I'm still thinking about this problem. For me, it's much simpler conceptually to think of this as a missing data problem. Andy Peytchev's paper makes this point. If I have the "right" structure for my data, then I can use imputation to address both nonresponse and measurement error.
If the measurement error is induced differently across different modes, then I need to have some cases that receive measurements in both modes. That way, I can measure differences between modes and use covariates to predict when this happens.
The covariates, as I discussed last week, should help identify which cases are susceptible to measurement error. There is some work on measuring whether someone is likely to be influenced by social desirability. I'm think that will be relevant for this situation. That sounds sort of like, "so you don't want me to tell me the truth about x, but at least you will tell me that you don't want to tell me that." Or something like that...
If the measurement error is induced differently across different modes, then I need to have some cases that receive measurements in both modes. That way, I can measure differences between modes and use covariates to predict when this happens.
The covariates, as I discussed last week, should help identify which cases are susceptible to measurement error. There is some work on measuring whether someone is likely to be influenced by social desirability. I'm think that will be relevant for this situation. That sounds sort of like, "so you don't want me to tell me the truth about x, but at least you will tell me that you don't want to tell me that." Or something like that...
I fully agree with this. It is a "potential outcome" problem. The trick really is to find covariates that either predict A. selection effects and outcome or B. measurement effects and outcome. In either case, we need variables that strongly relate to Y in both modes.
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