Have a look at the role of covariates in the hierarchical model: They are there to explain variation between individual partworth utility vectors - that is, basically a multivariate regression is conducted with the individual utilities being the dependent variables and the covariates being the independent variables. Therefore, from my point of view, the "strength" of a covariate has to be assessed regarding the individual utility values, not regarding shares.
Actually this brings me to the conclusion that you should also simulate the individual utilities using a multivariate regression.

The references are the following (taken from my Sawtooth paper):

Andrews, Rick L.; Ainslie, Andrew; Currim, Imran S. (2002a): An Empirical Com­parison of Logit Choice Models with Discrete Versus Continuous Representations of Heterogeneity. In: Journal of Marketing Research 39(4), 479-487.
Andrews, Rick L.; Ansari, Asim; Currim, Imran S. (2002b): Hierarchical Bayes versus Finite Mixture Conjoint Analysis Models: A Comparison of Fit, Prediction and Partworth Recovery. In: Journal of Marketing Research 39(1), 87-98.
Vriens, Marco; Wedel, Michel; Wilms, Tom (1996): Metric Conjoint Segmentation Methods: A Monte Carlo Comparison. In: Journal of Marketing Research 33(1), 73-85.


Von: Dimitri Liakhovitski [mailto:dimitri.dcm@gmail.com]
Gesendet: Donnerstag, 3. März 2011 16:15
An: Wirth, Ralph (GfK SE)
Cc: Chris Chapman; R DCM List
Betreff: Re: [R-sig-DCM] What is a strong covariate in CBC/HB?

That's funny - I was thinking about it and was about to address the same point (about shares), but Ralph was beat me to it.
Upon reflexion, my problem with using shares is this: if one tries to study covariates, defining the strengh of covariates in terms of changes in shares seems circular. It's like saying "Look, the shares for 2 groups are pretty different, so it's a strong covariate". Or: "Look, the shares for 2 groups are not very different, so it's not a strong covariate." Ralph is probably right that very different utilities might result in the same shares. In fact: what if even very strong covariates produce a negligible difference in shares between groups? It would be a finding by itself, and that's what we are trying to find out. Hence, we need a definition of covariate strength that is pure and not outcome (share) related. Makes sense?
Until now I have not been able to define the relative strength/weakness of a covariate other than by referring to the spread of the means (like in Scenario 2). Probably, variance is also an important consideration, as Ralph mentioned. Not sure what else...

Ralph, do you happen to have the titles of these references?
Thanks!

Dimitri

On Thu, Mar 3, 2011 at 3:57 AM, Wirth, Ralph (GfK SE) <ralph.wirth@gfk.com<mailto:ralph.wirth@gfk.com>> wrote:
I'm not sure if you can evaluate the strength of covariates based on preference shares. Completely different utility values can result in the same preference shares, but still: the groups are different with regard to their utilities, and this difference in utilities is what covariates should explain. Based on this reasoning, I'd say group 2's covariate is "stronger". It's basically an ANOVA/t-test logic: the higher the value of the test statistic, the stronger is the covariate that defines the investigated groups.

This issue reminds me a lot of the issue of how to simulate preference segments. You can find a lot information about how to simulate them (and about which segments are distinctive, which are not) in the papers of e.g. Vriens, Wedel, Wilms (I think 1997) and Andrews et al. (2002 or 2003).







-----Ursprüngliche Nachricht-----
Von: Chris Chapman [mailto:cnchapman@msn.com<mailto:cnchapman@msn.com>]
Gesendet: Donnerstag, 3. März 2011 00:14
An: Dimitri Liakhovitski; Wirth, Ralph (GfK SE)
Cc: R DCM List
Betreff: Re: [R-sig-DCM] What is a strong covariate in CBC/HB?

I think I'm missing something here ... but seems that it depends on the use.
If you're using those to estimate share of preference (as I would assume),
then Scenario 2 would be "stronger" because the predicted preference share
differences are larger: (excuse the formatting)

Scenario 1                Raw utilities
       A=Features 1+3  B=Features 2+4  exp(A)              exp(B)      sumPref
A       Pref B
Group 1     -1                  1                   0.367879441 2.7182818283.08616127   12%     88%Group 2          1                   -1              2.718281828
0.367879441     3.08616127      88%     12%

Scenario 2                Raw utilities
       A=Features 1+3  B=Features 2+4  exp(A)          exp(B)  sumPref
A       Pref B
Group 1     -2                  2               0.135335283                 7.3890560997.524391382      2%      98%Group 2           2                  -2              7.389056099
0.135335283     7.524391382     98%     2%

OTOH, if the use is to predict something else from the utilities
(questionable, since they probably should be zero-centered diffs instead),
then the utilities shown are just linear transforms of one another (assuming
the utilities have the same property as their means).
cor(mean.scenario1,mean.scenario2) = 1.0.   In that case, the predictions
would be the same either way, in which case they're equally "strong".

And of course, as you noted WRT Ralph's comment, it depends on variance.  If
the variance is equal in both cases (e.g., 1) then Scenario 2 could be
regarded as "stronger" because the mean difference has a larger effect size.

But I'm not sure if that's what you're asking ... :-)

--------------------------------------------------
From: "Dimitri Liakhovitski" <dimitri.dcm@gmail.com<mailto:dimitri.dcm@gmail.com>>
Sent: Wednesday, March 02, 2011 10:26 AM
To: "Wirth, Ralph (GfK SE)" <ralph.wirth@gfk.com<mailto:ralph.wirth@gfk.com>>
Cc: "R DCM List" <r-sig-dcm@r-project.org<mailto:r-sig-dcm@r-project.org>>
Subject: Re: [R-sig-DCM] What is a strong covariate in CBC/HB?

> I totally agree with everything you've written, Chris. But let me stick
> for
> a second to actual values of utilities. I am not asking: is this covariate
> strong/weak for practical reasons, but rather: which of the 2 is
> stronger/weaker.
>
> Now to what Ralph's written: So, the variance is small (within group)
> would
> imply a stronger covariate. Agreed. How about the means? For example,
> let's
> discuss only 1 attribute with 4 levels.
>
> Covariate Scenario 1:
> Group 1's beta means are -2, -1, +1, and +2.
> Group 2's beta means are +2, +1, -1, and -2.
>
> Covariate Scenario 2:
> Group 1's beta means are -4, -2, +2, and +4.
> Group 2's beta means are +4, +2, -2, and -4.
>
> Which covariate is stronger?
> Dimitri
>
>
> On Wed, Mar 2, 2011 at 1:07 PM, Wirth, Ralph (GfK SE)
> <ralph.wirth@gfk.com<mailto:ralph.wirth@gfk.com>>wrote:
>
>> I'd say if the groups that are defined by the covariate are very
>> different
>> with regard to their preferences (i.e. utility parameters) and at the
>> same
>> time the members of each group are very homogeneous, then the covariate
>> is a
>> "strong" covariate.
>>
>>
>>
>>
>>
>>
>>
>> -----Ursprüngliche Nachricht-----
>> Von: r-sig-dcm-bounces@r-project.org<mailto:r-sig-dcm-bounces@r-project.org> [mailto:
>> r-sig-dcm-bounces@r-project.org<mailto:r-sig-dcm-bounces@r-project.org>] Im Auftrag von Dimitri Liakhovitski
>> Gesendet: Mittwoch, 2. März 2011 19:04
>> An: R DCM List
>> Betreff: [R-sig-DCM] What is a strong covariate in CBC/HB?
>>
>> Guys, I could ask some of you individually but it's always better to talk
>> to
>> several smart guys at a time.
>> Some of you know I am working on a project for an ART-Forum which
>> explores
>> the issue of covariates in CBC/HB estimation: Is estimation with
>> covariates
>> useful?
>>
>> I have a question: what do YOU THINK is a "strong" covariate? (and let's
>> assume we know what the true utilities and their dependence on the
>> covariate
>> are)
>> I am looking less for a mathematical answer (although it's OK if it is)
>> and
>> more for a conceptual answer, maybe even with examples. Like: assume we
>> have
>> a covariate with only 2 levels (i.e., respondents belong to 2 groups):
>> what
>> should their preferences be like in order for us to be able to say: group
>> membership a strong covariate / weak covariate? And if we have 3 groups?
>> Also, I am looking less for a "correct" answer and more for your
>> opinions.
>>
>> Looking forward to your replies!
>>
>> Dimitri Liakhovitski
>> ninah.com<http://ninah.com>
>>
>>         [[alternative HTML version deleted]]
>>
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