IPTW Estimator
Under the assumption of exchangeability and positivity, we can estimate \({\rm E}(Y^1)\) as
\[ \frac{\sum^n_{i=1}I(A_i=1)\frac{Y_i}{\pi_i}}{\sum^n_{i=1}I(A_i=1)\frac{1}{\pi_i}} \]
where \(\pi_i=\Pr(A=1\mid X_i)\) is the propensity score.
Inverse Probability of Treatment Weighting (IPTW) uses the inverse of the probability of being treated as weight, that is, \(1/\pi_i\) for individual \(i\).
IPTW uses the weight to create a pseudo-population where the numbers of subjects in the treatment and control groups are balanced.
The numerator \(\sum^n_{i=1}I(A_i=1)\frac{Y_i}{\pi_i}\) is the sum of the Y’s in treated pseudo-population, while the denominator \(\sum^n_{i=1}I(A_i=1)\frac{1}{\pi_i}\) is the number of subjects in treated pseudo-population. Overall, we have the average value of the outcome if treated, i.e., \({\rm E}(Y^1)\).
Exchangeability is another way of saying ignorability given X.
Positivity suggests \(\pi_i\in(0,1)\) which is critical for this formula since \(\pi_i\) is in the denominator position.
Similarly, we can estimate \({\rm E}(Y^0)\) by
\[ \frac{\sum^n_{i=1}I(A_i=0)\frac{Y_i}{1-\pi_i}}{\sum^n_{i=1}I(A_i=0)\frac{1}{1-\pi_i}}. \]
Then, an average causal effect estimator is obtained by
\[ {\rm E}(Y^1)-{\rm E}(Y^0). \]
Large Weight Problem
When using IPTW estimation, the large weights can create a problem, i.e., large standard errors.
Imagine one person has a weight of 10000, which means this person represents 10,000 people. So much is riding on this one person. That creates noise in a sense.
[Further Intuition] We estimate the standard errors with bootstrapping. The value of our estimate in each of these bootstrap samples is going to depend a great deal on the inclusion/appearance of this one person. When this person is in, the parameter estimate will be pulled towards him, while when he is out, that is not going to happen. As a result, the parameter estimate is going to vary a lot in large part due to this one person.
Check the distribution of weights in plots.
Density plot with tick marks on the bottom
Scatter plot for a sorted weight data series from the smallest to the largest
Remedies for Large Weights
Trimming the Tails
Large weights occur at observations in the tails of the propensity score distribution
Trimming the tails can eliminate some of the extreme weights
A common strategy:
Remove treated subjects whose propensity scores are above the 98th percentile from the distribution among controls
Remove control subjects whose propensity scores are below the 2nd percentile from the distribution among treated subjects
Weight Truncation
Truncation Steps:
Determine a maximum allowable weight
Could be a specific value (e.g., 100)
Could be based on a percentile (e.g., 99th)
If one weight is greater than the maximum allowable, set it to the maximum allowable value
- For example, if your upper limit is 100, someone who weights 1000 will be set to 100 instead.
Truncation changes the data and hence introduces bias. So there involves a bias-variance trade-off:
Truncation: Bias, smaller variance
No Truncation: Unbiased, larger variance
Judge if the trade-off is worth based on the mean squared error. If truncating a small number of observations (not too many) that have extreme weights, you’ll probably be better off in terms of mean squared error.
Doubly Robust Estimators
IPTW Estimator: We can estimate \({\rm E}(Y^1)\) given by
\[ \frac{1}{n}\sum_{i=1}^n\frac{A_iY_i}{\pi_i(X_i)}, \]
where \(\pi_i(X_i)\) is the propensity score. This estimator is unbiased if \(\pi_i(X_i)\) is correctly specified.
\(\sum_{i=1}^n A_i\) counts the number of treated subjects.
\(1/\pi_i(X_i)\) is the weight used to constructing a pseudo-population. For example, there are 9 out of 10 subjects treated in a randomized trial. The total number \(n\) of subjects is 10 and the propensity score \(\pi_i\) is 0.9 for any \(i\). The IPTW estimator gives an expectation of \(Y^1\).
Regression-based Estimator: We can estimate \({\rm E}(Y^1)\) given by
\[ \frac{1}{n}\sum_{i=1}^n\left\{A_iY_i+(1-A_i)m_1(X_i)\right\} \]
where \(m_1(X)={\rm E}(Y\mid A=1,X)\).
- Instead of creating a pseudo-population, we use \(m_1(X_i)\) to predict the value of \(Y^1\) for subjects who are not treated.
Doubly Robust Estimator: It is an unbiased estimator if either the propensity score model or the outcome regression model are correctly specified.
\[ \frac{1}{n}\sum_{i=1}^n\Bigg\{\underbrace{ \frac{A_iY_i}{\pi_i(X_i)}}_{\rm IPTW} -\underbrace{ \frac{A_i-\pi_i(X_i)}{\pi_i(X_i)}m_1(X_i)}_{\rm Augmentation}\Bigg\} \]
This doubly robust estimator is also known as Augmented IPTW (AIPTW) estimators. In general, AIPTW estimators should be more efficient than regular IPTW estimators.