In Table S2 of their Appendix, Villa et al. fit the eight-dimensional factorial model host:sex:time to the first principal component values on 3096 lice. Show that this is equivalent to fitting four separate linear regressions \(E\left(Y_{u}ight)=\alpha+\beta t_{u}\), with one intercept and one slope for each of the disjoint subgroups, Fer.F, Fer.M, Gr.F, Gr.M. Feral and female are the reference levels, so \(\operatorname{sex}_{u}=1\) is the indicator vector for males. Deduce that the host:time coefficient is equal to the slope difference \(\beta_{\text {Gr.F }}-\beta_{\text {Fer.F }}\) restricted to female lice. The fitted value is 0.009 . What is the fitted slope difference for male lice?

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MS Host 12 Aviary 290 Lineage 83 Residual 111 8 aviary lineage resid 36 1024 324 80 100 52 56 Variance-component estimates (x10) 2.1 3.2 10.7 10.4 9.5 6.5 6.7 -1.1 2.8 3.3 3.8 2.8 2.2 111.3 53.1 59.4 67.3 56.9 52.5 0 6 12 340 190 204 408 85 85 52 60 Time t in months 24 30 1095 690 493 333 87 56 18 996 575 94 68 5.5 56.1 42 3448 528 79 57 10.6 0.3 58.4 48 1506 350 152 64 6.6 12.1 63.3