Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 5K (Statistical Modelling)

Write $Y_{gj}$ for the yield in replicate $j$ of genotype $g\in \{aa,Aa,AA\}$. Model 1 is the one-way cell-means modelThus its three coefficients estimate the three genotype means directly.

Put $D_i=1\{{\text{count}}_i\ge 1\}$. Model 2 isIt imposes ${\mu }_{Aa}={\mu }_{AA}={\beta }_0+{\beta }_1$ and ${\mu }_{aa}={\beta }_0$. 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.

The displayed analysis of variance compares Model 2 with the unrestricted three-mean Model 1. It testswhich is precisely full dominance: once an $A$ allele is present, a second copy has no further effect. Under $H_0$,The p-value $0.01249$ rejects full dominance at the $5\%$ level, providing evidence that the heterozygote and $AA$ means differ.

No dominance says that mean yield is affine in the number of $A$ alleles:Equivalently, $2{\mu }_{Aa}={\mu }_{aa}+{\mu }_{AA}$. Fit that reduced model and compare it with Model 1 using

r
model3 <- lm(yield ~ count, data = potato)
anova(model3, model1)

The resulting one-degree-of-freedom F-test tests the no-dominance constraint against unrestricted genotype means.