Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 13J / b

Table of contents
- Solution b

Solution

Model 2 is obtained from model 3 by setting

β_{0} = 0

. Model 3 is contained in model 1 by choosing the symmetric block parameters

β_{k ℓ}^{(1)} = {β_{k} + β_{ℓ}, 2 β_{k} + β_{0}, k \neq = ℓ, k = ℓ .

Hence

model 2 \subset model 3 \subset model 1 .

Assuming every college-pair cell is observed and the models are identifiable, model 1 has

C (C + 1) /2

parameters and model 3 has

C + 1

. The likelihood-ratio test statistic for model 3 against model 1 is

D = 2 {ℓ_{1} (β_{1}) - ℓ_{3} (β_{3})} .

Under model 3 and the regularity conditions of Wilks theorem,

D d χ_{C (C + 1) /2 - (C + 1)}^{2} = χ_{(C - 2) (C + 1) /2}^{2} .

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