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Look into this preprint referenced in this tweet for recent ideas around Bayesian survival analysis.

Consider referencing Popoviciu's inequality on variances to define the upper limit of the variance for bounded continuous variables. https://en.wikipedia.org/wiki/Popoviciu%27s_inequality_on_variances. Consider the CES-D scale in Chapter 5. The lowest score is 0 and the highest score is 80. Using the formula, we can compute the maximum variance and standard deviation like so.

highest <- 80
lowest <- 0

# max variance
(highest - lowest)^2 / 4

# max sd
sqrt((highest - lowest)^2 / 4)

In this case, the maximum values are 1600 and 40, respectively.

You can confirm the inequality with sample statistics like so.

var(rep(c(0, 80), each = 1e5))
sd(rep(c(0, 80), each = 1e5))

A nice thing about the sample statistic version is it gives an intuition about what the expected standard deviation would be if, say, you expected a flat distribution for the CES-D.

sd(rep(0:80, each = 1e5))

In this case, the standard deviation is about 23.4.

• Add a basic line plot of the opposites_pp data at the top of the chapter, something like

opposites_pp %>% 
  ggplot(aes(x = time, y = opp, group = id)) +
  geom_line(size = 1/4, alpha = 1/4)

• Somewhere toward the end of the chapter, explicitly mention how the alternatives to the standard multilevel model of change do not have id-specific equations (what frequentists would call empirical Bayes).