statisticalrethinkin..

7.3. CONTINUOUS INTERACTIONS 239water.c = -1water.c = 0water.c = 1blooms0 100 200 300blooms0 100 200 300blooms0 100 200 300-1 0 1shade (centered)-1 0 1shade (centered)-1 0 1shade (centered)water.c = -1water.c = 0water.c = 1blooms0 100 200 300blooms0 100 200 300blooms0 100 200 300-1 0 1shade (centered)-1 0 1shade (centered)-1 0 1shade (centered)FIGURE 7.7. Triptych plot of predicted blooms across water treatments,without (top row) and with (bottom row) an interaction effect. Blue pointsin each plot are data. e solid line is the posterior mean and the dashedlines give 95% interval of the mean. Top row: Without the interaction,model m7.8. Each of the three plots of blooms against shade level is fora different water level. e slope of the regression line in each case is exactlythe same, because there is no interaction in this model. Bottom row:With the interaction, model m7.9. Now the slope of blooms against shadehas a different value in each plot.the mean water value in the middle plot, increasing shade clearly reduces the size of blooms.Going from the least (−1) to the most shade, blooms are approximately halved in size. Atthe highest water value, on the right, the slope becomes even more negative, and shade is aneven stronger predictor of smaller blooms, as shade increases. Going from the least to themost shade, bloom size declines by about two-thirds.What is going on here? e likely explanation for these results is that tulips need bothwater and light to produce blooms. At low water levels, shade can’t have much of an effect,because the tulips don’t have enough water to produce blooms anyway. At higher water levels,shade can matter more, because the tulips have enough water to produce some blooms.At very high water levels, water is no longer limiting the blooms very much, and so shade canhave a much more dramatic impact on the outcome. e same explanation works symmetricallyfor shade. If there isn’t enough light, then more water hardly helps. You could remakeFIGURE 7.7 with water on the horizontal axes and shade level varied from le to right, if you’dlike to visualize the model predictions that way.