This problem introduces a random effects model for the one-way layout. Consider a balanc...

This problem introduces a random effects model for the one-way layout. Consider a balanced one-way layout in which the I groups being compared are regarded as being a sample from some larger population. The random effects model is

To fix these ideas, we can consider an example from Davies (1960). The variation of the strength (coloring power) of a dyestuff from one manufactured batch to another was studied. Strength was measured by dyeing a square of cloth with a standard concentration of dyestuff under carefully controlled conditions and visually comparing the result with a standard. The result was numerically scored by a technician. Large samples were taken from six batches of a dyestuff; each sample was well mixed, and from each six subsamples were taken. These 36 subsamples were submitted to the laboratory in random order over a period of several weeks for testing as described. The percentage strengths of the dyestuff are given in the following table.

b. Suppose that the samples had not been mixed, but that duplicate measurements had been made on each subsample. Formulate a model that also incorporates variability between subsamples. How could the parameters of this model be estimated?