Statistics 850 yandell
Homework # 8-Due Mon 17 Apr
This problem comes directly from examples in Chapters 18-20 of Milliken and Johnson. The SAS runs are available in the public data directory /p/stat/Data/MJ as files of the form wheat.* and random.*. There are a number of errors in the MJ book. For instance, the E(MS) below Table 18.1 are not all correct. Keep this in mind as you compare your findings with the text. Note also that their notation changes quite a bit in this region of the text.
1. Consider the one-way random experiment concerning wheat varieties.
(a) Justify in words the one-way random model for this experiment.
Why is it appropriate.
(b) Write down the model with all assumptions. Indicate in a small
table the sample sizes.
(c) Test whether there is any variation among wheat varieties in terms
of insect damage.
(d) Estimate the (two) variance components using one of the four methods
discussed in class. Briefly justify your choice of method-are you
selecting it because it is easy to explain to scientist or because it
has some (specified) ``optimal'' properties?
(e) Construct a 95% confidence interval for each variance component
using the estimate from (d). Justify your approximation, whether it
be based on chi-square or normal distribution (see ch. 20 of MJ).
(f) Estimate the grand mean insect damage for wheat.
(g) Estimate the variance of the grand mean. Careful-the design is
not balanced and you may need to make some adjustments. See details
in my chapter.
(h) Construct a 95% confidence interval for the grand mean.
2. Consider the two-way random experiment labelled simply as row and
column treatments.
(a) Describe in words an experiment resulting in this data.
That is, identify realistic treatments instead of ``row'' and
``column''. Justify your choice as a random rather than fixed model.
(b) Write down the model with all assumptions. Indicate in a small
table the sample sizes.
(c) Test whether there is any variation among row and column
treatments, and any evidence of row*column interaction.
(d) Why are the Type I estimates of main effects variance components
negative? Is this consistent with the results of tests from part (c)?
(d) Estimate the interaction and error variance components using
ML or REML. Briefly justify your choice of method-are you
selecting it because it is easy to explain to scientist or because it
has some (specified) ``optimal'' properties?
(e) Construct a 95% confidence interval for the two variance components
estimated in (d). Justify your approximation, whether it
be based on chi-square or normal distribution (see ch. 20 of MJ).
(f) Estimate the grand mean.
(g) Determine the variance of the mean if all variance components were
known for this experiment. Careful-the design is
not balanced and you may need to make some adjustments. Thus my notes
provide a start, but you must
adjust for unequal sample sizes. [Hint: use the model and determine
what the sample grand mean (
) is in terms
of model components. Then find its variance.]
(h) Estimate the variance of the grand mean using the variance
component estimates. [Do not try to construct a confidence
interval-you would need to estimate df which is rather complicated!]