The theme of this homework is tadpoles. You must keep them alive.
1. Conduct a prior predictive simulation for the Reedfrog model. By this I mean to simulate the prior distribution of tank survival probabilities αj. Start by using this prior:
αj ∼ Normal(¯α,σ) α¯ ∼ Normal(0, 1) σ ∼ Exponential(1)
Be sure to transform the αj values to the probability scale for plotting and summary. How does increasing the width of the prior on σ change the prior distribution of αj? You might try Exponential(10) and Exponential(0.1) for example.
2. Revisit the Reedfrog survival data, data(reedfrogs). Start with the varying effects model from the book and lecture. Then modify it to estimate the causal effects ofthetreatmentvariablespredandsize, includinghowsizemightmodifytheeffect of predation. An easy approach is to estimate an effect for each combination of pred and size. Justify your model with a DAG of this experiment.
3. Now estimate the causal effect of density on survival. Consider whether pred modifies the effect of density. There are several good ways to include density in your Binomial GLM. You could treat it as a continuous regression variable (possibly standardized). Or you could convert it to an ordered category (with three levels). Compare the σ (tank standard deviation) posterior distribution to σ from your model in Problem 2. How are they different? Why?
4-OPTIONAL CHALLENGE. Return to the Trolley data, data(Trolley), from Chapter 12. Define and fit a varying intercepts model for these data. By this I mean to add an intercept parameter for the individual participants to the linear model. Cluster the varying intercepts on individual participants, as indicated by the unique values in the id variable. Include action, intention, and contact as treatment effects of interest. Compare the varying intercepts model and a model that ignores individuals. What is the impact of individual variation in these data?
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