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I am having difficulty in fitting a model on data. Basically, I have data about the evaluation of phenotypic property (i.e. hard) of 65 palm trees by 5 judges. As an evaluation scheme, each judge provides score to each sample. For 3 judges sample data look like this:

Judge       Product                  Hard
aa             1                      5
ab             1                      6
ac             1                      3
aa             1                      7 
ab             1                      5
ac             1                      4
aa             2                      5
ab             2                      8
ac             2                      6
aa             2                      7
ab             2                      4
ac             2                      4 

Yij=αi+βiθj+εij
i = judge, j = product

Here, αi is judge main coefficients, i is judge coefficients due to difference in their scoring pattern and θj is product coefficients and εi is assessor dependent.

I was trying to fit this model using lme function in R, but difficulty I am facing to fit the interaction term because model here fitted for parameters rather than co-variates.

This model looks quite accurate for my kind of data. I have seen Bayesian version (http://www.r-bloggers.com/extending-the-sensory-profiling-data-model/) of it and I don't know how to do using mixed-modelling approach or in a frequentist way.

My queries here are:

a) What can be an appropriate method to fit this kind of model? I had referred so much literature where description about iterative generalized least squares, multi-level model, separate regression model, weighted least-square model are given. But still I am not getting how to use and fit estimated value of parameters in interaction terms and get separate coefficients for both interaction parameters?

b) How can I get heterogeneous error in this form?

c) which R package can I use?

Ben Bolker
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maddy
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  • I think this is going to be difficult to do in a package that uses closed-form solutions (e.g. `nlme` or `lme4`). It would be relatively easy if it were *either* entirely additive (a_i + beta_i*(known covariate) + eps_{ij}) *or* entirely multiplicative (log(Y_{ij}) = a_i + theta_j + eps_{ij}), but not in between. I think you're stuck with Bayesian methods (WinBUGS/JAGS/Stan/roll your own) or something like [AD Model Builder](http://admb-project.org) – Ben Bolker Jun 26 '14 at 17:55
  • PS presumably you have more than 3 judges and 2 products in your data set -- otherwise you will have to do strong regularization/prior-setting to get any kind of sensible answer – Ben Bolker Jun 26 '14 at 17:56

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