Question

"In an experiment where several treatments are compared with a control, it may be desirable to replicate the control more than the experimental treatments, since the control enters into every difference investigated. Suppose each of m experimental treatments is replicated t times while the control is replicated c times. Let Y_ij denote the j-th observation on the i-th experimental treatment, j=1,⋯,t, i=1,⋯,m, and let Y_0j denote the j-th observation on the control, j=1,⋯,c. Assume that Y_ij = τ_i + ε_ij, i=0,⋯,m, where ε_ij are independent and identically distributed (iid) N(0,σ^2) variables.

Question: Find the distribution of the least squares estimates of θ_i = τ_i - τ_0, i=1,⋯,m. Provide the distribution of these estimates in terms of their mean and variance."

Answer 1

The distribution of the least squares estimates of θi can be approximated by a normal distribution with a mean equal to the difference between the mean values of the experimental treatment and the control group, and a variance dependent on the variances of the treatments and control.

Step-by-step explanation:

The distribution of the least squares estimates of θi = τi - τ0, i=1,⋯,m, can be approximated by a normal distribution. The mean of the estimates, denoted as μθi, is equal to the difference between the mean values of the experimental treatment and the control group, i.e., μθi = τi - τ0. The variance of the estimates, denoted as σ2θi, is given by σ2θi = σ2 + σ20/c, where σ2 is the variance of the experimental treatments and σ20 is the variance of the control group.