Previously: Fun With Pyro

In a typical clinical trial, the difference between the treatment group and the control group is measured with p-values. If the p-value for the “primary endpoint” is less than .05, the effect is said to be “statistically significant”; if not, then no effect has been found. There are lots of issues with this, but one of them is conflating not finding any evidence with finding negative evidence (evidence against the theory being tested). A small, poorly-controlled, or badly measured study can’t tell us anything either way, but a large, well-done study can provide strong *negative* evidence – “no evidence for X” is not the same thing as “evidence for no X”. With Pyro, this can be measured in a clean. quantitative way.

To illustrate, we can run simulations using the last post’s model of the Pikachu virus, only this time, the simulated treatment and control data will all be randomly drawn from the same distribution. Since there is no difference in the data, on average, the probability we assign to “no effect” should increase from the baseline 80% after a trial is run, and it should increase more with larger trials that provide stronger negative evidence. However, this is only an average. Sometimes, by luck, the data drawn will make it look like there is an effect, and p(no effect) will actually go down. To smooth this out, we can simulate a large number of trials and average them, but running lots and lots of simulations would take a long time. Luckily, PyTorch’s batch_shape feature and the efficiency of modern SIMD architectures makes it (relatively) easy to run them all in parallel:

```
def mixture_n(which_world, dists):
"""
Generalizes MaskedMixture to n > 2 components.
"""
assert dists
if len(dists) == 1: return dists[0]
assert which_world.max() < len(dists)
return reduce(lambda a, b: dist.MaskedMixture(
which_world == b[0], a[1] if isinstance(a, tuple) else a, b[1]),
enumerate(dists))
def model(patients):
patients = patients.clone()
scaling = common.make_tensor(100.0)
patients /= scaling
n_trials = patients.size()[2]
p_effect_sizes = common.make_tensor(PRIORS['p_effect_sizes'], n_trials)
mode_effect_size = common.make_tensor(PRIORS['mode_effect_size'], n_trials)
effect_k = common.make_tensor(PRIORS['effect_k'], n_trials)
small_effect_k = common.make_tensor(PRIORS['small_effect_k'], n_trials)
scale = common.make_tensor(PRIORS['scale']) / scaling
distributions = make_distributions(mode_effect_size, effect_k, small_effect_k)
with pyro.plate("world"):
which_world = pyro.sample("which_world", dist.Categorical(p_effect_sizes))
effect_size_raw = pyro.sample(
"effect_size", common.mixture_n(which_world, distributions))
effect_size = effect_size_raw * scale
mode0 = common.make_tensor(0.5, n_trials)
k0 = common.make_tensor(8.1269, n_trials)
# beta reparameterized
# https://bit.ly/2ZqjILG
original_dist = common.beta(mode0, k0)
new_dist = common.beta(mode0 - effect_size, k0)
with pyro.plate("trial"):
with pyro.plate("trial2"):
pyro.sample("severity_trial", new_dist, obs=patients[1])
with pyro.plate("test1"):
with pyro.plate("test2"):
pyro.sample("severity_test", original_dist, obs=patients[0])
```

We can then graph out the strength of negative evidence, averaged over a thousand simulated trials, for a range of different possible sample sizes:

(Note that the prior is a bit under 83% here, as this graph includes all probability density with effect size < 1. This is an average of probabilities rather than an average of log-odds, which might have been more representative.)

This is also a good scalability test; my desktop got up to one million data points before the simulation even started to slow down. At that point, I switched to CUDA, which got up to ten million data points before fully stressing the GPU.

If I’m reading your code right, you started with an assumption that if there is an effect, it’s one of two hardcoded sizes. Without an assumption like that, negative evidence is a lot harder to get or work with.

The effect size isn’t hard-coded; the prior distribution is a mixture of a “no effect” distribution, a “small effect” distribution, and a “significant effect” distribution, but the latter two are continuous over possible effect sizes (and overlapping). Here’s the graph of the prior, from the last post:

The parameters for the prior are fixed, but the posterior parameters (part of the variational distribution or “guide”) are adjusted to find the best fit. I didn’t include that here since it was in the last post, but here it is FWIW: