Estimating Interactions with design with several restriction

[JvB]

Hi,

I´ve used the following code to generate my design:

Design
;alts = alt1*, alt2*, SQ*
;rows = 14
;block = 2
;eff = (mnl,d)
;alg = mfederov
;require:
SQ.amount = 3, SQ.period = 3
;reject:
alt1.amount = 3 AND alt1.period = 3,
alt2.amount = 3 AND alt2.period = 3
;model:
U(alt1) = b1[-0.4862] * contrib[1.8,3.3,4.8](5,5,4)
+ b2.dummy[1.4049|1.0240|0.4527] * amount[0,1,2,3] ? 0 = 300, 1 = 600, 2 = 900, 3 = unlimited amount
+ b3.dummy[1.0609|0.5951|0.2733] * period[0,1,2,3] ? 0 = 12, 1 = 42, 2 = 72, 3 = unlimited period
/
U(alt2) = b1 * contrib
+ b2 * amount
+ b3 * period
/
U(SQ) = b0[-0.9618]
+ b1 * contrib_sq[1.6]
+ b2 * amount
+ b3 * period
$

Would it be possible to estimate interaction effects between attribute 2 "amount" and attribute 3 "period" even if I have several restrictions in my design regarding these attributes?
And if yes, probably way higher sample sizes are necessary for estimation of interaction effects, right?

Any advice is highly appreciated.
Thanks and best,
J.

[Michiel Bliemer]

You can try adding these interaction effects in all utility functions:

+ i1 * amount.dummy[0] * period.dummy[0]
+ i2 * amount.dummy[0] * period.dummy[1]
+ i3 * amount.dummy[0] * period.dummy[2]
+ i4 * amount.dummy[1] * period.dummy[0]
+ i5 * amount.dummy[1] * period.dummy[1]
+ i6 * amount.dummy[1] * period.dummy[2]
+ i7 * amount.dummy[2] * period.dummy[0]
+ i8 * amount.dummy[2] * period.dummy[1]
+ i9 * amount.dummy[2] * period.dummy[2]

To be able estimate these interaction effects, you need to increase the number of rows in your design (since all pairs of attribute levels between amount and period need to appear sufficiently within your design). I tried and it worked when I added the first 8 interactions, but when I add the last one the model is no longer identified (Undefined D-error). It may be good idea anyway to increase the number of rows to ensure that you have enough variation in the data to estimate all kinds of interaction effects later on.

Michiel

[JvB]

Hi Michiel,

thank you very much for your advice.
So generally speaking, the restrictions in a design do not per se cause a problem to estimate interactions between the restricted attributes/levels just in my case there seems to be an issue with the ninth interaction effect?!
And also generally speaking, and also with my design, if I add the interactions the minmal sample size requirements are higher that with main effects only?

Thanks and best,
J.

[Michiel Bliemer]

Correct, restrictions in a design do not per se cause problems with adding interaction effects.

Adding interactions at the design stage means that Ngene will try to optimise the design to also estimate parameters of interaction effects. To do so, it needs to balance Fisher information on parameters of main effects and parameters of interaction effects. In other words, with the same amount of information captured in a choice task you need to estimate more parameters, so the amount of information per parameter decreases and therefore the sample size requirements are expected to increase.

Michiel