Proc REG. How do I transform ln confidence limits back to original arithmetic units ?

[Fluvio1]

I have performed OLS regression on ln transformed x and y variables, and re-transformed the results (ln_pred) to the original units using Duan's Smearing Estimate (my own code). The latter is done to minimize re-transformation bias.   I am not sure how to re-transform the ln confidence limits (Ln_Upper and Ln_Lower 95%) back to the original units--do I apply the smearing estimate or simply exponentiate the ln confidence limit values ?

[SteveDenham · Accepted solution]

I haven't done this, but if I had, I probably would have used PROC NLIN, NLMIXED, GENMOD or GLIMMIX. For manually transformed values in PROC REG or GLM, it is nice to know that there is a relatively simple unbiased back-transformation that doesn't appear to be as volatile as the expected value estimator for a log-normal distribution.

See this site: https://stat.ethz.ch/education/semesters/as2015/asr/Script_v151119.pdf The authors argue that since the confidence bounds are estimated percentiles of the distribution, the naive back-transformation (simple exponentiation) is certainly adequate and appropriate.

At least until the smeared estimate is outside of the confidence interval...

SteveDenham

[Fluvio1]

Thank you Steve. However it does look  like some of the Duan's  smeared estimates are outside the confidence limits. If I compute the quantiles for the exponentiated confidence limits, they  are highly skewed:

Not sure how to handle this ?

[Rick_SAS]

If (L, U) is a (1-alpha) CI for a parameter, p, then (eta(L), eta(U)) is a (1-alpha) CI for the parameter, eta(p), for any strictly monotone increasing continuous transformation, eta. That's because 

P(L <= X <= U) = P(eta(L) <= eta(X) <= eta(U))

for a continuous r.v X. So, yes, you can apply the inverse transformation to back-transform the estimates, including CIs.

[SteveDenham]

Thanks, @Rick_SAS . That property was what I was thinking about when I said that if a smeared estimate is used for the expected value (= eta in your post), then it should also be applied to the confidence bounds. The Swiss Tech paper implies that a direct exponentiation should be adequate.  Thoughts?

SteveDenham 

[Rick_SAS]

I had never heard of Duan's smearing estimator until this post, and I have no experience with it. I quickly glanced at the paper you cited. It looks to me like they transformed X and Y variables by using LOG and then backtransformed the estimates by using inverse-LOG = EXP. Yes, that is valid and standard.

Maybe I am wrong, but even though the paper mentions Duan's work, I don't think they actually use it. Those paragraphs seem to be inserted (maybe at the suggestion of a referee) to let the reader know that there is an alternative, which they authors do not use.

I think it is important to point out that there is a difference between a generalized linear model with a log-link and a linear model on the log-transformed data. See Error distributions and exponential regression models - The DO Loop (sas.com)
which compares
GENMOD

model y = x / dist=normal link=log;

to GLM 

model logY = x;

The OP seems to be using the latter. Which model is correct depends on the error distribution, which is perhaps why the OP mentions the smearing estimate. I do not know the correct model for the OP's data. Graphing the distribution of the residuals might be prudent. Domain knowledge might also help.

[Fluvio1]

Thanks again Rick.  I attached the original paper by Duan that defines the smearing estimator. Also see below:

[Fluvio1]

Thank you Rick for the insightful explanation.