I heard if the observed data that will be used in the inverse probability weighting method is too small, the estimator based on the weighting will have a large variance.
Could you explain why that is so?
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1 Answer
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In general, a small data set yields high variance in effect estimates. Weighting decreases the effective sample size of the weighted sample, which further increases the variance. The effective sample size is defined as $$ESS = \frac{(\sum{w})^2}{\sum{w^2}},$$ and when the weights are scaled to add up to the sample size $n$, is equal to $$\frac{n}{1+var(w)},$$where $var()$ is the population variance (i.e., $\frac{1}{n}\sum{(w-\bar{w})^2}$). From that formula, it's clear that if the weights are any different from each other, $var(w)$ will increase, and $ESS < n$.
In addition to this, small data sets increase the tendency of propensity score models to overfit, yielding very high and very low propensity scores. These themselves can dramatically increase the variance of weights, which will affect the effect sample size as previously described.
