Dear Stephane Hess,
I estimated a mixed logit model with lognormal random parameters.
According to the manual, I used the following expression:
exp(mu_log_beta + sigma_log_beta_inter * draws_beta_inter)
Furthermore, I used the Delta method to calculate the mean and standard deviation of the lognormal coefficient "beta".
Using the Delta method in Apollo, I also obtain the robust tratio for "SD for exp(N( mu_log_beta , sigma_log_beta_inter )".
Now I am struggling with the interpretation of the results. According to the estimation results, "sigma_log_beta_inter" is significantly different from zero. Therefore, I conclude that significant interindividual heterogeneity is present. However, the tratio from the Delta method does not confirm this result. The SD of the lognormal "beta" is not significantly different from zero.
Now I wonder if I should use the one or the other for my statement on the significance of interindividual heterogeneity. Is there an easy answer?
Kind regards
Andy
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Delta method for lognormal parameters

 Site Admin
 Posts: 558
 Joined: 24 Apr 2020, 16:29
Re: Delta method for lognormal parameters
Andy
could you please show the outputs so we can help. We need to see the estimates, the robust covariance matrix, and also the outputs of the delta method
Thanks
Stephane
could you please show the outputs so we can help. We need to see the estimates, the robust covariance matrix, and also the outputs of the delta method
Thanks
Stephane
Re: Delta method for lognormal parameters
Dear Stephane,
Thanks a lot. Here they are.
I am sorry for the bad format. I did not know how to include tables here.
Estimates:
Estimate Rob.std.err. Rob.t.ratio(0)
mu_log_beta 0.0685 0.1716 0.4
sigma_log_beta_inter 0.5892 0.2025 2.91
Robust covariance matrix:
mu_log_beta sigma_log_beta_inter
mu_log_beta 0.0294 0.0252
sigma_log_beta_inter 0.0252 0.0410
Delta method:
Please note: I did not apply the delta method directly after estimation but reloaded the model using "apollo_loadModel" at a later point in time
> deltaMethod_settings < list(operation="lognormal", parName1="mu_log_beta", parName2="sigma_log_beta_inter")
> apollo_deltaMethod(model, deltaMethod_settings)
WARNING: Could not retrieve apollo_control.
Assuming this model was not estimated using HB.
Running Delta method computations
Value Robust s.e. Rob tratio (0)
Mean for exp(N( mu_log_beta , sigma_log_beta_inter ) 1.2739 0.3450 3.69
SD for exp(N( mu_log_beta , sigma_log_beta_inter ) 0.8206 0.5425 1.51
Kind regards
Andy
Thanks a lot. Here they are.
I am sorry for the bad format. I did not know how to include tables here.
Estimates:
Estimate Rob.std.err. Rob.t.ratio(0)
mu_log_beta 0.0685 0.1716 0.4
sigma_log_beta_inter 0.5892 0.2025 2.91
Robust covariance matrix:
mu_log_beta sigma_log_beta_inter
mu_log_beta 0.0294 0.0252
sigma_log_beta_inter 0.0252 0.0410
Delta method:
Please note: I did not apply the delta method directly after estimation but reloaded the model using "apollo_loadModel" at a later point in time
> deltaMethod_settings < list(operation="lognormal", parName1="mu_log_beta", parName2="sigma_log_beta_inter")
> apollo_deltaMethod(model, deltaMethod_settings)
WARNING: Could not retrieve apollo_control.
Assuming this model was not estimated using HB.
Running Delta method computations
Value Robust s.e. Rob tratio (0)
Mean for exp(N( mu_log_beta , sigma_log_beta_inter ) 1.2739 0.3450 3.69
SD for exp(N( mu_log_beta , sigma_log_beta_inter ) 0.8206 0.5425 1.51
Kind regards
Andy

 Site Admin
 Posts: 558
 Joined: 24 Apr 2020, 16:29
Re: Delta method for lognormal parameters
Hi Andy
the output makes sense. The mean and standard deviation for a Lognormal distribution are BOTH a function of BOTH the mean and the standard deviation of the underlying Normal. The mean for the underlying Normal has a high standard error in your case and this then translates into the higher standard error for the standard deviation for the Lognormal.
Of course, if your model is improving fit over a model without heterogeneity, then you can still use that as evidence that you are improving the model by including the heterogeneity
Stephane
the output makes sense. The mean and standard deviation for a Lognormal distribution are BOTH a function of BOTH the mean and the standard deviation of the underlying Normal. The mean for the underlying Normal has a high standard error in your case and this then translates into the higher standard error for the standard deviation for the Lognormal.
Of course, if your model is improving fit over a model without heterogeneity, then you can still use that as evidence that you are improving the model by including the heterogeneity
Stephane
Re: Delta method for lognormal parameters
Dear Stephane,
Thank you very much for your prompt and helpful reply.
Kind regards
Andy
Thank you very much for your prompt and helpful reply.
Kind regards
Andy