Antibody effects are often tested by observing reactions at concentrations which are multiples of each other  eg 2x, 4x,8x, 16x  or 10x, 100x,1000x. This implies the logarithms of the concentrations increase arithmetically.In a series of observations the logarithms give a more symmetical distribution for whicn an arithmetic mean is appropriate. The arithmetic mean of the logs is the log geometric mean(GM)

log GM = 1/n { logx1 + logx2 + logx3.....logxn }

            = log[{x1x2x3   xn} 1/n.]

so GM ={x1 x2 x3 ....xn} 1/n

The technique can be used for other observations following a log normal type of distribution, eg occupational and environmental dust  exposures

See eg Armitage and Berry Statistical Methods in Medical Research 2nd ed 1987 p 31-33.

Dear James,

Thank you a lot. It really helps.

In some situations the geometric mean is the only appropriate mean when you want to average ratios. Many papers are dedicated to this issue.

Fleming, P. J., & Wallace, J. J. (1986). How not to lie with statistics: thecorrectwaytosummarizebenchmarkresults.CommunicationsoftheACM,29(3),218–221

Spizman, L., & Weinstein, M. (2008). A note on utilizing the geometricmean:when,whyandhowtheforensiceconomistshouldemploythegeometricmean.JournalofLegalEconomics,15(1),43–5

Some examples can be found in our paper (section 2):

https://www.researchgate.net/publication/257026708_Measuring_Forecasting_Accuracy_The_Case_Of_Judgmental_Adjustments_To_Sku-Level_Demand_Forecasts

Article Measuring Forecasting Accuracy: The Case Of Judgmental Adjus...

Use of the geometric mean implies that the data are not normally distributed and that the arithmetic mean is not a good indicator of central tendency. The data are natural log transformed, the mean of these transformed data is calculated, and the geometric mean is equal to the the mean of natural log transformed data exponentiated.  Often non-normally distributed data are natural log transformed for analyses via linear regression or t-tests, the geometric mean is then presented to report results.  If the raw data are natural log distributed then the geometric mean and the median of the raw data are often very similar in value.    

The arithmetic mean is usually what is used to estimate mathematical expectation of the population. The advantage of the arithmetic mean is that it gives us an unbiased estimation. When data distribution is heavy-tailed, the arithmetic mean becomes very inefficient (a lot of data is required to get reliable estimates), but, nonetheless, the arithmetic mean remains an unbiased estimator of the population mean.

The geometric mean does not give us an unbiased estimator of the population mean and it is used for different purposes. If we average ratios, we often need the mean of log-transformed values because of the nature of our data. However, if log-transformed values have a non-Gaussian distribution, the arithmetic mean of log-transformed values is also not efficient.

Therefore, if you want to average ratios efficiently, you have to

1) log-transform your data

2) calculate trimmed mean of transformed values (usually 2 percent trim is sufficient and the percentage of trimming (it can be between 1 and 5) does not influence the quality of the estimator very seriously)

3) exponentiate the trimmed mean

If CIs are needed, we need to use additional methods based on finding CIs for log-transformed data and then exponentiating them,

Thank you for all your inputs.

But GMTs are notorious to hide low responders to vaccines, be careful when interpreting results.

Hi Bharti,

Could you elaborate on your statement a little bit? It is interesting to know and good to see an example.

Thank you.

GMTs hide non responders to vaccine suppose 10 people have good vaccine response and three donot have a threshold vaccine response GMTs would hide all non responders and show all have good response. So they are deceptive.

good information

In human beings even one non response can be life threatening for him so GMT may be academic but not practical in human trials.

As GMT hides non- responders also, so what do you recommended for practical trials. We mostly use CV % and mean. Is it reliable.

It would depend on what we are looking for and how non responders are going to affect with the results of our analysis.

Omesh

Well we are only looking for monitoring titre of the group.But we are still interested in individual titre, like how many are showing low titre.

To conclude group titres are ok but they may be dangerous in case an outlier is not responding to say rabies vaccine which can be life threatening for him/her. So interpretation of results matters if this is a health group or patients.

This is very good discussion if any one can help me in suggesting the analysis of 4-fold rise of anti-Vi serum immunoglobulin G (IgG)

enzyme-linked immunosorbent assay titers over baseline (seroconversion [SCN]) 42 days after immunization.

Interesting information as GMT will hide non responder in case of antibodies titration and evaluation.

Explanations of the terms you're looking for can be found here. https://www.who.int/biologicals/expert_committee/Clinical_changes_IK_final.pdf

This is a very good reference:

DOI: 10.1016/S1081-1206(10)60595-9

Olivier J, Johnson WD, Marshall GD. The logarithmic transformation and the geometric mean in reporting experimental IgE results: what are they and when and why to use them? Ann Allergy Asthma Immunol. 2008 Apr;100(4):333-7. doi: 10.1016/S1081-1206(10)60595-9. Erratum in: Ann Allergy Asthma Immunol. 2008 Jun;100(6):625-6. PMID: 18450118.