Miscelany for Feb 21

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# Miscelany for Feb 21
### Jean Morrison
### University of Michigan
### 2022-21-02 (updated: 2022-02-21)

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# Bang and Robins Estimator Correction

- You may have noticed that when HR describe the Bang and Robins (2005) estimator they say that we should add a new covariate to the regression which is equal to the PS weight for individuals with `$A = 1$` and minus the propensity score weight for individuals with `$A = 0$`, i.e.
`$$R_i = \frac{A_i}{\hat{\pi}_i} - \frac{1-A}{1-\hat{\pi}_i}$$`

- If you read the Bang and Robins (2005) paper, the special covariate is given as 
`$$R_i = \frac{A_i}{\hat{\pi}_i} + \frac{1-A}{1-\hat{\pi}_i}$$`

- I have been telling you the second version.

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# Bang and Robins Estimator Correction

- You may have found in the homework that the version we have been using does not work well in the homework. 
  + This is because it is wrong. 
  
- As it turns out a correction was issued for the original paper. 
  + The correction is weirdly extremely hard to find (at least it was for me), but is now linked on the course website.

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# Review Exercise

- Write down the definition of sequential exchangeability for a static treatment strategy.

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# Review Exercise

- Does static sequential exchangeability hold in the DAG below? `$L_0$` and `$L_1$` are observed but `$U_1$` is not.

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# Review Exercise

- Write down the definition of sequential exchangeability for a dynamic strategy `$g$`.

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# Review Exercise

- Draw a SWIG for the DAG below under a dynamic strategy in which `$g_k$` depends on `$A_{k-1}$` and `$L_{k-1}$`.

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# Review Exercise

- Using your SWIG, does sequential exchangeability for `$Y(g)$` hold for the DAG below and a dynamic strategy in which `$g_k$` depends on `$A_{k-1}$` and `$L_{k-1}$`?