Possible bias when fitting linear regression with errors in variables using TMB

[DM Gillis]

I am trying to develop a simple errors in variables (measurement error) example using TMB,  by including erros in X as a random variable. It is not specified as Random in making the objective function, since I do not want its effects to be integrated out. The slope returned is an improvement from the OLS solution, but the 95% CI falls short or the "true" value (simulated data) rather than being centered around it. I have posted details with a full code listing on Stack Overflow:

https://stackoverflow.com/questions/76705890/possible-bias-when-fitting-linear-regression-with-errors-in-variables-using-tmb

since my attempt to post it all here was rejected by the server. Any suggestions would be appreciated. Thank you for taking thetime to consider my problem.

Regards, Darren

[Hans Skaug]

Hi, I think this is more a statistical theory question than a question about TMB. There are different

assumptions that can be made for this model:

https://en.wikipedia.org/wiki/Errors-in-variables_models#Terminology_and_assumptions

Your approach seems to add one parameter (del) per observation, which may be problematic.

[DM Gillis]

Thank you for the quick response.  There may be different model assumptions that work better, but I am interested in this form:

https://en.wikipedia.org/wiki/Errors-in-variables_models#Simple_linear_model

 https://wikimedia.org/api/rest_v1/media/math/render/svg/72c8882e35c59a196c8115548e47a19b0555360e

because it would be straight forward to extend it to other models besides linear regression. However, I am not sure that either 1) this form can be coded in TMB, and 2) if I have done it correctly. In this code, both have random components: Y (represented in the classic errors) and X (the del's) . If this formulation cannot be executed in TMB then I will seek other solutions (JAGS or STAN, perhaps), but I was hoping to stay in the world of likelihood (and learn some TMB at the same time).  Thanks again for taking the time to review this issue.

Regards, Darren

[Mollie Brooks]

In response to Han’s comment, I tried running it with del as a random variable and didn’t do any better. I don’t know why, because the Wikipedia article seems to say it should be identifiable in this case.

Obj <- MakeADFun(data=Data, 

parameters=Params, 

random=c("del"),

DLL=model_name,

silent=FALSE)

Mollie

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[DM Gillis]

Yes, when I do the same it returns the OLS slope, which makes sense to me since setting it to "random" means it will be integrated out and should not impact the estimates (just their SEs). However, I want it to have an impact and move the slope around the true value of 2 :(  Perhaps TMB only works with latent random variables/effects and not variation added to predictors that are not not LRVs to be integrated out.Thanks for taking the time to check and enjoy the weekend! 

Darren