Homework problems for Random Effects
1. A genetics study with beef animals consisted of several sires each mated to a separate group of dams. The matings that resulted in male progeny calves were used for an inheritance study of birth weights. The birth weights of eight male calves in each of five sire groups follow. ( Source:
Dr. S. DeNise, Department of Animal Sciences, University of Arizona )
Sire
177
200
201
Birthweights
61, 100, 56, 113, 99, 103, 75, 62
75, 102, 95, 103, 98, 115, 98, 94
58, 60, 60, 57, 57, 59, 54, 100
202
203
57, 56, 67, 59, 58, 121, 101, 101
59, 46, 120, 115, 115, 93, 105, 75 a. Assume a random model for this study. Perform the analysis of variance, write and test the hypotheses, and show the expected mean squares. b. Estimate the components of variance for sires and progeny within sires, and determine the
90% confidence interval estimates. c. Estimate the intraclass correlation coefficient, and give the 90% confidence interval estimate.
2. A plant pathologist took four 3-pound samples from 50-ton lots of cottonseed accumulated at
5
6
7
8 various cotton gins during the ginning season. The samples of seed were analyzed in the laboratory for Aflatoxin, which is a toxin produced by organisms associated with the seeds. The
Aflatoxin concentrations in parts per billion for samples from eight lots of cottonseed follow.
( Source: Dr. T. Russell, Department of Plant Pathology, University of Arizona )
2
3
4
Lot
1
Aflatoxin (ppb)
39, 57, 63, 66
56, 13, 25, 31
64, 83, 88, 71
29, 55, 21, 51
38, 66, 53, 81
11, 49, 34, 10
23, 0, 5, 20
10, 11, 23, 37 a. Assume the lots are random effects. Perform the analysis of variance, write and test the hypotheses, and show the expected mean squares. b. Estimate the components of variance. c. Use the estimates in part b to estimate the total variation in Aflatoxin ( y ). d. What percent of the total variation in Aflatoxin can be attributed to variation among lots?
3. Cholesterol was measured in the serum samples of five randomly selected patients from a pool of patients. Two independent replicate tubes were prepared for each patient for each of four runs on a spectrophotometer. The objective of the study was to determine whether the relative cholesterol measurements for patients were consistent from run to run in the clinic. The data are

4
3 mg/dl of cholesterol in the replicate samples from each patient on each run. ( Source: Dr. J.
Anderson, Beckman Instruments, Inc )
Patient
Run
1
2
1 2 3
167.3 186.7 100
4 5
214.5 148.5
166.7 184.2 107.9 215.3 148.5
179.6 193.8 111.6 228.9 158.6
175.3 198.9 114.4 220.4 154.7
169.4 179.4 105.9 208.2 144.7
165.9 177.6 104.1 207.1 145.9
177.7 190.4 113.4 221 156.1
177.1 192.4 114.6 219.7 151 a. Perform the analysis of variance for this data set, write and test the hypotheses, and show the expected mean squares (assume run and patient are random effects). b. Estimate the components of variance for runs, patients, and interaction.
4. An animal scientist conducted an experiment to study the effect of water quality on feedlot performance of steer calves. Four water quality treatments were used for the experiment. The water sources were designated as normal (N) and saline (S). The saline water was formulated to approximate the mineral concentrations in some underground water sources utilized in practice for watering livestock. Four combinations of water used in two consecutive 56-day periods of the experiment were N-N, N-S, S-N, and S-S. The feeding trial consisted of the four water treatments with two replicate pens of animals for each treatment in a completely randomized design. The trial was conducted on two separate occasions (two consecutive summers). The resulting design is a factorial arrangement of four water treatments and two summers. The water treatments are considered fixed effects and summers are considered random effects. The data are the average daily gains for the 16 pens of steers. ( Source: Dr. D. Ray, Department of Animal
Sciences, University of Arizona )
2
Summer
1
Water
N-N N-S S-N S-S
2.65 2.46 2.56 2.43
2.53
2.25
2.2
2.36
1.95
2.25
2.38
2.01
1.98
2.5
2.14
2.37 a. Perform the analysis of variance for this data set, write and test all appropriate hypotheses, and show the expected mean squares (be careful with stating and interpreting the hypotheses). b. Estimate appropriate variance components. c. If appropriate, conduct multiple comparisons.