delta <- 0.5
d_ipsi_down <- function(data, trt) {
eps <- runif(nrow(data), 0, 1)
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
Dı́az, Iván, Katherine L Hoffman, and Nima S Hejazi. 2024. “Causal Survival Analysis Under Competing Risks Using Longitudinal Modified Treatment Policies.” Lifetime Data Analysis 30 (1): 213–36.
Dı́az, Iván, Nicholas Williams, Katherine L Hoffman, and Edward J Schenck. 2023. “Nonparametric Causal Effects Based on Longitudinal Modified Treatment Policies.” Journal of the American Statistical Association 118 (542): 846–57.
Kennedy, Edward H. 2019. “Nonparametric Causal Effects Based on Incremental Propensity Score Interventions.” Journal of the American Statistical Association 114 (526): 645–56.
Wen, Lan, Julia L Marcus, and Jessica G Young. 2023. “Intervention Treatment Distributions That Depend on the Observed Treatment Process and Model Double Robustness in Causal Survival Analysis.” Statistical Methods in Medical Research 32 (3): 509–23.