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Such missing data scenarios leave researchers with the choice of dropping cases entirely when they have missing responses to some questions, or alternatively, finding an imputation solution to deal with missing information.
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We develop an imputation-based solution to using mismeasured covariates in propensity score methods that provide an estimate of a causal treatment effect, and use it to estimate the effects of living in a disadvantaged neighborhood on adolescent mental health and substance use.
Thus, no imputation in the solution dominates another imputation in the solution.
Thus, any imputation outside the solution is dominated by one inside the solution.
In addition, any imputation outside the solution is dominated by one in the solution, because the two players with the lowest payoffs must each get less than 1/2; clearly, this imputation is dominated by an imputation in the solution in which these two players each get 1/2.
It is stable because, for the members of the coalition, any imputation outside the solution is dominated by and is therefore less attractive than an imputation within the solution.
For any imputation within the solution, player A always gets 1/4 and therefore cannot gain.
When player A gets less than 1/4, say 1/8, he always does better with an imputation in the solution.
When players B and C do not divide the 7/8 equally, the player who gets the smaller amount can always do better with an imputation in the solution.
When A gets more than 1/4, players B and C share less than 3/4 and, therefore, can do better with an imputation within the solution.
Players B and C now have more to share; but no matter how they split the new total of 7/8, there is an imputation in the solution that one of them will prefer.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com