9.12 A researcher wants to do an experiment using mice as the experimental units. She comes to you for advice on how to set up the experiment. Here is the situation: She wants to compare the effects of feeding three levels of calcium, say 0, 1, 2 units, on certain bone measurements (which can be observed only after the mice have been sacrificed). She has available 4 litters of 6 mice each, each litter containing 3 females and 3 males. Each litter comes from a different breed of mice. (i) Give two experimental designs that might be suitable for this experiment. For each design

(a) state any assumptions that would have to be made.

(b) Give the associated linear model.

(c) Sketch the ANOVA (source of variation, d.f., E(MS)).

(d) State what inferences can be made. (ii) Explain how you would investigate the question that the two sexes react differently to the calcium treatments. (iii) The experimenter plans to make duplicate measurements on each animal. For both designs [given in (i)] give the estimators for the linear and quadratic effect of calcium and their standard error.