The distribution of random terms

[infiniteugi]

Hello,

I used Apollo to develop a hybrid choice model for my study to submit a peer-reviewed journal. I got a critique on the methodology regarding the distribution of random terms in the LV model as follows:

"The random terms in the latent variables are following the normal distribution N(0,1), but it is not correct. They need to be distributed with variances different from 1 and the variances need to be estimated."

I'm not sure if he/she is correct, and if it is correct how can I fix this? Please give me some advice.

Thank you very much.

[dpalma]

Hi,

There are two possible normalisation approaches when working with Integrated Choice and Latent Variable (ICLV) model, also called hybrid choice models:

• To fix one of the coefficients in the measurement equation to the value 1, while leaving all parameters in the structural equation free.

• To fix the variance of the structural equation error term to the value 1, while leaving all parameters in the measurement equation free.

In the choice modelling literature, the preferred method is the second (fixing the std dev of the structural equation). You can see Raveau et al. (2012) for details.

Now, I don't know how your model is formulated, so I can't say what would be the best way to reply to your reviewer, but the reference I mentioned is a good place for you to start looking.

Cheers
David

[infiniteugi]

Hello David,

Thank you for your response. Could you please explain more about the distribution of the random terms?

I followed the Apollo manual (mostly example #26), and my code is as follows:

Code: Select all

apollo_draws = list(
  interDrawsType="sobol",
  interNDraws=500
  interNormDraws = c("eta_tl", "eta_qb", "eta_av")
)

### Create random parameters
apollo_randCoeff=function(apollo_beta, apollo_inputs){
  randcoeff = list()
  
  randcoeff[["LV_tl"]] = 
    gamma_physical_tl*physical +
    gamma_visual_tl*visual +
    gamma_age1834_tl*age1834 +
    gamma_age3544_tl*age3544 +
    gamma_age4554_tl*age4554 +
    gamma_age5564_tl*age5564 +
    gamma_age65_tl*age65 +
    gamma_no_car_tl*no_car +
    gamma_dist_bus_tl*dist_bus + 
    eta_tl
  
  randcoeff[["LV_qb"]] =  
    gamma_physical_qb*physical +
    gamma_visual_qb*visual +
    gamma_no_car_qb*no_car + 
    gamma_avg_p_sw_qb*avg_p_sw + 
    gamma_p_com_pub_qb*p_com_pub +
    gamma_medinc_qb*medinc +
    eta_qb
  
  randcoeff[["LV_av"]] = 
    gamma_physical_av*physical +
    gamma_visual_av*visual +
    gamma_age1834_av*age1834 +
    gamma_age3544_av*age3544 +
    gamma_age4554_av*age4554 +
    gamma_age5564_av*age5564 +
    gamma_age65_av*age65 +
    gamma_college_av*college +
    gamma_no_car_av*no_car +
    gamma_LV_tl_av*randcoeff[["LV_tl"]] +
    gamma_LV_qb_av*randcoeff[["LV_qb"]] +
    eta_av
  
  return(randcoeff)
}

I thought that the random components, in this case, follow the normal distribution. But the reviewer said that "The random terms in the latent variables are said to be distributed N(0,1), this is not correct, they need to be distributed with variances different from 1 and the variances need to be estimated. The structural equations of the LV do also need to include a constant and these need to be estimated."

Did I misunderstand the reviewer's comment?

Please advise me...

Thank you very much.

[stephanehess]

Hi

in relation to the reviewer's comment:

"The random terms in the latent variables are said to be distributed N(0,1), this is not correct, they need to be distributed with variances different from 1 and the variances need to be estimated."

--> This depends on whether you are estimating the variances in your measurement model. See the discussion in https://doi.org/10.1007/s11116-011-9351-z

"The structural equations of the LV do also need to include a constant and these need to be estimated"

--> this is not correct in general

Stephane