1. A drug company wants to study three new types of medicine (T) for the treatment...

 

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1. A drug company wants to study three new types of medicine (T) for the treatment of a certain kind of skin disease. The response variable Y in this study is "time to heal" (in days); i.e., how long would it take for the patient to recover after receiving the new medicine. It is also believed that Y is linearly related to the age X of the patient. All together 47 patients are involved in this study and the first 6 are listed below. Patient Number "time to heal" i 1 2 3 4 1 46.5 22.1 26.3 29.2 47.6 30.0 1 age Type of Medicine X 42 18 29 31 39 21 # T 1 1 2 2 3 3 In below X is considered as a continuous regressor while T is considered as a factor with 3 levels. The following three models, all with an intercept term were fitted: Model 1: General; i.e., different medicines have different intercepts and slopes Model 2: Parallel; i.e., all medicines have the same slope but different intercepts Model 3: Common regression; i.e., all medicines have the same slope and intercept (a) Write down the regression model for Model 1. Also write down the first six rows for the corresponding model matrix X. Note that you need to define new indicator variable(s). (b) Write down the regression model for Model 2. Also write down the first six rows for the corresponding model matrix X. (c) The residual sum of squares (RSS) for Models 1, 2 and 3 are, respectively, 40.3, 41.4, and 939.8. First test if Model 1 or Model 3 is preferred. Then test if Model 1 or Model 2 is preferred. 1. A drug company wants to study three new types of medicine (T) for the treatment of a certain kind of skin disease. The response variable Y in this study is "time to heal" (in days); i.e., how long would it take for the patient to recover after receiving the new medicine. It is also believed that Y is linearly related to the age X of the patient. All together 47 patients are involved in this study and the first 6 are listed below. Patient Number "time to heal" i 1 2 3 4 1 46.5 22.1 26.3 29.2 47.6 30.0 1 age Type of Medicine X 42 18 29 31 39 21 # T 1 1 2 2 3 3 In below X is considered as a continuous regressor while T is considered as a factor with 3 levels. The following three models, all with an intercept term were fitted: Model 1: General; i.e., different medicines have different intercepts and slopes Model 2: Parallel; i.e., all medicines have the same slope but different intercepts Model 3: Common regression; i.e., all medicines have the same slope and intercept (a) Write down the regression model for Model 1. Also write down the first six rows for the corresponding model matrix X. Note that you need to define new indicator variable(s). (b) Write down the regression model for Model 2. Also write down the first six rows for the corresponding model matrix X. (c) The residual sum of squares (RSS) for Models 1, 2 and 3 are, respectively, 40.3, 41.4, and 939.8. First test if Model 1 or Model 3 is preferred. Then test if Model 1 or Model 2 is preferred.