I'm doing some simulation work with the classical CBC-HB-Model:

1) I generated hierarchical part worths ( alpha ) and a covariance matrix ( D )

2) I generated individual part worths (Beta_i) from Normal(alpha,D)

3) For a given (balanced overlap) design ( X ) I calculated individual choice probabilities

p_i=exp(X*Beta_i)/sum(exp(X*Beta_i))

4) With these choice probabilities I generated individual chocies y_i from the multinomial distribution

So the parameter generated in 1) and 2) are the true values to be recovered in the estimation.

The estimation has been done with X and y_i as input and default settings with 200000 iterations (with 100000 as burn-in).

The Beta and alpha estimates from CBC/HB are very high correlated with the true values but they are all larger by the factor =~1.45, i.e. estimate=~ 1.45* (true value).

My question is where this scaling factor is coming from? (It's different from the theoretical scaling factor sqrt(pi^2/6)=1.28). Is this value to be expected in all simulation studies with CBC/HB?

Many thanks.

Vladimir

Except for correlations what mesaures comparing true and estimated values would you use to show/track the convergence?