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Miscellaneous Notes

As a researcher, the most importat consideration is “How can I best separate the treatment assignment from potential outcomes?”

  • “Block what you can, Randomize what you cannot”
  • “No Causation without Manipulation”

Time-varying / Person-time Notation​

It’s just notation but worth noting (hehe).

Example: one patient (i=1i=1), a physician, BP checked monthly at t=0,1,2t=0,1,2.

  • t=0t=0: physician gives a high dose.
  • t=1t=1: BP is HIGH. Switch to a low dose.
  • t=2t=2: BP is OK. Stop.

Now answer the usual questions:

  • Units? ωt\omega_t, t=0,1,2t=0,1,2. The unit is the patient at time t (a person-time), not the person. One person, three units!
  • Treatment? A(ωt)A(\omega_t). Treatment written as a function of the unit. Can have more than two levels (high dose, low dose, stop).
  • Potential Outcome? Yt(A(ωt−1))Y_t(A(\omega_{t-1})). Outcome at time tt under the treatment at t−1t-1. Treatment goes in parentheses instead of a superscript, and the subscript is time, not person. Note the lag.

Coding the outcome:

Yt(A(ωt−1))={0BP HIGH or LOW (not OK)1BP right (OK)Y_t(A(\omega_{t-1})) = \begin{cases} 0 & \text{BP HIGH or LOW (not OK)} \\ 1 & \text{BP right (OK)} \end{cases}
No carryover

In full, the outcome at t=2t=2 could depend on the whole treatment history:

Y2(A(ω1))=Y2(A(ω1),A(ω0))Y_2(A(\omega_1)) = Y_2(A(\omega_1), \cancel{A(\omega_0)})

Crossing out A(ω0)A(\omega_0) is an assumption: the dose at t=0t=0 has no carryover effect on the outcome at t=2t=2. Only the most recent treatment matters. This is what lets you write the shorter Yt(A(ωt−1))Y_t(A(\omega_{t-1})).

So what’s the Causal Effect formula used?

Y2(A(ω1))−Y1(A(ω0))=1−0=1Y_2(A(\omega_1)) - Y_1(A(\omega_0)) = 1 - 0 = 1

Low dose at t=1t=1 gives OK at t=2t=2. High dose at t=0t=0 gives HIGH at t=1t=1. So the effect is 1? Careful: this is a before/after comparison of two different units (ω1\omega_1 vs ω0\omega_0), not two potential outcomes of the same unit. The true individual causal effect would be Y2(low)−Y2(high)Y_2(\text{low}) - Y_2(\text{high}) for ω1\omega_1, and you only ever see one of those (the fundamental problem again).

The difference above counts as causal only under extra assumptions: no carryover, and stability over time (the unit at t=1t=1 is “the same” as the unit at t=0t=0).

The ‘Sharp’ Null (Fisher)​

The idea here is “How much do you care about the distribution of the counterfactual?” You want model-free? (Remember that a model is a set of distributions!)

Now here, you’re saying there is no effect for the given individual

H0:τi=Yi1−Yi0=0H_0: \tau_{i} = Y_i^1 - Y_i^0 = 0

TODO: See the cola and calcium example.

TODO: See the steps in experiments bro. Again. What you want from a test statistic is high power. Sample means is one thing. But there are very sophistacated ones; you should check for applicability.
See the idea here is to permute and shuffle and compute E[Y1] - E[Y0]. You compute the test statistic assumign that the outcome remains fixed no matter the treatment assigned. This gives you a distribution. So in the cola and water example, the statistic is 6.875. This is not zero. What does this mean? You plot these differences. You will get a nice bell curve. (So this means that model-based inference would work well.) THI SIS IMPORTANT. Now you compare this random shit to your test distribution and COMPUTE THAT P-VALUE! “Exact” in Fisher’s Exact test yo. Jesus.

TODO: “Model based inference” — what other things exist?

Fisher is useful for some early evidence, whether it’s “worth it” to continue.

Books​

“Causal Inference In Statistics — A Primer” by the GOAT Himself. Read this first.

Graphs​

Videos​

Many of these are introductory.

Other​

About that pMp^M mess: When you say pMd(Y=y)=0.75p^{M_d}(Y = y) = 0.75, you’re saying “I subjectively believe that when the model takes decision D=dD=d, I will see event Y=yY=y.