Solution

Write

Y_{g j}

for the yield in replicate

j

of genotype

g \in {aa, A a, AA}

. Model 1 is the one-way cell-means model

Y_{g j} = μ_{g} + ε_{g j}, ε_{g j} \sim iid N (0, σ^{2}) .

(18)

Thus its three coefficients estimate the three genotype means directly.

Put

D_{i} = 1 {count_{i} \geq 1}

. Model 2 is

Y_{i} = β_{0} + β_{1} D_{i} + ε_{i}, ε_{i} \sim iid N (0, σ^{2}) .

(19)

It imposes

μ_{A a} = μ_{AA} = β_{0} + β_{1}

and

μ_{aa} = β_{0}

. The reported finite-sample p-values assume independent normal errors, a common unknown variance in every genotype, fixed full-rank design matrices, and correctness of the relevant mean model. Random assignment of genotypes supports interpreting the fitted differences as treatment effects.

Solved by gpt-5.6-sol high.

 Ancestors (10)

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