```
<- 0.5
delta <- function(data, trt) {
d_ipsi_down <- runif(nrow(data), 0, 1)
eps ifelse(eps < delta, data[[trt]], 0)
}
```

# Incremental Propensity Score Interventions

\[ \renewcommand{\P}{\mathsf{P}} \newcommand{\m}{\mathsf{m}} \newcommand{\p}{\mathsf{p}} \newcommand{\q}{\mathsf{q}} \newcommand{\bb}{\mathsf{b}} \newcommand{\g}{\mathsf{g}} \newcommand{\rr}{\mathsf{r}} \newcommand{\IF}{\mathbb{IF}} \newcommand{\dd}{\mathsf{d}} \newcommand{\Pn}{$\mathsf{P}_n$} \newcommand{\E}{\mathsf{E}} \]

Recall that an incremental propensity score intervention (IPSI) is a hypothetical intervention where the conditional probability of treatment is shifted. We defined an IPSI, based on the risk ratio, that *increased* the likelihood of treatment as

\[ \dd_t(a_t, h_t, \epsilon_t) = \begin{cases} a_t &\text{ if } \epsilon_t < \delta \\ 1 &\text{ otherwise.} \end{cases} \]Assume we want to increase the likelihood of initiating treatment by 2-fold. We can implement this in R with

### Problem 1

What if we wanted to *decrease* the likelihood of initiating treatment by 2-fold? Implement this shift function in R. As a reminder

\[ \dd_t(a_t, h_t, \epsilon_t) = \begin{cases} a_t &\text{ if } \epsilon_t < \delta \\ 0 &\text{ otherwise.} \end{cases} \]

`lmtp`

already implements `d_ipsi_up`

and `d_ipsi_down`

as a single shift function factory, `ipsi()`

!

Risk ratio IPSIs that increase the likelihood of treatment should be specified with a value greater than 1 (i.e. a risk ratio IPSI that increases the likelihood of treatment 2-fold is equivalent to

`ipsi(2)`

).IPSIs that decrease the likelihood of treatment should be specified with a value less than 1 (i.e. a risk ratio IPSI that decreases the likelihood of treatment 2-fold is equivalent to

`ipsi(0.5)`

)

Let’s apply the shift function that increases the likelihood of initiating treatment by 2-fold to `covid`

dataset.

Estimating the effect of incremental propensity scores based on the risk ratio are easy to do with `lmtp`

. Just use the `ipsi()`

function as the input for the `shift`

argument!

## References

*Lifetime Data Analysis*30 (1): 213–36.

*Journal of the American Statistical Association*118 (542): 846–57.

*Journal of the American Statistical Association*114 (526): 645–56.

*Statistical Methods in Medical Research*32 (3): 509–23.