Dear Stephane and David,
I wonder if the function of calculating the confidence interval for WTP has already been built into the Apollo package. It seems the DeltaMethod only prints out s.e and t-ratio for WTP. Could you please tell me, or invent something like KrCI or BootCI in package "DCchoice"?
Best,
Xian
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WTP intervals
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stephanehess
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Re: WTP intervals
Dear Xian
you can use the function apollo_bootstrap to produce bootstrap estimates, and you can then use these for WTP.
I know some people use Krinsky & Robb to produce confidence intervals for WTP, but this should NOT be done as you're simulating the ratio of two normals using that approach while the estimates only have the property of asymptotic normality. You can look at the discussions in this paper: http://www.stephanehess.me.uk/papers/wo ... r_2020.pdf
Best wishes
Stephane
you can use the function apollo_bootstrap to produce bootstrap estimates, and you can then use these for WTP.
I know some people use Krinsky & Robb to produce confidence intervals for WTP, but this should NOT be done as you're simulating the ratio of two normals using that approach while the estimates only have the property of asymptotic normality. You can look at the discussions in this paper: http://www.stephanehess.me.uk/papers/wo ... r_2020.pdf
Best wishes
Stephane
Re: WTP intervals
Dear Stephane,
Thank you for your reply! I'd like to know if the DeltaMethod function could be utilized to compute a ratio between a sum of two parameters and the monetary parameter? For example, I have estimated b_safe and b_female*safe. I would like to compute the WTP of women for safety, so the ratio should be -(b_safe+b_female*safe)/b_price. Right now I tried x1=-b_safe/b_price and x2=-b_female*safe/b_price separately, and got the associated standard errors to compute the standard error of x1+x2 by pen and paper. I got a rather big standard error so I become suspicious.
Best,
Xian
Thank you for your reply! I'd like to know if the DeltaMethod function could be utilized to compute a ratio between a sum of two parameters and the monetary parameter? For example, I have estimated b_safe and b_female*safe. I would like to compute the WTP of women for safety, so the ratio should be -(b_safe+b_female*safe)/b_price. Right now I tried x1=-b_safe/b_price and x2=-b_female*safe/b_price separately, and got the associated standard errors to compute the standard error of x1+x2 by pen and paper. I got a rather big standard error so I become suspicious.
Best,
Xian
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stephanehess
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Re: WTP intervals
Hi
yes, the Delta method can be used for any functions of estimated parameters. But only a few are implemented in apollo_deltaMethod. You can follow the approach set out in this paper to do it easily yourself: http://www.stephanehess.me.uk/papers/jo ... B_2012.pdf
Best wishes
Stephane
yes, the Delta method can be used for any functions of estimated parameters. But only a few are implemented in apollo_deltaMethod. You can follow the approach set out in this paper to do it easily yourself: http://www.stephanehess.me.uk/papers/jo ... B_2012.pdf
Best wishes
Stephane
Re: WTP intervals
Dear Stephane,
Sorry to bother you again. since Delta Method only works out s.e, should I use "mean+-1.96*s.e" to calculate the interval? I don't think this formula suits multinomial logit as this assumes normal distribution instead of asymptotic normal of parameters. Could you please suggest me the right way to calculate the interval? This question troubles me for a long time, so I will very appreciate it if you can give me any advice!!
Best,
Xian
Sorry to bother you again. since Delta Method only works out s.e, should I use "mean+-1.96*s.e" to calculate the interval? I don't think this formula suits multinomial logit as this assumes normal distribution instead of asymptotic normal of parameters. Could you please suggest me the right way to calculate the interval? This question troubles me for a long time, so I will very appreciate it if you can give me any advice!!
Best,
Xian
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stephanehess
- Site Admin
- Posts: 998
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Re: WTP intervals
Hi Xian
The use of +-C*s.e. (where C is the critical value for whatever level you are interested in, e.g. 1.96 for two-sided 95%) is not affected by whether you use it for an individual parameter or for a transformation of parameters such as WTP, with the s.e. given by the Delta method.
The question is not so much about the delta method, but whether this way to calculate confidence intervals (which assumes symmetry etc) is always appropriate. In cases with reasonable significance levels, the use of this calculation is generally fine. However, you are right that the property of normality may not extend far away from the MLE. This is why other approaches have been put forward for calculating confidence intervals, such as in the work of Armstrong et al. (2001), or using bootstrapping. These techniques are of course more cumbersome to use. But again, this is a general point, and not one that only applies to the delta method std errors, but others too
Stephane
The use of +-C*s.e. (where C is the critical value for whatever level you are interested in, e.g. 1.96 for two-sided 95%) is not affected by whether you use it for an individual parameter or for a transformation of parameters such as WTP, with the s.e. given by the Delta method.
The question is not so much about the delta method, but whether this way to calculate confidence intervals (which assumes symmetry etc) is always appropriate. In cases with reasonable significance levels, the use of this calculation is generally fine. However, you are right that the property of normality may not extend far away from the MLE. This is why other approaches have been put forward for calculating confidence intervals, such as in the work of Armstrong et al. (2001), or using bootstrapping. These techniques are of course more cumbersome to use. But again, this is a general point, and not one that only applies to the delta method std errors, but others too
Stephane
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Anna_Edenbrandt
- Posts: 6
- Joined: 27 Apr 2020, 20:01
Re: WTP intervals
Dear Stephane
Thanks for sharing your working paper above. I found it very interesting and useful!
As mentioned in the paper it is common to use the method proposed by Krinsky & Robb to produce confidence intervals for WTP-estimates. It is also common to use the draws obtained with this method to test for differences between WTP-estimates (from different models) with the complete combinatorial method (Poe et al 2005). Poe et al. argue that it is not appropriate to test for differences by examining if the confidence intervals overlap or not.
Since you argue against using the method proposed in Krinsky & Robb, I wonder if you recommend a different method than that proposed in Poe et al. for testing for differences in wtp-estimates across models?
Many thanks!
Regards,
Anna
Thanks for sharing your working paper above. I found it very interesting and useful!
As mentioned in the paper it is common to use the method proposed by Krinsky & Robb to produce confidence intervals for WTP-estimates. It is also common to use the draws obtained with this method to test for differences between WTP-estimates (from different models) with the complete combinatorial method (Poe et al 2005). Poe et al. argue that it is not appropriate to test for differences by examining if the confidence intervals overlap or not.
Since you argue against using the method proposed in Krinsky & Robb, I wonder if you recommend a different method than that proposed in Poe et al. for testing for differences in wtp-estimates across models?
Many thanks!
Regards,
Anna
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stephanehess
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Re: WTP intervals
Anna
I agree that it is not appropriate to test for differences by looking if confidence intervals overlap. You could see whether the estimated wtp from one model falls into the confidence interval from another, but even that ignores correlation which clearly exists by being based on the same data. The same is the case if you take the estimate of WTP in model 2, say WTP2, and use the delta method to see if this value is different from the value obtained in model 1.
Stephane
I agree that it is not appropriate to test for differences by looking if confidence intervals overlap. You could see whether the estimated wtp from one model falls into the confidence interval from another, but even that ignores correlation which clearly exists by being based on the same data. The same is the case if you take the estimate of WTP in model 2, say WTP2, and use the delta method to see if this value is different from the value obtained in model 1.
Stephane
Re: WTP intervals
Dear Stephane and Anna,
First, thanks Anna for bringing out the question. Actually that's the fundamental question has troubled me for long. That's why I keep asking Stephane about WTP intervals (Thank you so much for your patience!!). While the coefficient estimated of the interaction term indicates that females perceive the attribute significantly different from males, female's WTP interval for that attribute overlaps with male's. I tried several methods to calculate the WTP intervals, but the overlapping always exists.
Poe et al 2005 shared by Anna is quite inspiring, thank you again!
First, thanks Anna for bringing out the question. Actually that's the fundamental question has troubled me for long. That's why I keep asking Stephane about WTP intervals (Thank you so much for your patience!!). While the coefficient estimated of the interaction term indicates that females perceive the attribute significantly different from males, female's WTP interval for that attribute overlaps with male's. I tried several methods to calculate the WTP intervals, but the overlapping always exists.
Poe et al 2005 shared by Anna is quite inspiring, thank you again!