STA442/1008 Assignment 8

Quiz in Tutorial on Friday March 16th

  1. Consider a two-factor analysis of variance in which each factor has two levels. Use this regression model for the problem:

    E[Y|X] = β0 + β1d1 + β2d2 + β3d1d2

    where d1 and d2 are dummy variables.

    1. Make a two-by-two table showing the four treatment means in terms of β values. Use effect coding (the scheme with the minus ones). In terms of the β values, state the null hypothesis you would use to test for
      1. Main effect of the first factor
      2. Main effect of the second factor
      3. Interaction
    2. Make another two-by-two table showing the four treatment means in terms of β values. Use indicator dummy variables (just zeros and ones ones). In terms of the β values, state the null hypothesis you would use to test for
      1. Main effect of the first factor
      2. Main effect of the second factor
      3. Interaction
    3. Which dummy variable scheme do you like more for this purpose? Why?

      

  2. I know this is pretty gruesome, but the data are real -- from the U of T School of Dentistry.

    An experiment in dentistry seeks to test the effectiveness of a drug (HEBP) that is supposed to help dental implants become more firmly attached to the jaw bone. This is an initial test on animals. False teeth were implanted into the leg bones of rabbits, and the rabbits were randomly assigned to receive either the drug or a saline solution (placebo). Technicians administering the drug were blind to experimental condition.

    Rabbits were also randomly assigned to be "sacrificed" after either 3, 6, 9 or 12 days. At that time, the implants were pulled out of the bone by a machine that measures force in newtons and stiffness in newtons/mm. For both of these measurements, higher values indicate more healing. A measure of "pre-load stiffness" in newtons/mm is also available for each animal. This may be another indicator of how firmly the false tooth was implanted into the bone, but it might even be a covariate. Nobody can seem to remember what "preload" means, so we'll ignore this variable for now.

    The data are available in the file bunnies.data. The variables are

    1. Identification code
    2. Time (3,6,9,12 days of healing)
    3. Drug (1=HEBP, 0=saline solution)
    4. Stiffness in newtons/mm
    5. Force in newtons
    6. Preload stiffness in newtons/mm

    Please do the following.

    1. Classify the factors as within cases or between cases.
    2. Use proc freq to find out how many rabbits are in each experimental condition.
    3. Using proc glm, conduct a two-way ANOVA, with force as the dependent variable. Use the means statement to get cell means and marginal means. Be prepared to answer the following questions about each of the significance tests that SAS produces by default (I count 4 default tests).
      1. What is the value of the test statistic? The answer is a number from your printout.
      2. What proportion of the remaining variation is explained? Better use proc iml.
      3. What is the p-value? The answer is a number from your printout.
      4. Is the result statistically significant at the 0.05 level? Yes or No.
      5. What, if anything, do you conclude? This is not the place for statistical jargon. "What do you conclude" means say something about the drug, healing, time -- something like that.
    4. I know this is a bit redundant with the preceding question, but did the drug work? If the results justify an answer, then answer Yes or No.
    5. Now, make a table with a row for each treatment combination. Give the coefficients of the constrast or contrasts that would be used to test for
      1. The main effect of Drug
      2. The main effect(s) of Time
      3. The Drug by Time interaction.
    6. Make another table with a row for each treatment combination. Make columns showing the dummy variables for effect coding.
    7. Give E[Y|X=x] for a regression model with both main effects and the interaction. Use your variable names from the preceding question.
    8. In terms of the β values of your regression model, give the null hypothesis you would test in order to answer each of the following questions.
      1. Averaging across time periods, is there a differnece between the drug and placebo in mean force required to extract the tooth?
      2. Averaging across drug and placebo, is does elapsed time affect the mean force required to extract the tooth?
      3. Does the effect of the drug depend upon elapsed time?
    9. Now please return to SAS. Using proc reg and cell means coding, conduct tests to answer the following questions. Just do regular one-at-a-time (custom) tests. Don't bother with any Bonferroni correction this time. Just consider one dependent variable: Force. As usual, we are guided by the alpha = 0.05 significnce level.
      1. Are the marginal means different at 3 and 6 days?
      2. Are the marginal means different at 6 and 9 days?
      3. Are the marginal means different at 9 and 12 days?
      4. Is there a difference between Drug and Placebo just at 3 days?
      5. Is there a difference between Drug and Placebo just at 6 days?
      6. Is there a difference between Drug and Placebo just at 9 days?
      7. Is there a difference between Drug and Placebo just at 12 days?
      Be able to answer questions like these for each test:
      1. What is the value of the test statistic? The answer is a number.
      2. What is the p-value? The answer is a number.
      3. Is the result statistically significant at the 0.05 level? Yes or No.
      4. What, if anything, do you conclude? This is not the place for statistical jargon. "What do you conclude" means say something about the drug, healing, time -- something like that.

 

 

Please bring your log file and your list file to the quiz. As usual, answers to the questions are not to be handed in. They are just practice for the quiz. Please do not write anything on your printouts except your name and student number. It is okay to highlight the list file.