Blocking

[mdann]

Hello,

my question relates to building blocks in choice experiments.

Please imagine following scenarios:

- bayesian effcient design with 1 block including 12 choice sets, d-error=0,84
- bayesian efficient design with 2 blocks each with 12 choice sets (together 24 choice sets), d-error=0,44

Should I chose the alternative with the lower d-error or is there a justification (besides that I need a smaller sample by using 1 block) to chose the alternative with fewer choice sets, but higher d-error?

Thank you in advance
Micha

[Michiel Bliemer]

The smaller design with 12 choice tasks and D-error of 0.84 is more efficient per choice task.

[mdann]

Can you please explain this? Is there a parameter, a calculation which argues that this is more efficient?

[Michiel Bliemer]

You have to compare 24 choice tasks with 24 choice tasks.
So for the first design, you have to consider the D-error for a design with twice the same block of 12. The D-error of one block of 12 rows is 0.84, while the D-error for the second design (with 24 rows) is 0.44.

Suppose that A is the asymptotic variance-covariance (AVC) matrix of the first design with 12 rows. Then two blocks yielding 24 rows will have an AVC matrix of A/2.
The D-error is the determinant of the AVC matrix, i.e. det(A) = 0.84. From the general computational rules of the determinant, we know that det(A/2) = (1/2)^K * det(A), where K is the number of parameters (attributes) you are estimating. I do not know what K is in your case, but K>=2, so det(A/2) <= (1/2)^2 * det(A) = det(A)/4 = 0.21. Since 0.21 < 0.44, you are better off using twice the 12 choice tasks in design 1 than 24 choice tasks in design 2.

[Michiel Bliemer]

I made a mistake in the calculation here, I forgot to take into account that the D-error takes the power 1/K of the determinant. The result is still that your design with 12 choice tasks is more efficient than your design with 24 choice tasks, but the difference is not that great.

So the calculation simply becomes 0.84 / 2 = 0.42. So compared to a D-error of 0.44, it is still slightly more efficient per choice task.