Include quadratic terms in experimental design

[dce_learner]

I'm designing a discrete choice experiment with four attributes: three topic-related attributes and one cost attribute. Each attribute has four levels. I want to estimate a quadratic utility function of the form:

V_i = beta_1 [Attribute1]_i + beta_2 [Attribute1]_i ^2 + beta_3 [Attribute2]_i + beta_4 [Attribute2 %]_i ^2 + beta_5 [Attribute3]_i + beta_6 [Attribute3]_i ^2 + beta_7[Cost ~ lognormal]

I saw in the Ngene manual (Section 5.6) that only simple linear functions are supported via fcn(...). Specifically, it states that "Currently only simple linear functions are supported in Ngene, therefore only plus ( + ), minus ( - ), attribute names and constants are allowed."

Does this mean I cannot include squared terms in the utility function (I tried it in Ngene, and it failed)? If so, are there any workarounds or supported methods for implementing quadratic terms in Ngene?

Alternatively, I can generate an efficient design using only the linear terms, under the assumption that the variation in the linear attribute will still carry the relevant information needed to estimate a quadratic term during model estimation, given the deterministic relationship between an attribute and its square. That is, I’d estimate the utility function with both linear and squared terms even if the design only explicitly includes the linear levels. However, if the quadratic terms are not explicitly included in the design, it isn't included in the Fisher information matrix and will affect the efficiency of the design.

Any insights or advice would be greatly appreciated!

[Michiel Bliemer]

Assuming that your attributes are numerical (not categorical), you can easily include quadratic terms into the utility function as follows:

;model:
U(...) = b1 * X1[...] + b2 * X1 * X1 + etc.

A quadratic term is simply an interaction with itself.

Michiel

[dce_learner]

Thanks Michael. I did do this. However, D-error becomes pretty big, which seems to confirm my thought on the loss of efficiency, given that the quadratic term is a deterministic function of the linear term itself.

This is why I was curious if you've had any experience with it and/or if you'd recommend choosing the design based on just linear terms and then adding the quadratic term to the utility function in the estimation phase. Sorry if my question didn't make it super clear.

[Michiel Bliemer]

What do you call a big D-error? Did you use any non-zero prior for the quadratic term?