be sure to clearly state what the baseline group is for each odds ratio). [Hint: the connection between conditional odds ratio and the value of parameter] [20 pts] 2. The file leukemia. dat contains data on the survival of 33 leukemia patients as a function of their white blood cell (wbc) count and the existence of a certain mor- phological characteristic in the cells, referred to as either AG positive or negative (AG: 1-positive, 0-negative). The binary response is survival of at least 52 weeks beyond the time of diagnosis (Survival: 1=Yes, 0=No). (a) Fit a model with separate slopes and intercepts for AG positive and negative patients (i.e., a model with AG:wbc interaction and main effects). What is the statistical significance of the interaction term? [10 pts] (b) Fit the model in part a) again using a log-transformation of wbc. Compare this model with the model in part a), which fit is better and why? [10 pts] (c) For the model in part b), calculate and plot predicted probabilities of survival as a function of AG and wbc (HINT: There should be two curves, one for AG positive and one for AG negative). By how much do the log of odds increase for each unit increase in log wbc counts for the AG positive group, and for the AG negative group? Taking all of the above together, interpret the nature of the interaction between AG and wbc. [15 pts] (d) Continuing with the model in part b), fit a model without the interaction term, and test for statistical significance of the interaction using a likelihood ratio test. Test the significance of AG and log wbc in the main effects model simultaneously by using the likelihood ratio test and Wald test. [15 pts]