# lower bound of rlh in dual response none settings

How to calculate the lower bound (i.e. random selection / null model) of rlh in CBC/HB dual response none settings.

Do you just add one to the number of stimuli in one choice set?

Thanks a lot in advance.
asked Dec 14, 2011

That's a tricky question.  The CBC/HB documentation (Appendix J) gives some hints:

"Our approach is quite simple: we model the two choices (the forced choice among alternatives, and the buy/no buy follow-up) as independent choice tasks. In the first task, the choice is among available products (without a "None" alternative available). In the second task, the choice is among available products and the "None" alternative. Failure to pick the "None" alternative in the second stage (a "buy" indication) results in a redundant task. All information may be represented using just the second stage choice task. With that one task, we can indicate the available options, which option the respondent chose, and the fact that the respondent rejected the "None" alternative. Therefore, we omit the redundant first-stage task. "

So, let's imagine a CBC dataset where there are 4 alternatives per task, and 12 tasks, each followed by a dual-response none question.  Let's imagine that the respondent didn't pick the None alternative for six of the tasks.  That gives us six 4-concept tasks and twelve 5-concept tasks for this respondent, for a total of 18 coded tasks in the data.

Now, for each 4-alternative task, we know the null likelihood is 0.25; and for each 5-alternative task the null likelihood is 0.20.

So, this respondent's null likelihood would be:  [(0.25)^6 x (0.20)^12]^(1/18) = 0.215443469

(In other words, we are taking the geometric mean across six 0.25s and twelve 0.20s, which rounded to two decimal places is 0.22.)

As you can see, the null RLH will differ by respondent, depending on how many times the respondent used the "none" alternative in the second stage question.
answered Dec 14, 2011 by Platinum (163,515 points)