Last updated: 2018-05-21

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    Rmd 9548ba6 stephens999 2018-05-21 workflowr::wflow_publish(“eight_schools.Rmd”)
    Rmd 6fbf7cb stephens999 2018-04-11 add eight schools

Introduction

Here I apply the adaptive shrinkage EB method to the eight schools data at (http://andrewgelman.com/2014/01/21/everything-need-know-bayesian-statistics-learned-eight-schools/)

Enter the data

x = c(28,8,-3,7,-1,1,18,12)
s = c(15,10,16,11,9,11,10,18)

Plot the data

Notice how the intervals very much overlap… suggesting the data may be consistent with no differences

library(rmeta)
metaplot(x, s)

Apply adaptive shrinkage

Here we apply the adaptive shrinkage package, which solves the normal means problem with unimodal prior. By default the mode is 0, which is not appropriate for these data. So we estimate the mode:

library(ashr)
a = ash(x,s,mode="estimate")
get_pm(a) # get the posterior mean
[1] 7.685617 7.685617 7.685617 7.685617 7.685617 7.685617 7.685617 7.685617

We can see that ash shrinks all the estimates to the same value, meaning the data are “most” consistent with no variation in effect.

Check adaptive shrinkage

We might be worried maybe that we mis-used the software, or that it is not working right. Does it always overshrink like that?

When you get an unexpected result it is important to go back and check your understanding. A good way to do this is to make a prediction and then test it.

For example, here I predict that if the standard errors for these data had been much smaller (with same x) then ash should not shrink as much, because the data would be more convincing of true variation in effect.

I can test this prediction by dividing s by 10 say:

metaplot(x,s/10)

a = ash(x,s/10,mode="estimate")
get_pm(a)
[1] 26.766778  7.971568 -2.957616  7.918593 -1.000000  1.000000 17.999958
[8]  9.721773

We can see my prediction was correct. So this gives me more confidence that I used the software correctly and that it is behaving sensibly.

Remember: The most valuable opportunties to learn come from when you see something that you did not expect! Do not ignore unexpected results! They will sometimes just be silly errors, but other times they will reflect a gap in your understanding. And - sometimes - a gap in the understanding of the whole research community.

Session information

sessionInfo()
R version 3.3.2 (2016-10-31)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X El Capitan 10.11.6

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] ashr_2.2-7 rmeta_3.0 

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.16      knitr_1.20        whisker_0.3-2    
 [4] magrittr_1.5      workflowr_1.0.1   REBayes_1.3      
 [7] MASS_7.3-49       pscl_1.5.2        doParallel_1.0.11
[10] SQUAREM_2017.10-1 lattice_0.20-35   foreach_1.4.4    
[13] stringr_1.3.0     tools_3.3.2       parallel_3.3.2   
[16] grid_3.3.2        R.oo_1.22.0       git2r_0.21.0     
[19] htmltools_0.3.6   iterators_1.0.9   assertthat_0.2.0 
[22] yaml_2.1.18       rprojroot_1.3-2   digest_0.6.15    
[25] Matrix_1.2-14     codetools_0.2-15  R.utils_2.6.0    
[28] evaluate_0.10.1   rmarkdown_1.9     stringi_1.1.7    
[31] Rmosek_7.1.2      backports_1.1.2   R.methodsS3_1.7.1
[34] truncnorm_1.0-7

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