Resolving the issue of undefined model with many constraints

[ndurn]

Hi everyone

I have a question about a model with constraints we are trying to run in Ngene but results in undefined D-error. We have proceeded with a few approaches to resolve the issue but have not managed to find a solution. We would really appreciate your input on this.

The main code used for the model is the following:

Design
;alts = alt1*, alt2*
;rows = 48
;block=4
;eff = (mnl,d)
;alg = mfederov(candidates = 5000)

;reject:

alt1.var4=3 and alt1.var5=6,
alt2.var4=3 and alt2.var5=6,

alt1.var4=4 and alt1.var5=6,
alt2.var4=4 and alt2.var5=6,

alt1.var4=2 and alt1.var5=8,
alt1.var4=2 and alt1.var5=9,
alt2.var4=2 and alt2.var5=8,
alt2.var4=2 and alt2.var5=9

;model:
U(alt1) = b1.dummy[0] * var1[1,2] + b2.dummy[-0.002|-0.001] *
var2[0.85,0.90,0.95] + b3.dummy[-0.002|-0.001]* var3[0.80,0.90,1] +
b4.dummy[0.002|0.001] * var4[2,3,4]
+ b5.dummy[0.002|0.001] * var5[6,8,9] + b6.dummy[-0.002|-0.001] * var6[3,10,18]

/

U(alt2) = b1 * var1 + b2 * Var2 + b3 * Var3 + b4 * Var4 +b5 * var5 + b6 * Var6
$

Running the model as it appears here results in an undefined D-error. The issue seems to derive from imposing constraints on two variables var4 and var5. We have implemented two strategies to resolve the issue but neither has worked.

- We generated in Ngene an orthogonal design of 5000 paired choice tasks, transferred it to excel to conduct further manipulations by assigning the value of 6 in var5 each time var4 had the value of 2 according to the constraints. In addition each time var4 had the value of either 3 or 4, two random variables of 0,1 (one for each alternative) were used to assign either 8 or 9 to var5. Then we used the command ;alg = mfederov(candidates = externalcandidateset_5000_paired_choice_tasks). After running the model it led to an undefined D-error.

- We generated the full factorial design externally and filtered out all the implausible combinations based on the constraints as they appear in the above code. Then we used the command ;alg = mfederov(candidates = externalcandidateset_full_factorial) but the model resulted again in an undefined D-error.

- We treated the constrained variables as continuous and the model ran without problems.

Do you think that the problem lies in perfect linearity of 2 attributes levels? What would be the best way to go to retain the constraints if possible? Furthermore, would you regard the continuous approach as a valid method of proceeding under the current constraints?

Best wishes,
Nick

[Michiel Bliemer]

The reason why you get an undefined D-error is because it is not possible the estimate all dummy coded parameters with the constraints you have imposed. The parameters are simply not identifiable because it creates too strong correlations across the dummy coded variables. You essentially have 4 constraints (applied to alt1 and alt2). You can impose 3 constraints and still have a model with dummy coded paraneters that can be estimated. However, adding the fourth constraint creates too may correlations in the data such that you can no longer estimate all parameters.

The specified model cannot be estimated using any design that satisfies the specified constraints, so the only option you have is to (i) remove one of the constraints, (ii) use linear coding for var4 or var5, or (iii) add another level if that makes sense, e.g. var5[6,7,8,9]. If these variables are continuous variables then of course that should not be a problem at all, however, if these variables are categorical then linear coding is not really appropriate.

Michiel