Motivating example

• Suppose there is a single confounder $X$, with propensity scores $$P(A=1\mid X=1)=0.1,P(A=1\mid X=0)=0.8$$

• In propensity score matching, for subjects with $X=1$, 1 out of 9 controls will be matched to the treated

  • Thus, 1 person in the treated group counts the same as 9 people from the control group

  • So rather than matching, we could use all data, but down-weight each control subject to be just 1/9 of the treated subject

Inverse probability of treatment weighting (IPTW)

• IPTW weights: inverse of the probability of treatment received

  • For treated subjects, weight by $1/P(A=1\mid X)$
  • For control subjects, weight by $1/P(A=0\mid X)$

• In the previous example

  • For $X=1$, the weight for a treated subject is $1/0.1=10$, and the weight for a control subject is $1/0.9=\frac{10}{9}$

  • For $X=0$, the weight for a treated subject is $1/0.8=\frac{5}{4}$, and the weight for a control subject is $1/0.2=5$

• Motivation: in survey sampling, it is common to oversample some subpopulation, and then use Horvitz-Thompson estimator to estimate population means

Pseudo population

• IPTW creates a pseudo-population where treatment assignment no longer depend on $X$

  • So there is no confounding in the pseudo-population

• In the original population, some people were more likely to get treated based on their $X$’s

• In the pseudo-population, everyone is equally likely to get treated, regardless of their $X$’s

Estimation with IPTW

• We can estimate $E\left(Y^1\right)$ as below $$\frac{{\sum }_{i=1}^n\frac{1}{{\pi }_i}{A}_i{Y}_i}{{\sum }_{i=1}^n\frac{1}{{\pi }_i}{A}_i}$$

  • where ${\pi }_i=P(A_i=1|X_i)$ is the propensity score
  • The numerator is the sum of $Y$’s in treated pseudo-population
  • The denominator is the number of subjects in treated pseudo-population

• We can estimate $E\left(Y^0\right)$ as below $$\frac{{\sum }_{i=1}^n\frac{1}{1-{\pi }_i}(1-A_i)Y_i}{{\sum }_{i=1}^n\frac{1}{1-{\pi }_i}(1-A_i)}$$

• Average treatment effect: $E\left(Y^1\right)-E\left(Y^0\right)$

Marginal structural models

• Marginal structural models (MSM): a model for the mean of the potential outcomes

• Marginal: not conditional on the confounders (population average)

• Structural: for potential outcomes, not observed outcomes

Linear MSM and logistic MSM

• Linear MSM $$E\left(Y^a\right)={\psi }_0+{\psi }_{1}a,a=0,1$$

  • $E\left(Y^0\right)={\psi }_0$, $E\left(Y^0\right)={\psi }_0+{\psi }_1$
  • So the average causal effect $$E\left(Y^1\right)-E\left(Y^0\right)={\psi }_1$$

• Logistic MSM $$logit\left\{E\right(Y^a\left)\right\}={\psi }_0+{\psi }_{1}a,a=0,1$$

  • So the causal odds ratio $$\frac{\frac{P(Y^1=1)}{1-P(Y^1=1)}}{\frac{P(Y^0=1)}{1-P(Y^0=1)}}={\psi }_1$$

MSM with effect modification

• Suppose $V$ is a variable that modifies the effect of $A$

• A linear MSM with effect modification $$E(Y^a\mid V)={\psi }_0+{\psi }_{1}a+{\psi }_{3}V+{\psi }_{4}aV,a=0,1$$

  • So the average causal effect $$E\left(Y^1\right)-E\left(Y^0\right)={\psi }_1+{\psi }_{4}V$$
• General MSM $$g\left\{E\right(Y^a\mid V\left)\right\}=h(a,V;\psi )$$
  • $g\left(\right)$: link function
  • $h\left(\right)$: a function specifying parametric from of $a$ and $V$ (typically additive, linear)

MSM estimation using pseudo-population

• Because of confounding, MSM $$g\left\{E\right(Y^a\mid V\left)\right\}={\psi }_0+{\psi }_{1}a$$ is difference from GLM (generalized linear model) $$g\left\{E\right(Y_i\mid A_i\left)\right\}={\psi }_0+{\psi }_1{A}_i$$

• Pseudo-population (obtained from IPTW) is free of confounding

  • We therefore estimate MSM by solving GLM with IPTW

MSM estimation steps

1. Estimate propensity score, using logistic regression

2. Create weights

   • Inverse of propensity score for treated subjects
   • Inverse of one minus propensity score for control subjects

3. Specify the MSM of interest

4. Use software to fit a weighted generalized linear model

5. Use asymptotic (sandwich) variance estimator

   • This accounts for fact that pseudo-population might be larger than sample size

Bootstrap

• We may also use bootstrap to estimate standard error

• Bootstrap steps

  1. Randomly sample with replacement from the original sample

  2. Estimate parameters

  3. Repeat steps 1 and 2 many times

  4. Use the standard deviation of the bootstrap estimates as an estimate of the standard error

Assessing covariate balance with weights

Problems and remedies for large weights