Confident Interval of the ratio of two random parameters using Mixlogit
Posted: 28 Jan 2021, 10:25
Hello Stephane and David,
Currently, I am running a mixlogit using Apollo, and I want to compute the confidence interval of the ratio of the random parameters. In particular, I want to calculate the standard error of a normally distributed parameter divided by a log-normally distributed one (in Utility Space). If I am not mistaken, please correct me if I am wrong; the way to go should be the one proposed by Bliemer & Rose (2013) because, as they state in the article:
(A better quality document is available here: https://www.overleaf.com/read/bggymcbqtstx )
That being said, a Monte Carlo simulation is needed to compute the standard error of the ratio of the two variables using draws from z_{c} and z_{k} and then averaging them. I was wondering if this is possible on Apollo at the moment.
Finally, I also checked the `apollo::apollo_deltaMethod()`, but I in the source code I only found the exact Delta method for non-random parameters provided by Daly, Hess & de Jong, 2012:
Thank you for your help!
Currently, I am running a mixlogit using Apollo, and I want to compute the confidence interval of the ratio of the random parameters. In particular, I want to calculate the standard error of a normally distributed parameter divided by a log-normally distributed one (in Utility Space). If I am not mistaken, please correct me if I am wrong; the way to go should be the one proposed by Bliemer & Rose (2013) because, as they state in the article:
Unfortunately, the case I want wasn't provided by the authors in their original article; regardless, following their notation, it should be like the following:Although standard errors, and therefore confidence intervals, of the moments of several random distributions can be determined analytically
(see Daly, Hess & de Jong, 2012), there exist no analytical expressions for determining the confidence intervals for WTPs that are defined as ratios of random coefficients. Even in the case of a random coefficient in the numerator and a fixed coefficient in the denominator, determining the confidence interval of WTP requires simulation.
(A better quality document is available here: https://www.overleaf.com/read/bggymcbqtstx )
That being said, a Monte Carlo simulation is needed to compute the standard error of the ratio of the two variables using draws from z_{c} and z_{k} and then averaging them. I was wondering if this is possible on Apollo at the moment.
Finally, I also checked the `apollo::apollo_deltaMethod()`, but I in the source code I only found the exact Delta method for non-random parameters provided by Daly, Hess & de Jong, 2012:
Code: Select all
if (operation == "ratio") {
v = est[parName1]/est[parName2]
se = sqrt(v^2 * (robvarcov[parName1, parName1]/(est[parName1]^2) +
robvarcov[parName2, parName2]/(est[parName2]^2) -
2 * robvarcov[parName1, parName2]/(est[parName1] *
est[parName2])))
operation_name = paste("Ratio of ", parName1name,
" and ", parName2name, ": ", sep = "")
}
- Bliemer, M. C., & Rose, J. M. (2013). Confidence intervals of willingness-to-pay for random coefficient logit models. Transportation Research Part B: Methodological, 58, 199-214.
Daly, A., Hess, S., de Jong, G., 2012. Calculating errors for measures derived from choice modelling estimates. Transportation Research Part B 46, 333-341.