Hi Barbara,

I am co-author of the paper entitled "Interpretation of effects in linear, logistic and Poisson regression models with transformed variables: guidance for practice and software", which recently has been accepted by the journal Epidemiology. The paper shows how to interpret these models in the original scale of the variable and provides illustrative examples, including the log transformed X in a linear regression model, as is your case. The paper is still in press but if you are interested, I can give you a pre-print copy under request. In addition, the paper is accompanied by the R package "tlm" which is available on http://cran.r-project.org/web/packages/tlm/.

Regards,

Jose

Anyway, if you want only to find the inflection point of a titration curve, then you can use two methods, the Extremum Surface Estimator (ESE) & the Extremum Distance Estimator (EDE) for computing it.

Article: https://www.researchgate.net/publication/228059400  or http://arxiv.org/abs/1206.5478v2

R implementation: http://cran.r-project.org/web/packages/inflection/index.html

or here: https://www.researchgate.net/publication/235791114

Thank you

Article Developing methods for identifying the inflection point of a...

Data Find inflection point using R

Nice question!

The log concentrations should be averaged by the arithmetic mean. Another way to the very same result is to the the concentrations, calculate the geometric mean and then log this result.

exp ( average of the logs ) == geometric mean

log( geometric mean) == average of the logs

@Demetris, it is possible, that I used the word titration curve inappropriately. I removed the tag.

@ Jochen Wilhelm, I am very glad and thankful that the question caught your attention.

My method is a real-time PCR.

For some reason, people always calculate arithmetic mean of concentrations, they even put it into qPCR guidelines (MIQE). The only place I have seen this aproach opposed and prefering geometric mean is a booklet from Quiagen company ( http://www.qiagen.com/sk/resources/resourcedetail?id=58d4a7d9-287f-4b01-85c3-5cb83db2228b&lang=en ). Since company booklet is not very impressive as a reference, do you know of any authoritatively-looking paper or book I could quote in my article ?

EDIT: What I said about MIQE is probably incorrect. When I was re-reading the guidelines recently, I did not see any recommendation to do arithmetic mean of concentrations.

Look here:

http://www.bio-rad.com/webroot/web/pdf/lsr/literature/Bulletin_5279.pdf

At the page 94 of pdf.

Hi Barbara,

giving a good referece specifically to this topic is difficult, because it is mainly common sence and very basic statistical thinking.

The easy line of argumatation is that the distribution of concentrations is typically right skewed and the distribution of the log concentration is unimodal symmetric (-> normal distribution), so standard statistical procedures that are often based on the assumption of normal distributed data can sensibly applied to the log data. Only for those who are not comfortable with interpreting log values, one can antilo the results to get it back to the original scale - and there the average of the logs is simply the geometric mean.

You can find this argumentation for instance in the book "Statistics in Clinical Vaccine Trias" from Jozef Nauta (Springer Verlag, 2011). You can find it in Google books, Chapter 3.2 is about this topic.

A "harder" approach is to derive the distribution of concentrations based on gene activity and regulatory networks. There are some papers you can find, e.g.

http://www.pnas.org/content/105/45/17256.full

and especially this one:

http://bernstein.harvard.edu/papers/nir_prl_2006.pdf

The demonstrate by theoretical considerations that the concentrations of stochastic regulatory processes like for mRNAs and proteins, the their distribution is of the gamme family, of which the log-normal distribution is a special case and a good approximation.

I hope this helps.

Hi Barbara,

I am co-author of the paper entitled "Interpretation of effects in linear, logistic and Poisson regression models with transformed variables: guidance for practice and software", which recently has been accepted by the journal Epidemiology. The paper shows how to interpret these models in the original scale of the variable and provides illustrative examples, including the log transformed X in a linear regression model, as is your case. The paper is still in press but if you are interested, I can give you a pre-print copy under request. In addition, the paper is accompanied by the R package "tlm" which is available on http://cran.r-project.org/web/packages/tlm/.

Regards,

Jose

If I am understanding your question correctly, the following reference may help.  http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2350653/

The link also has a section of "Selected References" at the bottom of the page with more information than the first Statistical Note.  The R package by Jose sounds like it will be very helpful as well.

if you deal with concentrations, you might also think of compositional data analysis and log-ratio transformations instead of the log-transformation, i.e. in this case the average of log-ratios.

@ Everybody, thank You very much for Your input !

I eventually used these citation to support geometric mean of concentrations in real-time PCR.

Larionov A, Krause A, Miller W. A standard curve based method for relative

real time PCR data processing. BMC Bioinformatics. 2005 Mar 21;6:62.

http://www.ncbi.nlm.nih.gov/pubmed/15780134

Scholtens IM, Kok EJ, Hougs L, Molenaar B, Thissen JT, van der Voet H.

Increased efficacy for in-house validation of real-time PCR GMO detection

methods. Anal Bioanal Chem. 2010 Mar;396(6):2213-27.

http://www.ncbi.nlm.nih.gov/pubmed/20012027