Certain natural populations of the plant Arabidopsis halleri have two genetically encoded leaf types. Some individuals have trichomes on the leaves (“hairy”) and others lack trichomes (“naked”). One hypothesis to explain the persistence of both leaf types over time is that if either leaf type becomes more abundant than the other in the population, herbivores target it and reduce its abundance relative to the other leaf type. Sato and Kudoh (2017) designed an experiment to test this idea. They created plots with mixtures of the two leaf types of A. halleri plants. In some plots, hairy plants were more abundant than naked plants (H>N). In the other plots, naked plants were more abundant than hairy plants (N>H). They introduced the natural beetle herbivore Phaedon brassicae to the plots and recorded the average amount of leaf area lost to the beetles. The hypothesis predicts that the amount of leaf area lost to hairy and naked plants should depend on the relative abundance of the two leaf types. In other words, there should be an interaction between the two factors “leaf type” and “relative abundance.” The ANOVA results are below.

a. The F test of the interaction term compares the fits of the two models to the data. The full model includes the main effects and the interaction. The null model for the test includes only the main effects and not the interaction term. The fits of the two models to the data are shown in the figure below. Lines connect the fitted means for the two model fits (solid=hairy, dashed=naked). Which panel (left or right) shows the fit of the full model, with the interaction included? Explain how you decided this.

b. Examine the accompanying ANOVA table. Was the interaction term statistically significant?

Explain how you decided this.

c. Which main effects were statistically significant, and which were not?

d. Can we conclude from the ANOVA table that the factor leaf type makes no statistically significant contribution to leaf area loss by herbivores? Why or why not?

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Source of variation ABUNDANCE ABUNDANCE LEAFTYPELEAFTYPE ABUNDANCE* LEAFTYPE ABUNDANCE* LEAFTYPE ResidualsResiduals TotalTotal Sum of squares did f 0.046610.04661 0.000000.00000 0.054970.05497 Mean squares 11 0.046610.04661 FF 5.97595.9759 0.467930.46793 6060 0.467930.46793 0.569510.56951 6363 PP 0.0170.01 11 0.000000.00000 0.00040.0004 0.9840.98 11 0.054970.05497 7.0489 7.0489 0.0100.01