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Homework 4

Due Feb 11 on canvas

Conceptual Exercises

(Chap 11, p. 338) 1, 2, 5, 6, 7, 8,

Do the above conceptual exercises from the Sleuth but do not hand them in as part of your homework. The answers can be found at the end of the chapter, but make an honest effort to answer them before looking. They are a useful way to gauge how much you understood and can show up on exams.

1

This homework asks you to reproduce the analysis of the second case study in Chapter 11. You can read more about this case study here

library(Sleuth3)
case1102$ratio <- with(case1102, Brain/Liver)

  1. Create a plot of the logarithm of the ratio of brain to liver antibodies against the sacrifice time, using different colours to code the treatment cateogories. Comment on the relationship between the response and the design variables.

  2. For each covariate (Sex, Weight, Loss and Tumor) create a plot to examine the relationship with the response (ratio of brain to liver antibodies). Do there seem to be any relationships? (No regression models!! No residual plots!! Think boxplots and/or scatterplots)

  3. For each covariate use plots or summary statistics to explore if there are relationships between the covariate and the treatment assignment.

  4. Fit the tentative model: \[\begin{eqnarray} \mu\{\text{log(anitbody ratio)} \, &|& \, \text{TIME},\text{ TREAT, DAYS, FEM, weight, loss, tumor}\} = \\ &&\text{TIME} + \text{ TREAT} + (\text{TIME} \times \text{ TREAT}) + \text{DAYS} + \text{FEM} + \text{weight} + \text{loss} + \text{tumor} \end{eqnarray}\] Be careful that you specify that both sacrifice time (TIME) and days after innoculation (DAYS) as entering the model as categories.

  5. Examine the residuals versus fitted values plot for evidence of violations of the assumptions, and identify the two unusual observations.

  6. Examine case influence statistics for of the fit. Do the unusaual observations seem influential?

  7. Construct partial residuals plots for the sacrifice time, sex and days after innoculation variables. Comment on whether the relationship between the response and the design variables seems to change after accounting for the covariates. And whether the relationship between the covariates and the response, remains after accounting for the treatment variables.

  8. Fit the following simpler models: Model 1: \[ \mu\{\text{log(anitbody ratio)} \, | \, \text{TIME},\text{ TREAT, DAYS, FEM, weight, loss, tumor}\} = \text{TIME} + \text{ TREAT} + (\text{TIME} \times \text{ TREAT}) + \text{DAYS} + \text{FEM} \] Model 2: \[ \mu\{\text{log(anitbody ratio)} \, | \, \text{TIME},\text{ TREAT, DAYS, FEM, weight, loss, tumor}\} = \text{TIME} + \text{ TREAT} + (\text{TIME} \times \text{ TREAT}) \] Model 3: \[ \mu\{\text{log(anitbody ratio)} \, | \, \text{TIME},\text{ TREAT, DAYS, FEM, weight, loss, tumor}\} = \text{DAYS} + \text{FEM} \] Use them to reach the conclusions that:

    • The covariates are not significant when the design variables are also included (F-test comparing Models 1 & 2)
    • The design variables are significant when the covariates are also included (F-tests comparing Models 1 & 3)
    • The conclusions regarding the design variables depend very little on whether the covariates are in the model (compare the estimates of the time, treat and time-treat interaction between Models 1 & 2).

  9. Using model 2, test whether the interaction term is significant using an F-test.

  10. Finally using the model \[ \mu\{\text{log(anitbody ratio)} \, | \, \text{TIME},\text{ TREAT, DAYS, FEM, weight, loss, tumor}\} = \text{TIME} + \text{ TREAT} \]

report a confidence interval for the difference in mean log ratio between the BD and NS treatments.