Bayesian random-effects meta-analysis with selection models (publication bias)

Modeling
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5

Yes, w(x) needs to be a probability, but they don’t need (should not) to sum to 1.0.

The assumption applied here is that when the p-value was smaller than 0.05, all the experiments will be published (that is to say the probability of publishing here is 100%, i.e., w[2]). When one experiment’s p-value was larger than 0.05, the probability of publishing was a fixed probability, which should be smaller than 100% w[1]. Their sum is not 1. Hope it makes sense.

I guess the probabilities that one experiment’s p-value was larger or smaller than 0.05 sum to 1.0, but it is not w(x) here.

Just another update. It seems that I found the bug in the original code:

...
      // (bias-related)
      target += normal_lpdf(Y[n] | mu[n], se[n]);
      target += log(omega[I[n]]);
      // denominators
      target += - log_sum_exp(
      normal_lcdf(1.96 * se[n] | mu[n], se[n]) + log(omega[1]),
      normal_lccdf(1.96 * se[n] | mu[n], se[n]) + log(omega[2])
      );
...

Once I use 1.96 * se[n] (instead of 1.96; I used the wrong value to calculate the CDF), the model can converge now. The results are similar to my last answer. I’m not sure if I fully understand the differences, but it seems that they are equivalent (this and the last answer).

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