7: MORE ASSUMPTIONS
Section 7 adds some additional a priori assumptions that can allow us to estimate in the context where we don't have all necessary confounders.
We have the beautifully named: A-complete Stage 0 PL-sufficient reduced graph of R CISTG A
We start with some rules for reducing graph G_A to a counterpart G_B. Honestly the language in this section isn't clear to me despite reading it several times...
I do think the graphs help a bit though. To me it seems we are narrowing the space of the problem. We are going from multiple divisions at t_1 and t_2 to only considering the divisions at t_2 for a single branch. The reduced STG is a single branch
The purpose of our graph reduction is that G_A is identifiable given G_B and the listed assumption R
So we can think about G_B as either everyone being set to the same value at t_1 OR everyone was off their assigned protocol (and then returned to it at t_2).
The section ends with the note that we can see if the observed G_B is compatible with a proposed randomization scheme. Section 8 goes on to discuss this and when standard analyses (ignoring time-vary confounding) are valid
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a 🧵 on M-Estimation and why I think its a valuable tool that epidemiologist should be using more often
M-Estimation is a general approach of defining an estimator as the solution to estimating equations like the following. Importantly, obs are independent and \psi is a known function that doesn't depend on i or n
I think its a great tool for two reasons: (1) the ability to stack estimating equations together, and (2) the sandwich variance
Big fan of the "I forced a bot to [...] over 1000" memes. But most of those posts are fake (i.e. human-generated). That's why I decided to make a real one
So I forced a bot to read over 1000 PubMed abstracts in order to generate new abstracts
Basically, I pulled a random sample of 5000 abstracts from PubMed using the search terms: (causal inference) AND English[Language]
A random sample of the returned abstracts was used to train a recurrent neural network (RNN)
Basically, a sequence of 40 characters is used to predict the next character. This process can then be repeated with the new character to generate a whole new sentence
So you give the machine a starting point, set a 'creativity dial', and let it go
Herd immunity is a far squishier concept then many seem to be describing in their "shielding" or "stratified herd immunity" plans. Here is the formula for herd immunity threshold for a SIR model
where \beta is the effective contact rate, N is the number of individuals, and r is the inverse of the duration
The threshold says if are above that level the disease will disappear / we expect no outbreaks of disease. However, that threshold is neither sufficient nor necessary
To show this, let's talk about a perfect vaccine. If you get this vaccine you are perfectly protected from the infection and thus cannot transmit it (everything also applies to imperfect vaccines but it's messier)
Blue circles are vaccinated individuals and red are unvaccinated
8: WHEN CAN I IGNORE THE METHODOLOGISTS
Section 8 discusses when standard analytic approaches are fine (aka time-varying confounding isn't as issue for us). Keeping with the occupation theme, it is presented in the context of when employment history can be ignored
First we go through the simpler case of point-exposures (ie only treatment assignment at baseline matters). Note that while we get something similar to the modern definition, I don't think the differentiation from colliders is quite there yet (in the language)
Generalization of the point-exposure definition of confounding to time-varying exposures isn't direct
6: NONPARAM TESTS
Section 6 goes through the sharp null hypothesis (that no effect of exposure on any individual). Note that this is weaker than the null of no _average_ effect in the population
Another way of thinking about this is if there is no individual causal effect (ICE) then there must be no average causal effect (ACE). The reverse (no ACE then no ICE) is not guaranteed
Robins provides us with the G-null hypothesis as a means of assessing the sharp null (the g-null is that call causal parameters are 0)
5: ESTIMATION
After a little hiatus, back to discussing Robins 1986 (with a new keyboard)! Robins starts by reminding us (me) that we are assuming the super-population model for inference
If we had a infinite n in our study, we could use NPMLE. However, time-varying exposures have a particular large number of possible intervention plans. We probably don't have anywhere near enough obs to consider all the possible plans
Instead we use a parametric projection of the time-varying variables. We hope that the parametric projection is sufficiently flexible to approx the true density function (it is why it is best to include as many splines and interaction terms as feasible)