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STAT 741 SAMPLE PROBLEMS
Note: Some problems were taken from previous STAT 741 courses using different notation and textbooks.
- Consider the following incomplete experimental design in three fac- tors, each at two levels. In regression notation, the model is expressed as a regression model in variables xj = ±1, j = 1, 2 , 3. The data follow a main effects model Y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + e. (a) Suppose the design matrix is
X =
.
This is a 2^3 III− 1 fractional factorial design. Its defining relation is I = ABC. Write the vector of observations as Y = [a, b, c, abc]T^ , following the notation that whenever a appears, Factor A is set at its high level, and whenever a is absent, Factor A is set at its low level, etc. For example, a represents the observation β 0 + β 1 − β 2 − β 3 + e. Find the least squares estimates of the βj and calculate Var βˆj , j = 0 ,... , 3.
(b) Suppose that the AB interaction term β 12 x 1 x 2 is also present in the model. Which main effect terms, if any, can be estimated using this design?
(c) An alternative design (the one-at-a-time design) is given by the matrix
Z =
with (ZT^ Z)−^1 =
.
For this design, the observation vector is [a, b, c, (1)]T^. Assuming the main effects model, verify that the estimate of the coefficient βj , j = 1, 2 , 3, involves the difference of only two observations. Which design is preferable? Explain.
- Consider the balanced two way ANOVA model
Yijk = μ + αi + βj + γij + eijk,
i = 1,... , I; j = 1,... , J; k = 1,... , K.
(a) Assuming that the interaction parameters are all zero, show that Y¯i·· + Y¯·j· − Y¯··· is the least squares estimator of the cell mean μij = E[Yijk].
(b) In an actual data set, the following ANOVA table was obtained. The entries are Type I sums of squares.
Source SS d.f. F A 896.75 3 15. B 46.50 2 1. A*B 641.50 6 5. Error 455.04 24
The F statistics for A and A*B are significant at the 0.001 level. Should these results be used to test for main effects? For interactions? Explain your answers.
(c) Assuming that the interaction terms are nonzero, is the contrast α 1 − α 2 estimable?
- Consider the balanced nested model
Yijkl = μ + αAi + α B(A) ij +^ α
C(AB) ijk +^ eijkl,
where i = 1,... , I, j = 1,... , J, k = 1,... , K and l = 1,... , L. All param- eters represent fixed effects, B is nested within A, and C is nested within A and B. The eijkl are i.i.d N (0, σ^2 ).
(a) Set up the ANOVA table, including formulas for all sums of squares, expected mean squares, and degrees of freedom.
(b) What is the F statistic for testing H 0 : αAi = 0, i = 1,... , I, against a general alternative? What is its distribution under the null hypothesis?
(c) What is the F statistic for testing H 0 : αC ijk(AB ) = 0, k = 1,... , K, against a general alternative? What is its distribution under the null hypothesis?
(d) How, if at all, would your answers to (b) and (c) differ if the block effect was thought to be a fixed effect?
- Consider the two way mixed model Yij = τi + bj + eij , i = 1,... , I, j = 1,... , J, where bj ∼ N (0, σ^2 b ), eij ∼ N (0, σ e^2 ), and the bj and eij are mutually independent.
(a) Write out the usual ANOVA table for this problem, including the sums of squares, degrees of freedom and expected mean squares. With no assumptions on the parameters, what is the joint distribution of the sums of squares and the sample treatment means Y¯i·?
(b) How would you test the hypothesis of no treatment differences, that is, H 0 : τ 1 = · · · = τI? What is the distribution of your test statistic, under both H 0 and the alternative?
(c) How would you test H 0 : τ 1 = · · · = τI = 0? What is the distribution of your test statistic, under both H 0 and the alternative?
- Let Yij = μ+ai +eij , i = 1,... , I, j = 1,... , J, be data from a one-way random effects ANOVA, where the ai are i.i.d. N (0, σ a^2 ) and the eij are i.i.d. N (0, σ^2 e ).
(a) Write out the usual ANOVA table and compute the expected mean squares, E(M SA) and E(M SE ).
(b) Find the distribution of the statistic F = M SA/M SE under general conditions.
(c) Find a 1 − α confidence interval for the intraclass correlation coefficient
ρ =
σ^2 a σ^2 a + σ^2 e
- In an agricultural study, the weight in pounds (Y ) and age in weeks (x) were recorded for samples of turkeys selected from three different treatment groups. The following (full) model was fitted to the data:
Yij = β 0 + β 1 xij + α 1 zij + α 2 wij + eij ,
where i = 1, 2 , 3 indexes treatment groups, j = 1,... , Ji indexes turkeys within group, zij = I{i = 1}, wij = I{i = 2}, and I{·} denotes the indicator function of an event. The sample sizes were J 1 = 4, J 2 = 4, and J 3 = 5. Least squares analysis of this model yielded R^2 = 97.94%. By contrast, when the simple linear regression model (reduced model)
Yij = β 0 ∗ + β 1 ∗ xij + eij
was fitted to the data, it was found that R^2 = 64.77%.
(a) The experimenters claimed that the large differences in R^2 showed that the treatment differences were significant. Can this statement be ver- ified? If so, calculate an appropriate test statistic and give its distri- bution under the null hypothesis of no treatment differences. If not, explain why not.
(b) How would you test whether the mean difference between Groups 1 and 2 was nonzero, assuming this comparison had been planned in advance? Would the same testing procedure be used if this comparison was suggested by examination of the data?
- Suppose we wish to compare the effects of three drugs on people by measuring some response Y. Let Yij be the response of the jth person taking the ith drug, i = 1, 2 , 3, j = 1,... , J. Assume all the error terms are independent N (0, σ^2 ).
(a) Describe a one-way ANOVA model appropriate for this problem.
(b) Suppose we know that the effect of a drug depends quadratically on the age of the person. Explain how to model this problem.
(c) Suppose there is no interaction between the age and the type of drug. Explain how to model this problem.
(d) Summarize your models in (a), (b), (c) in matrix form.
- Let Yijk, i = 1,... , I, j = 1,... , J, satisfy the mixed effects model
Yijk = μ + αi + βj + cij + eijk
where the cij are i.i.d. N (0, σ C^2 ), the eijk are i.i.d. N (0, σ^2 e ), and the cij and eijk are independent.
