Your last two questions can be answered with YES:

- Do you mean you would average the experiments before 2^?

- Or can you also use the geometric mean after 2^?

All calculations, all summaries and statistics should be done, as you call it, "before 2^". You can also calculate the geom.mean "after 2^", what gives the very same result (please look at the fomulas of geo.mean and arith.mean and the law of logarithms, especially that log(a*b) = log(a)+log(b) and log(a^r) = r*log(a)).

The advantage to work "before 2^" is that you can simply calculate the average and the standard error, and the confidence interval (CI). If you work "after 2^", there is no defined standard error anymore. The only thing you could do is to go back to the logs ("before 2^") where you can calculate a CI and then take these limits (lower and upper) and transform them back to "after 2^". I always suggest to completely forget about everything "after 2^" and stay with what you have "before 2^". (*)

You can always give the median and the IQR, also when you work "after 2^". However, these are descriptive statistics, and they are not sufficient (i.e. they do not use all the information from the data). They are particularily useless when you have only little data (say, n < 20-50), what is quite typical.

You measure ct values. You take the difference between reference and target gene -> dct. You may have several samples of the same type/condition, so you can average these dct values. Now you have (averaged or not) dct values for your different conditions (e.g. treated and control), and you calculate the difference of these dct values -> ddct value. If you have several of such ddct values, you can average them.

Still these are all simple differences (or differences of differences of) of ct values. Such values (differences) can be averaged without changing the meaning.

This would be different for products, factors, or fold-changes (like 2^ddct). Factors have a skewed distribution, and high values will bias the average: the average overestimates the fold-change.

(*) Honestly, knowing that the fold-change is 8 ("after 2^") not provide any more biological insight than knowing that the ddct is 3 ("before 2^"). The problem is that many people *think* they would know more if the knew the fold-change, but this is not true, as long as one cannot precisely say what these fold-changes biolgically mean. The biological meaning or relevance is so much depending on many many other things and circumstances, that the actual numeric value that is used does anyway not provide any really useful information. I start becoming sceptical when people *think* they knew something, just because they see their beloved "fold-changes". I find it better to present results *not* in a way that typically leads to over-interpretations. I prefer making such over-interpretations difficult, so that the scientist/reader has to think harder about the real biology instead of relying on easy thoughts about familiar-looking quantities.

You surely can calculate the mean (average) of several ddct values. If you prefer using the 2^ddct values (i.e. "fold-changes" instead of "log fold-changes"), the average is not a very sensible statistic. Here, one should rather use the geometric mean. But this turns out to be 2^(average ddct). So one should generally avoid raising ddct-values to 2. This makes life a lot easier (especially when it comes to the calculation of confidence intervals or tests).

Did I understand you aright:

The arithmetic mean cannot be used with the ddct?

You would use the geometric mean?

You would prefer the geometric mean before the median?

Nevertheless would the median be also okay?

If I use the geometric mean is there something like the standard derivation or the standard error of the mean for the geometric mean? Or do I have to use the range of my data or the quintiles instead?

"[...] But this turns out to be 2^(average ddct)."

Do you mean you would average the experiments before 2^?

Or can you also use the geometric mean after 2^?

Thanks you in advance!

Your last two questions can be answered with YES:

- Do you mean you would average the experiments before 2^?

- Or can you also use the geometric mean after 2^?

All calculations, all summaries and statistics should be done, as you call it, "before 2^". You can also calculate the geom.mean "after 2^", what gives the very same result (please look at the fomulas of geo.mean and arith.mean and the law of logarithms, especially that log(a*b) = log(a)+log(b) and log(a^r) = r*log(a)).

The advantage to work "before 2^" is that you can simply calculate the average and the standard error, and the confidence interval (CI). If you work "after 2^", there is no defined standard error anymore. The only thing you could do is to go back to the logs ("before 2^") where you can calculate a CI and then take these limits (lower and upper) and transform them back to "after 2^". I always suggest to completely forget about everything "after 2^" and stay with what you have "before 2^". (*)

You can always give the median and the IQR, also when you work "after 2^". However, these are descriptive statistics, and they are not sufficient (i.e. they do not use all the information from the data). They are particularily useless when you have only little data (say, n < 20-50), what is quite typical.

You measure ct values. You take the difference between reference and target gene -> dct. You may have several samples of the same type/condition, so you can average these dct values. Now you have (averaged or not) dct values for your different conditions (e.g. treated and control), and you calculate the difference of these dct values -> ddct value. If you have several of such ddct values, you can average them.

Still these are all simple differences (or differences of differences of) of ct values. Such values (differences) can be averaged without changing the meaning.

This would be different for products, factors, or fold-changes (like 2^ddct). Factors have a skewed distribution, and high values will bias the average: the average overestimates the fold-change.

(*) Honestly, knowing that the fold-change is 8 ("after 2^") not provide any more biological insight than knowing that the ddct is 3 ("before 2^"). The problem is that many people *think* they would know more if the knew the fold-change, but this is not true, as long as one cannot precisely say what these fold-changes biolgically mean. The biological meaning or relevance is so much depending on many many other things and circumstances, that the actual numeric value that is used does anyway not provide any really useful information. I start becoming sceptical when people *think* they knew something, just because they see their beloved "fold-changes". I find it better to present results *not* in a way that typically leads to over-interpretations. I prefer making such over-interpretations difficult, so that the scientist/reader has to think harder about the real biology instead of relying on easy thoughts about familiar-looking quantities.

Hi Sandra,

There should not be any problem with arithmetic mean after 2^ . However, it may not so obvious to use before 2^, As ddct assumed the same condition for all PCRs, when you repeat the experiment, the PCRs in different run are not in the same condition, so to average different runs before 2^ can add extra variation..

Yizhou, you are right that the mean can be taken from any kind of numbers. The problem, howerer, is to estimate the precision of the mean. Simply taking the mean of the 2^(d,d)ct values regularily leads to the wrong assumption that deviations to higher and to lower values are equally probable. It is generally a problem to provide correct intervals of reasonable or likely (or "not-too-unlikely") values for such estimates. This is much simpler (i.e. possible) to give such intervals for the geometric mean, what in turn is equivalen to the standard intervals of the arithemtric mean of the log values.