Topic 7 IP weighting for censoring

Learning Goals

1. Understand how IP weighting can be used to address the lack of generalizability caused by censoring

Discussion

Language for interpreting coefficients in MSMs

$$E[Y^a] = \beta_0 + \beta_1 a$$

• Context: Y is cholesterol, A is medication (a = 1: on medication)
• $\beta_0$: The average potential cholesterol level when all are not on medication
• $\beta_1$: The average causal effect: the change in average potential cholesterol level if everyone were on medication compared to if no one were on medication

Do we really care about the ACE?

• Depends on research goals, but yes, sometimes ACE is of interest.
  • Is it a nationwide policy? If so, then enacting the policy means “treating” everyone and not enacting means treating no one.
• May care about the ACE within subgroups. We’ve seen this with effect modification MSMs:

$$E[Y^a] = \beta_0 + \beta_1 a + \beta_2 M + \beta_3 a\times M$$

• Say $M$ indicates history of heart disease (1: history. 0: no history)
• Question: What are the interpretation of $\beta_1$ and $\beta_3$ in this model?

What is the difference between regular and IP-weighted GLMs?

• Our tree diagrams showed that weighting allows us to create “populations” of all treated and all untreated.
  • Our WHATIF book calls the combined population of the all treated and the all untreated a pseudopopulation
• We can model the outcomes in this pseudopopulation with standard regression models (marginal structural models).
  • Allows us to estimate causal quantities: $P(Y \mid \hbox{do}(A=a))$, $E[Y \mid \hbox{do}(A=a)]$ (in potential outcome notation: $P(Y^a)$, $E[Y^a]$)
  • Standard regression models without IP weights do not estimate these causal quantities.

Censoring and selection bias

Reminder: NHEFS investigation from last time

Research goal: What is the average causal effect of smoking cessation on weight gain at a follow-up visit about 10 years later?

• There were some people who did not have their weight measured at Visit 2. They must be excluded from our analysis.
  • Define a censoring indicator $C$ where $C = 1$ means censored/excluded and $C = 0$ means uncensored/included.
  • Our analysis only generalizes to uncensored individuals–the types of people who would stay in the study!

• Before: just wanted to find $Z$ such that conditional on $Z$ (within subsets defined by $Z$), the outcomes of the treated and untreated were comparable.
  • Could appropriately upweight the treated and untreated to get “populations” of all treated and all untreated.

• Now: also want to ensure that the outcomes in the censored and uncensored are comparable.
  • Can appropriately upweight the uncensored to represent everyone had they not been censored.
  • The censored get zero weight. (We don’t observe them!)

• When are the outcomes in the censored and uncensored comparable?
  • When variables in $A$ and $Z_2$ together d-separate $C$ and the outcome $Y$.
  • e.g., The DAG below. View $C$ as the new “treatment” variable.
  • Another way to put it: when we put a box around $C$ in the DAG, $Z$ must d-separate $A$ and $Y$ under the null.
    Same condition as before, now explicit about paths with conditioned-on colliders (non-backdoor paths).

Questions: $C$ is inherently conditioned on in our analysis ($C = 0$).

• What set $Z$ d-separates the treatment and outcome under the null?
• What is a set $Z_2$ such that $Z_2 \cup A$ (read $Z_2$ “union” $A$, or $Z_2$ and $A$ together) d-separate $C$ and the outcome?

Conclusion:

• $Z$ should include:
  sex, age, race, education, smokeintensity, smokeyrs, active, exercise, wt71, and VidGames
  (Same as before with the addition of VidGames)
• $Z$ blocks spurious, non-causal paths between $A$ (qsmk) and $Y$ (wt82_71)

• $Z_2$ should include VidGames
• Together with $A$, $Z_2$ blocks spurious, non-causal paths between $C$ and $Y$
• Note that $Z_2$ is contained within $Z$.
  • This will be the case when $A$ is a cause of $C$ (often true).

Ultimate goal: Want to upweight individuals using $P(A, C = 0 \mid Z)$

• Why? Same rationale as our tree diagram with additional $C = 1$ and $C = 0$ branches.
• Allows us to estimate $E[Y^{a, c = 0}]$: expected potential outcome under treatment $a$ and when no one has been censored ($C = 0$)

We can express $P(A, C = 0 \mid Z)$ as the following:

$$\begin{align*} P(A, C = 0 \mid Z) &= P(A, C = 0, Z)/P(Z) \\ &= P(A \mid Z) P(C = 0 \mid A, Z) \end{align*}$$

• Based on the last line, we see (1) the propensity score and (2) another “propensity” of being uncensored.
• Will want models for both.
• (The censoring probability is not typically called a propensity score.)

A little extra theory (don’t worry about the math part below if you haven’t taken Probability):

$Z$ may contain extra variables not in $Z_2$ if there are, say, many common causes of $A$ and $Y$, but only a few of these are a cause of $C$. Let’s say that these extra variables are in the set $Z_1$. ($Z = Z_1 \cup Z_2$)

If the variables in $Z_1$ are not causes of $C$, then $C$ and $Z_1$ are conditionally independent given $A$ and $Z_2$.

Going through the work below…

$$\begin{align*} P(A, C = 0 \mid Z) &= P(A, C = 0 \mid Z_1, Z_2) \\ &= P(A, C = 0, Z_1, Z_2)/P(Z_1, Z_2) \\ &= P(A \mid Z_1, Z_2) P(C = 0 \mid A, Z_1, Z_2) \\ &= P(A \mid Z_1, Z_2) P(C = 0 \mid A, Z_2) \end{align*}$$

…this implies that the propensity score model (for $A$) should depend on all variables in $Z$ (the set $Z_1$ and $Z_2$ together) and that the censoring probability model only need depend on $Z_2$.

Analysis plan:

Notation:

• $Z$ is the set that d-separates $A$ and $Y$ when $C$ is conditioned on.
• $Z_2$ (a subset of $Z$) that, together with $A$, d-separates $C$ and $Y$

1. Model the propensity scores $P(A\mid Z)$ using all variables $Z$ in the d-separating set. Create inverse probability weights $W^A = 1/P(A\mid Z)$ (treatment weights).

2. Model the probability of censoring $P(C \mid A, Z_2)$. Create inverse probability weights $W^C = 1/P(C = 0\mid A, Z_2)$ (censoring weights).

3. Create final weights (for treatment and censoring) $W^{A,C} = W^A\times W^C$.

4. Fit the desired marginal structural model with these final weights.

Exercises

A template Rmd is available here.

Setup

library(readr)
library(dplyr)
library(ggplot2)
library(geepack)

nhefs <- read_csv("https://cdn1.sph.harvard.edu/wp-content/uploads/sites/1268/1268/20/nhefs.csv")

Create a censoring indicator variable that will be TRUE if the individual was censored and FALSE otherwise.

nhefs$cens <- is.na(nhefs$wt82_71)

Throughout we’ll use the full nhefs dataset, and we’ll work from the DAG below ($C$ has been conditioned on):

Part 1: Modeling to obtain weights

1. Based on the DAG above, identify the set of variables $Z$ that d-separates qsmk and weight gain.

2. Fit an appropriate treatment propensity score model, and call this ps_treat_mod. For simplicity, assume that a quadratic relationship for the quantitative variables gives a good fit. Make sure to wrap categorical variables inside factor() in your model formula.

3. Add a new variable to your dataset called weights_treat that contains the IP weights for treatment.

4. Is it believable that quitting smoking (treatment) is a cause of censoring? How do the tabulations/calculations below help answer this?

   # P(censored | quitters) and P(censored | nonquitter)
   table(qsmk = nhefs$qsmk, cens = nhefs$cens)
   table(qsmk = nhefs$qsmk, cens = nhefs$cens) %>% prop.table(margin = 1)

5. How would you see if the data support the age --> C and education --> C arrows? Describe, but don’t actually perform this analysis.

6. Fit an appropriate model for the probability of censoring, and call this prob_cens_mod. For simplicity, assume that a quadratic relationship for the quantitative variables gives a good fit. Make sure to wrap categorical variables inside factor() in your model formula.

7. Add a new variable to your dataset called weights_cens that contains the IP weights for censoring. (Note that in R, the default for logistic regression is to model the probability of the outcome variable equaling 1.)

8. Add a final weight variable to your dataset called weight_TC.

Part 2: Fitting MSMs

Fit the following two MSMs ($A$ is qsmk, and for sex, 0 indicates males and 1 indicates females):

$$E[Y^{a, c=0}] = \beta_0 + \beta_1 a$$ $$E[Y^{a, c=0}] = \beta_0 + \beta_1 a + \beta_2\hbox{sex} + \beta_3 a\times\hbox{sex}$$

Note: In our previous analysis where we worked with nhefs_subs, we had actually fit the following:

$$E[Y^{a} \mid C = 0] = \beta_0 + \beta_1 a$$ $$E[Y^{a} \mid C = 0] = \beta_0 + \beta_1 a + \beta_2\hbox{sex} + \beta_3 a\times\hbox{sex}$$

Refit these models here using the relevant weights. How do your results compare?