Description

1. Revisit the Reed frog survival data, data(reedfrogs), and add the predation and size treatment variables to the varying intercepts model. Consider models with either predictor alone, both predictors, as well as a model including their interaction. What do you infer about the causal influence of these predictor variables? Also focus on the inferred variation across tanks (the σ across tanks). Explain why it changes as it does across models with different predictors included.
2. In 1980, a typical Bengali woman could have 5 or more children in her lifetime. By the year 2000, a typical Bengali woman had only 2 or 3. You’re going to look at a historical set of data, when contraception was widely available but many families chose not to use it. These data reside in data(bangladesh) and come from the 1988 Bangladesh Fertility Survey. Each row is one of 1934 women. There are six variables, but you can focus on two of them for this practice problem:
(1) district: ID number of administrative district each woman resided in
(2) use.contraception: An indicator (0/1) of whether the woman was using contraception
The first thing to do is ensure that the cluster variable, district, is a contiguous set of integers. Recall that these values will be index values inside the model. If there are gaps, you’ll have parameters for which there is no data to inform them. Worse, the model probably won’t run. Look at the unique values of the district variable:
sort(unique(d$district))
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 [26] 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
[51] 51 52 53 55 56 57 58 59 60 61
District 54 is absent. So district isn’t yet a good index variable, because it’s not contiguous. This is easy to fix. Just make a new variable that is contiguous. This is enough to do it:
d$district_id <- as.integer(as.factor(d$district)) sort(unique(d$district_id))
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 [26] 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
[51] 51 52 53 54 55 56 57 58 59 60
Now there are 60 values, contiguous integers 1 to 60.
Now, focus on predicting use.contraception, clustered by district_id. Fit both (1) a traditional fixed-effects model that uses an index variable for district and
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(2) a multilevel model with varying intercepts for district. Plot the predicted proportions of women in each district using contraception, for both the fixed-effects model and the varying-effects model. That is, make a plot in which district ID is on the horizontal axis and expected proportion using contraception is on the vertical. Make one plot for each model, or layer them on the same plot, as you prefer. How do the models disagree? Can you explain the pattern of disagreement? In particular, can you explain the most extreme cases of disagreement, both why they happen where they do and why the models reach different inferences?
3. 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 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 before. Compare the varying intercepts model and a model that ignores individuals, using both WAIC/LOO and posterior predictions.
What is the impact of individual variation in these data?