Yes.

If you have a CI on a log scale, you state that, on the long run, the log-CI covers the log value of the parameter of interest. By transforming back, your claim stays valid; because of the monoticity of the log-transform. Same goes for NHST.

I understand that is a monotonic transformation, however the CI and the hypothesis testing are done on value shrunk.

I would like to understand more, could you please provide some supporting papers or demonstration.

Thanks a lot.

Roberto -

I am not familiar with your subject matter application, but if the transformation is not to obtain linearity, but just to handle heteroscedaticity of y with regard to x, then I prefer to leave heteroscedasticity in the error structure where it naturally belongs.  A log may not be the appropriate transformation for that anyway.  If you use weighted least squares (WLS) regression, then there are several ways to estimate the coefficient of heteroscedasticity, and if you have multiple regression, then you can use a preliminary prediction for y (say, OLS) instead of x, as the size measure in the regression weight for WLS regression.  

I am not clear on your application, but if y and x are both skewed distributions, the relationship can be very much linear.  In fact, for establishment survey data, that is very useful, and I developed work in that area for many years. 

Hypothesis tests are often misinterpreted, and a confidence interval does not need to rely on normality. But even so, you can use the usual confidence interval 'formulas' as long as the Central Limit Theorem is applicable.

Cheers - Jim

PS -   I know of one good book that may be of use to you in your situation:

Carroll and Ruppert(1988), Transformation and Weighting in Regression, Chapman & Hall, Ltd. London, UK.

PSS -  Technically, for y, the dependent variable,  we are talking about prediction intervals. 

Thanks a lot.

Roberto Molteni As a Chemist I am sure you are aware of Calibration or inverse regression models. There are several approaches to dealing with this. See:

https://www.bing.com/search?q=Calibration+models&pc=U528&form=QBLH&sp=-1&ghc=1&pq=calibration+models&sc=3-18&qs=n&sk=&cvid=00ED03E51464493B9C1F7B382C958353

My personal favorite has been sect 11.6 of:

https://www.bing.com/search?q=Calibration+models+Ott++and+Longnecker+Introduction+to+Statistical+Methods+and+Data+analysis&qs=n&form=QBRE&sp=-1&pq=calibration+models+ott+and+longnecker+introduction+to+statistical+m&sc=4-67&sk=&cvid=B28288FFA12E43D8AFB4CE2BD7A9B304

which I used to teach out of. I would appreciate your thoughts on these approaches. Thank you very much. David Booth

Prof David Eugene Booth , my apologies for the delay in replying.

I completely agree that there are several approaches to dealing with calibration. Therefore, there are also lots of discussions, especially for Classical vs Inverse Regression approach, started mostly from the study “Classical and Inverse Regression Methods of Calibration” R. G. Krutchkoff, Technometrics, Vol. 9, No. 3 (Aug., 1967), pp. 425-439. However, the discussion had already been started slowly since ’30.

Classical approach: treats the observed measurements (Y) as the response and the standards (X) as the regressor in the calibration. Hereby, referring to the simplest and general univariate calibration, with X free of error (measured with negligible error). Then, the analyst must invert the resulting regression model in order to use the instrument to make actual measurements in practice.

Inverse Regression approach: treats the standards (X) as the response and the observed measurements (Y) as the regressor in the calibration.

There are pros & cons in both approaches. However, in general the classical calibration is still a valid approach in analytical chemistry for several statistical and practical reasons.

For example, one statistical reason is that the assumption of X free of error (measured with negligible error) and random error only in Y is quite true, meaning unbiased estimation of intercept and slope of the calibration curve (under the assumption of residual error independent and identically distributed).

One practical reason come from the necessity to be simple as much as possible, even relaxing variability and uncertainty assessment, just to be understandable for not scientific people in the legal dispute. In the Court, judges want everything black/white, guilty/guiltless.

And, to complicate our life, in Europe is still missing the full harmonization: “The present interpretation rules shall apply for the analytical result obtained on the sample for enforcement. In case of analysis for defence or reference purposes, the national rules shall apply” (Consolidate version 14.12.2019 of the Commission Reg. (EC) No 333/2007 of 28 March 2007 laying down the methods of sampling and analysis for the control of the levels of trace elements and processing contaminants in foodstuffs).

In conclusion, classical or inverse regression approach? I do not go through the several criteria for choosing between approaches, because at the moment there is not a definitive answer on which approach is best. However, as general answer: the classical should be preferred where measurement error of X is negligible in comparison with the error of Y (e.g. X values fixed by the chemist in controlled calibration). While the inverse regression should be preferred where measurement error X is not insignificant in comparison with the error of Y (e.g. X and Y randomly sampled from a population of values in natural calibration, with Y always the instrument response).

Of course, if prior information is available, a Bayesian approach could be an alternative.

Sorry for being long, but calibration is really important to get reliable data.

Best regards,

Roberto

Roberto Molteni Dear Dr. Thanks very much for one of the best summaries of the topic I have ever seen. This is much appreciated. David Booth

Roberto -

A friend on ResearchGate showed me some time ago how in analytical chemistry one might use regression, and an inverse regression, but I thought something else was being discussed here.

You noted that "...data are often analysed after a logarithmic transformation to overcome the drawback of heteroscedasticity and/or extremely skewed distribution." But I indicate in a response above that heteroscedasticity and skewed distributions are not a problem. This is exactly the situation with establishment survey data. In fact, it is skewness which can make heteroscedasticity so obvious. Actually, anytime that there is a large difference between predicted values, we may notice heteroscedasticity. (See reference below to "Essential Heteroscedasticity.")

All you need here is weighted least squares (WLS) regression where the regression coefficients are the result of simultaneously minimizing the sum of weighted squares of estimated residuals with regard to these coefficients, and the estimated residuals are factored into a random factor, and a nonrandom factor which is the regression weight raised to the negative 0.5.

Instead of a log transformation, or any transformation, to attempt to control for heteroscedasticity - where heteroscedasticity is a natural result of predictions being of different sizes (but heteroscedasticity can be made of larger or lesser magnitude from model and/or data issues) - we need to realize that the practice of assuming homoscedasticity, where the coefficient of heteroscedasticity is zero, is a convenient artificial construct, but it is routinely wrong.

We need to allow for heteroscedasticity in the range that has been long known from agricultural experiments, and justified by Ken Brewer, as noted in the following: https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity. More information is found in the following, including updates and references: https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression. Some examples of highly skewed data where the classical ratio estimator has a good coefficient of heteroscedasticity to often robustly deal with a vast production of official energy statistics on a burdensome schedule are found here: https://www.researchgate.net/publication/319914742_Quasi-Cutoff_Sampling_and_the_Classical_Ratio_Estimator_-_Application_to_Establishment_Surveys_for_Official_Statistics_at_the_US_Energy_Information_Administration_-_Historical_Development.

Log transformations used to 'deal with' heteroscedasticity can go too far, or not far enough. The coefficient of heteroscedasticity can be set at the right level. See examples in https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity, and you can use the following tool to do this with your data and model, when you have used OLS regression, but can enter a regression weight into your software and try again: https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx.

Cheers - Jim

Yes, the statistical conclusions and the limits of the confidence intervals are still valid if expressed in the back-transformed scale (if logs are used). This happens due to the invariance property.

You can find good references in Zellner, Handbook of Econometrics.

Thanks Prof. James R Knaub for the useful information on the heteroscedasticity and I agree that rationale for the transformation and an interpretation of the estimates of treatment effects based on transformed data should be provided.

However, the question was on if “Backtransformed CIs have the statistical properties for the statistics on the original scale”, therefore if interpretation based on data transformed back to the original scale has not drawbacks due to different limits of the confidence intervals.

Thanks Andrey Davydenko , do you have web-links directly related to your statement? They would be useful.

Roberto Molteni it's page 150 in Zellner:

https://books.google.com.tr/books?id=-G2R1pQAE_QC&pg=PA151&dq=zellner+handbook+of+econometrics+invariance+of+confidence+intervals&hl=ru&sa=X&ved=2ahUKEwix5IG2q5vsAhWI-ioKHRoKBu4Q6AEwAHoECAIQAg#v=onepage&q=invariance&f=false

Not sure if the link works though.

Will try to find more refs later.

Btw, we used this property of invariance in our chapter here:

Chapter Forecast Error Measures: Critical Review and Practical Recommendations

On page 3 of the above full-text document we say that if you log-transform your data to stabilize variance and obtain forecasts as the median of density forecast and then back-transform the forecast, it will still be the median of density forecast and it will still be optimal under linear loss.

Roberto -

Understood. I was just saying that if you did this to reduce heteroscedasticity, that you best not do that transformation in the first place.

Cheers - Jim

Roberto Molteni here's a freely available direct web-link:

http://www.biostathandbook.com/transformation.html

They give this example: "For example, the log transformed data above has a mean of 1.044 and a 95% confidence interval of ±0.344 log-transformed fish. The back-transformed mean would be 101.044=11.1 fish. The upper confidence limit would be 10(1.044+0.344)=24.4 fish, and the lower confidence limit would be 10(1.044-0.344)=5.0 fish. "

But I would say the CI they are talking about relates no the median, not the mean of the distribution of interest.

Thanks Andrey Davydenko for the useful links.

Following the example in the link, “If you were comparing the fish abundance in different watersheds, and you decided that log transformation was the best, you would do a one-way anova on the logs of fish abundance, and you would test the null hypothesis that the means of the log-transformed abundances were equal”.

Back to my original question, that was “Are the statistical conclusions and the limits of the confidence intervals still valid if expressed in the back-transformed scale?”.

Based on the back-transform confidence interval, that is not symmetric, I wonder if some conclusion in log scale and in original scale could be different.

Roberto Molteni I'm now looking at my answer and you question again and it seems your question is a bit trickier than I expected in the beginning. Basically, if you just transform back the CI for the mean, the result will not be valid. The mean becomes the median, so the CI for the mean will now relate to the CI for the median. But if you transform back the prediction intervals, the result will be correct.

If you want to transform back the CI for the mean, you'll need some special methods developed specifically for this task:

https://www.tandfonline.com/doi/full/10.1080/10691898.2005.11910638

So sorry for not understanding the question:

The limits of the confidence intervals are not valid if they relate to the mean. They are only valid if the relate to the median. If you want to work with the mean, you need special methods, such described in the above paper.

Roberto Molteni

P.S. Here's a paper showing that CI for the mean on the transformed scale relate to the median when back-transformed, page 4 last paragraph:

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwiV8uWjjJ3sAhWQzYUKHW9hCEUQFjAMegQIAxAC&url=http%3A%2F%2Fcourse1.winona.edu%2Fbdeppa%2FStat%2520115%2FTutorials-Handouts%2FConsequences%2520of%2520the%2520Log%2520Transformation%2520(Biometry).doc&usg=AOvVaw0rmNujhFmiSltJmVAsjHsx

Most papers, however, focus on finding CIs for the mean, which complicates things.. So returning back to your question: it depends on how you formulate your hypothesis.

Yes, I am interested to the mean.

Thanks Andrey Davydenko for the useful information.