Article in Quillette + Technical Details

Followers of my blog who haven’t already seen it might enjoy this opinion piece I wrote for Quillette magazine. I have wanted to write something like that for a while, but never thought I’d get around to it. I’m glad I proved myself wrong!

Here are the technical details of the calculations I did to get the results in the article. If you’re comfortable with an analytical derivation, see this one by Jared Tobin.

For both the Bayesian and frequentist calculations I used a binomial conditional prior for the data (aka likelihood) for the number of recoveries x:

p(x | \theta) \sim \textnormal{Binomial}(100, \theta).

For the Bayesian analysis the prior for \theta was a 50/50 mixture of a uniform prior from 0 to 1, and a delta function at \theta=0.7 (the old drug’s effectiveness):

p(\theta) = \frac{1}{2}\delta(\theta - 0.7) + \frac{1}{2}

where \theta \in [0, 1]. Is this prior debatable? Yes, just like the prior for the data. The conclusion of an argument can change if you change the premises.

Here’s R code to calculate the posterior:

# Theta values, prior, and likelihood function
# theta[7000] is basically 0.7
theta = seq(0, 1, length.out=10001)
likelihood = dbinom(83, prob=theta, size=100)

# Prior (discrete approx)
prior = rep(0.5/10000, 10001)
prior[7000] = 0.5

# Posterior
posterior = prior*likelihood/sum(prior*likelihood)

# Probability new drug is worse, same, better
print(sum(posterior[1:6999]))
print(posterior[7000])
print(sum(posterior[7001:10001]))

Here is R code for the (two-sided) p-value:

# Possible data sets
x = seq(0, 100)

# p(x | H0)
p = dbinom(x, prob=0.7, size=100)

# P(x <= 57 | H0)
prob1 = sum(p[x <= 57])

# P(x >= 83 | H0)
prob2 = sum(p[x >= 83])

# Print p-value
print(prob1 + prob2)
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About Brendon J. Brewer

I am a senior lecturer in the Department of Statistics at The University of Auckland. Any opinions expressed here are mine and are not endorsed by my employer.
This entry was posted in Inference, Personal. Bookmark the permalink.

One Response to Article in Quillette + Technical Details

  1. G. Belanger says:

    Thanks for that, especially the course notes: reading Jaynes is not something I can recommend to most people I meet who are interested in learning some more, although I find really amazing and inspiring myself.

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