# When calculating weights using inverse probability weighting, should the mean of the distribution of weights =1?

I am trying to calculate stabilized weights using inverse probability weighting by "dropout" from my cohort study to try to account for selection bias due to follow-up. I read from one source that the distribution of the stabilized weights should have a mean of 1, that the sum of the unstabilized weights should be double the size of the sum of the stabilized weights and that the range for the unstabilized weights should be greater than that of the stabilized weights. However, no matter what I've tried with my code, the average distribution of the stabilized weights is not 1. Are these criteria valid? I have not been able to find these criteria anywhere written anywhere else.

There is no reason that the sum of inverse probability weights should be equal to 1 but you can always normalize each weight by the total sum of weights to obtain a sum equal to 1

The true stabilized weights should have a mean of 1. This is explained in Hernán and Robins (2006). Their proof is the following:

$$E \left[\frac{P(A=1)}{P(A=1|L)}\right] = E \left\{E \left[ \frac{P(A=1)}{P(A=1|L)}|L\right] \right\}= E \left\{E \left[ \frac{P(A=1|L)}{P(A=1|L)}\right] \right\}=1$$

where $$A$$ is the treatment and $$L$$ is the covariate vector, and $$\frac{P(A=1)}{P(A=1|L)}$$ is the stabilized weight. Cole and Hernán (2008) use whether the estimated standardized weight has a mean of 1 as a diagnostic for the adequacy of the propensity score model.

In my experience, this is not a mainstream practice. The preferred way to assess the adequacy of the propensity score model is to see whether it yields balance between the groups on the covariates. As @mirimo mentioned, it's also always possible to force weights to have a given sum by normalizing them, which doesn't change their properties but does change their mean and standard deviation. The heuristic can be useful when the weights are estimated using propensity scores, but there are several other ways of estimating weights that don't require the estimation of the propensity score, and for these methods, the mean of the weights is arbitrary.

Cole, S. R., & Hernán, M. A. (2008). Constructing Inverse Probability Weights for Marginal Structural Models. American Journal of Epidemiology, 168(6), 656–664. https://doi.org/10.1093/aje/kwn164

Hernán, M. A., & Robins, J. M. (2006). Estimating causal effects from epidemiological data. Journal of Epidemiology and Community Health (1979-), 60(7), 578–586.

• Thank you for your response! – Courtney Aug 12 '20 at 8:57