Simultaneous estimation of parameters with potentially different scale

[maa033]

Hi. I have a dataset encompassing respondents who have answered the same questions/survey, including a Choice Experiment (CE), but have got different pre-survey information. I am interested in testing whether the two subsamples (A and B) state different preferences for the attributes in the CE. To this end I define separate attribute parameters for each sub-sample, and separate scale. Scale for sub-sample A is fixed at 1, while scale for subsample B is estimated as part of the model. The sub-sample specific attribute parameters and relative scale for subsample B is estimated jointly. The result yields a very low relative scale, and somewhat inflated attribute coefficients for subsample B. However, when I multiply the estimated parameters With the relative scale for subsample B, these attribute estimates are comparable With those for subsample A. Is this a good model specification given the aim of the analysis? Or is there a more efficient way to test parameter differences, taken into account possible scale differences? I have run models in WTP-Space on the two sub-samples separately, which partly confirsm the model results, given my interpretation above (multiplying attribute parameters and relative scale for one sub-sample).

this is my model specification. the variable Certain is a dummy indicating which info the respondents have got.


V = list()

V[['alt1']] = scale_base*Certain*(asc + cost_B*Cost1 + torsk_B*KT1 + laks_B*VL1 + bunn_B*HB1 + land_B*KL1)+
scale_treat*(1-Certain)*(cost_T*Cost1 + torsk_T*KT1 + laks_T*VL1 + bunn_T*HB1 + land_T*KL1)

V[['alt2']] = scale_base*Certain*(cost_B*Cost2 + torsk_B*KT2 + laks_B*VL2 + bunn_B*HB2 + land_B*KL2)+
scale_treat*(1-Certain)*(cost_T*Cost2 + torsk_T*KT2 + laks_T*VL2 + bunn_T*HB2 + land_T*KL2)


V[['alt3']] = scale_base*Certain* (cost_B*Cost3 + torsk_B*KT3 + laks_B*VL3 + bunn_B*HB3 + land_B*KL3)+
scale_treat*(1-Certain)*(cost_T*Cost3 + torsk_T*KT3 + laks_T*VL3 + bunn_T*HB3 + land_T*KL3)

[stephanehess]

Hi

in order to estimate scale differences, at least one of the parameters needs to be shared between the samples. That's not the case in your specification, so your model is overspecified

NB. You're also missing the ASC for the second group, but maybe this was intentional

Best wishes

Stephane

[maa033]

So would dividing the scale parameter for sample A in both brackets, even if it is fixed at 1, solve the problem?

[stephanehess]

No. You need to fix the scale parameter for one sample to 1 anyway. But you also need to keep one parameter generic across the two samples.

Think about it this way:

- You have 5 attributes that you're estimating beta parameters for in your data
- If you split your data into two groups, then you can estimate 5 parameters for each group, so 10 in total.

But what you are doing is that you're trying to estimate 11 parameters, 10 betas and one scale.

The scale parameter is used to capture the difference in scale between the two groups. But if all the betas that this scale parameter multiplies are specific to group 2, then it has no way to capture scale differences.

You can test this easily empirically. Your model will stay the same if you fix both scales to 1 in this case.

So what you need to do is to make one parameter generic between the two groups, such as using a common cost coefficient.

You also need to decide what you're doing about the asc. Was it meant to be in both groups?

[maa033]

Thanks for the reply.
Could the common variable be the asc? This is meant to be included in the utility of both sub-samples.
And do I use the relative scale for the second sub-sample, i.e. scale_treat/scale_base instead of just scale_treat?

[stephanehess]

The ASC could take that role for a start.

You just use scale_treat. scale_treat/scale_base is going to be the same anyway as you must fix scale_base to 1.

Then once you have estimated this first model, you can see whether some of the other betas should be the same between groups by conducting statistical tests for the difference

[maa033]

Works!!
I just have to clarify; the estimates from this model can be interpreted as marginal (attribute) Utilities for each of the two sub-groups? I don't need to multiply estimated attribute parameters by the estimated scale for sub-sample B?

thanks for replies.

[stephanehess]

Yes, you have to:

Let's say you have:

V[['alt1']] = scale_base*Certain*(asc + cost_B*Cost1 + torsk_B*KT1 + laks_B*VL1 + bunn_B*HB1 + land_B*KL1)+
scale_treat*(1-Certain)*(cost_T*Cost1 + torsk_T*KT1 + laks_T*VL1 + bunn_T*HB1 + land_T*KL1)

then the marginal impact of Cost1 on the utility is given by the partial derivative against Cost1

So for someone in the Certain group, that will be scale_base*cost_B, and in the other group, it will be scale_treat*Cost1

[bartosz]

Hi,

If I may bump this topic, I would like to ask about the scale parameter (as in example_22). Following the "artificial tree structure" approach by Bradley and Daly (1992) ro Louviere et al. (2000), is the parameter that we get from Apollo, the inclusive value of the "nest", which means one needs to inverse it, to obtain the scale parameter, or is it the true scale parameter that can be used directly?

In the latter case, assuming we normalized mu_RP to 1, and estimated mu_SP, will it be correct to calculate the ratio of variance in both datasets in the following way?

variance_RP / variance_SP = (mu_SP / mu_RP)^2

Cheers
Bartosz

[stephanehess]

Bartosz

it's the direct estimate of the scale parameter. So you can use the calculation you show below

Stephane