It is completely irrelevant what is avereaged first and what next, and if you first subtract and then average or the otjher way around. It's simple algebra: it all leads to the same result.

(a - b + c - d)/n = (a+c)/n - (b+d)/n = (a-b)/n + (c-d)/n = (a+b-c)/n - d/n = ...

The important point is from what data you get the variance. For an unpaired analysis, you get the variance from the ΔCt values, and there should be one ΔCt value per biological sample/replicate (technical reps must be averaged). If the samples are paired, there might be a component of that variance shared between paired samples. This shared part can be "eliminated" by subtracting the paired values, what means that for each pair of samples (like a tumor and an adjacent tissue sample from the same patient) an individual ΔΔCt is calculated, and the variance is estimated from these ΔΔCt values. The cost of this procedure is that you loose half of the degrees of freedom (depending on the amount of shared variance this can be worth it).

So your solution 1B for the analysis of paired samples is correct.

Putting together:

Technical replicates must always be averaged. This (usually) means that you average the replicate Ct values measured fror the same gene from the same biological specimen, so you end up with a single ΔCt value per specimen.

For unpaired samples of sizes n and m, you get n and m ΔCt values, respectively. The typical task is to test if the mean difference beween these two samples of ΔCt values is zero. This is a straight-forward two-sample t-test. The observed mean difference is nothing else but the (mean) ΔΔCt value (which value can be obtained in several different ways; the most intuitive way to my opinion is to calculate the mean ΔCt for each group and to subtract those). If you use a ststs software to compare the two groups of ΔCt values, this software will give you the mean difference (=ΔΔCt) together wit its standard error and confidence interval.

For paired samples you have n ΔCt values of each group, and there is a pairing structure (each one ΔCt of one group is "paired" with a ΔCt of the other group). Actually, there is no need to do anything further to these values. In contrast to the unpaired analysis one can use the paired-sample t-test to test for a mean difference of zero. If desired, the ΔCt values may be pairwise subtracted, yielding n ΔΔCt values. Testing that the mean ΔΔCt is zero with a one-sample t-test is equivalent to the paired-sample t-test. The mean ΔΔCt can be calculated just like for unpaired samples or, if the n ΔΔCt values have already been calculated, by averaging those. If you use a ststs software to compare the two groups of (paired) ΔCt values, this software will give you the mean difference (=ΔΔCt) together wit its standard error and confidence interval.

NB: The analysis of ΔCt values is in no way different from the analysis of any other normally distributed variable. For instance, if you would analyze the difference in pH values or in OD values or in blood pressure etc, the procedure would exactly be the same.

It is completely irrelevant what is avereaged first and what next, and if you first subtract and then average or the otjher way around. It's simple algebra: it all leads to the same result.

(a - b + c - d)/n = (a+c)/n - (b+d)/n = (a-b)/n + (c-d)/n = (a+b-c)/n - d/n = ...

The important point is from what data you get the variance. For an unpaired analysis, you get the variance from the ΔCt values, and there should be one ΔCt value per biological sample/replicate (technical reps must be averaged). If the samples are paired, there might be a component of that variance shared between paired samples. This shared part can be "eliminated" by subtracting the paired values, what means that for each pair of samples (like a tumor and an adjacent tissue sample from the same patient) an individual ΔΔCt is calculated, and the variance is estimated from these ΔΔCt values. The cost of this procedure is that you loose half of the degrees of freedom (depending on the amount of shared variance this can be worth it).

So your solution 1B for the analysis of paired samples is correct.

Putting together:

Technical replicates must always be averaged. This (usually) means that you average the replicate Ct values measured fror the same gene from the same biological specimen, so you end up with a single ΔCt value per specimen.

For unpaired samples of sizes n and m, you get n and m ΔCt values, respectively. The typical task is to test if the mean difference beween these two samples of ΔCt values is zero. This is a straight-forward two-sample t-test. The observed mean difference is nothing else but the (mean) ΔΔCt value (which value can be obtained in several different ways; the most intuitive way to my opinion is to calculate the mean ΔCt for each group and to subtract those). If you use a ststs software to compare the two groups of ΔCt values, this software will give you the mean difference (=ΔΔCt) together wit its standard error and confidence interval.

For paired samples you have n ΔCt values of each group, and there is a pairing structure (each one ΔCt of one group is "paired" with a ΔCt of the other group). Actually, there is no need to do anything further to these values. In contrast to the unpaired analysis one can use the paired-sample t-test to test for a mean difference of zero. If desired, the ΔCt values may be pairwise subtracted, yielding n ΔΔCt values. Testing that the mean ΔΔCt is zero with a one-sample t-test is equivalent to the paired-sample t-test. The mean ΔΔCt can be calculated just like for unpaired samples or, if the n ΔΔCt values have already been calculated, by averaging those. If you use a ststs software to compare the two groups of (paired) ΔCt values, this software will give you the mean difference (=ΔΔCt) together wit its standard error and confidence interval.

NB: The analysis of ΔCt values is in no way different from the analysis of any other normally distributed variable. For instance, if you would analyze the difference in pH values or in OD values or in blood pressure etc, the procedure would exactly be the same.

Dr. Jochen Wilhelm thank you very much for your answer. It was really helpful!

I just want to reemphasise this incredibly important statement:

"NB: The analysis of ΔCt values is in no way different from the analysis of any other normally distributed variable. For instance, if you would analyze the difference in pH values or in OD values or in blood pressure etc, the procedure would exactly be the same."

There is nothing magic about qPCR, and nothing arcane about dCt values. A dCt is literally a normalised expression value that happens to be in log2 format.

I see, time after time, questions regarding pairing and inversion and statistical analysis of ddCt values and all of them, all of them, render the question far, far more complicated than it needs to be.

qPCR uses an exponential process to generate meaningful quantitative data: the numbers it spits out are, by the very nature of the process, inverse log2 values. This means

1) High numbers = low expression, low numbers = high expression, and

2) A Cq of 23 represents twice as much target molecule as a Cq of 24, and a Cq of 22 is four times as much.

3) Subtracting log values is the same as dividing linear values.

If you are comfortable with logarithms, this presents no issues. A lot of people who publish qPCR methods are very, very comfortable with logarithms, and they write manuscripts that reflect this. A lot of people who actually do qPCR are less comfortable with logarithms, and thus the many questions in this vein. I would, in all honesty, count myself in this latter category.

If you are not careful, it is very easy to lose all sense of what your numbers mean, because the ddCt method, for example, involves normalising your individual sample Cqs (GOI Cq - refgene Cq) to get dCt, then directly subtracting these in paired formats to get ddCt, all without ever leaving log2 format. And maybe throw in some averaging, too, because why not?

You may, at the end of all of this, be looking at "-2.13, -1.17, 0.10" and thinking "is...is this significant or not? And is...is it up or down?"

You may also make critical mistakes that are not immediately obvious, because the format serves to obfuscate the true meaning of your data.

So. If you are entirely happy with log values, and understand, implicitly, what they mean at every step, then stick with log values.

My advice, however, would be to convert all your Cq data to linear values as soon as you can, and then do all subsequent analysis in this format. As Jochen Wilhelm notes, it's just...data: there's nothing special about it. Linearised qPCR data is no different from qPCR data left in log2 format, it's just a lot easier (so, so much easier) for those of us without log-compatible brains to grasp at a glance.

These are "relative quantities" (RQ), and they are so much easier to deal with.

The equation is literally "efficiency(lowest Cq - sample Cq) ", where efficiency is the extent of doubling each cycle: assuming your PCR isn't awful, this is basically "2".

So if, for your GOI, all your samples give values between 21 and 24, take your lowest (21), which represents your highest expression, and express all the others as fractions of this.

Thus, highest expression sample: 2(21-21), which is 20, and anything to the power of zero is 1. Your highest expression sample thus becomes 1.

A sample with Cq of 23 becomes: 2(21-23), or 2(-2), i.e. 0.25. Looking at the data, yeah: that sample had a Cq 2 cycles higher, which does mean it's about a quarter of the levels, but now you have this information immediately available in an easy to understand format.

No more log values. You do this ONCE, for all your data, and after that everything is linear, using normal linear comparisons like you use for everything else.

The only, only foible worth addressing is in generating your normalisation factor: for more than one reference gene (and you should absolutely use more than one), once linearised you should generate the geometric mean, not the arithmetic mean (so square root of(A*B), rather than 0.5*(A+B)). This is to make sure that variation in either direction is afforded equal power (and is equivalent to arithmetic averaging of log2 values).

I've attached an excel file with some made-up numbers using ddCt and RQ methods: the end result is exactly the same, but for me it's so, so much easier to see how to do stats on the RQ values. (Edit: and I've just realised I forgot to point out that with RQ, the pairing you might employ for ddCt can be assessed at the end: instead of just doing a straight Ttest or whatever, do a paired Ttest -again, once it's linear data, treat it just like any other normal linear data -if paired, assess as paired, if not, don't).

At the end of the day, Jochen is really good at stats: his answer is excellent.

I'm rubbish at stats (and worse at logs), so I have learned many tricks to make that deficiency less of a liability. If you lean more to my end of the statistical spectrum, I hope this may be of help.

John Hildyard , thank you for re-emphasizing my statement.

I must make two comments. They are not meant to critisize you. They should just provoke thinking and discussing the points. If it sounds personal or impolite, it's due to my limited skills in English. I honour your opinion, please don't get me wrong.

I agree that some people obviousely have problems with logarithms (what seems a serious limitation to me whenworking na natural/life sciences). However, I don't understand the fixed direction you look at it. You say that it helps to make sense of the numbers when you go to the "linear scale".* But the linear scale you mean is the scale of the sequence concentration; this is the exponential scale of the Ct values. There is a linear scale of the Ct-values (that refers to the log scale of the concentration), too. There is not "one right - one wrong". They are both right, and I wonder why you don't see that you sacrifice the lienarity of Ct by wanting the linearity of conc. You would never even think about this when you were dealing with pH values, or with OD values, or with dB values, or virus titers, etc. Do you really calculate c(H+) from pH if you want to check if something changes the acidity of a solution/environment, because your brain cannot handle logs?

Then there is one imho relevant piece you forgot to mention: the dCts you propose are differences between two measurements of the same gene (in different samples). This way, the dCt in fact represents a relative expression, but is does not normalize to differences in the amount of sample being present in the PCRs (i.e.there is no reference gene involved). The typical understanding is that a dCt is the difference between two genes (goi ad ref) in the same sample.

In fact, if you first subtract the "highest Ct value", you loose the possibility to normalize to a reference in a meaningful way. This is surely problematic.

Another point refers to your advice to work on the exponential scale of the Cts throughout. You are not really fully consequent here. You should already start with y = 2(-Ct) and continue from there. Insteah of 2(a-b) your calculation should be ya/yb.

Finally one note to the stats: the log of geometric mean of 2x is the arithmetic mean of log(x). So by suggesting to use the geomatric mean on exponentiated Ct values is similar to averaging the Ct values. This is ok. The problem starts when you need the variance or standard error of such geometric means, e.g. because you want to judge the statistical significance of a difference of geometric means. An apporpriate way would be to log the values and get the statistical significance for the transformed value. But, hey, these are the Ct values...so finally you need to undo everything you did before.

---

*I personally don't get the point why 0.25 should be easier to be recognized as "less concentration" as -2. What is so difficult to understand the sign as the direction and the value as the amount?

Hi Jochen, thanks for the reply.

" the dCts you propose are differences between two measurements of the same gene (in different samples). This way, the dCt in fact represents a relative expression, but is does not normalize to differences in the amount of sample being present in the PCRs (i.e.there is no reference gene involved)."

This is a consequence of me posting late at night and not being clear. I'm pretty careful not to call any of my RQ values dCts (because they're not dCts, they're RQs), and under the method I suggest you never actually see the "lowest Cq - sample Cq" value because you're immediately raising 2 to the power of it. Yes, these are non-normalised, within-gene values, but the purpose of this transformation is to put all your samples on a linear range as early as possible, both GOI and refgenes (you lose any sense of expression magnitude that raw Cq values indicate, but you lose those via the dCt method too).

The reference genes are of course applied (see the attached excel sheet), but you do not need to do this before linearising: you can do it afterwards, and after linearising the normalisation is more conventional (i.e. "divide by loading control value").

My argument, in essence, is that converting all data (GOI and ref genes) to linear quantities puts all your data in realms that many people are more comfortable with.

If you were quantifying western blot densitometry, for example, you would get your per-sample quantities for your protein of interest (which are linear values), and divide these by your per-sample loading control quantities (which are also linear values), and very few people ever seem to have particular difficulty conceptualising this.

If you convert your qPCR data to linear quantities, you can do exactly the same, and in my experience this usually reduces random errors creeping in because people are just...more comfortable with linear values.

In cases, for example, where GOI and refgene Cq values are similar, the values following the dCt normalising (GOI - ref) can produce "-2.3, -0.45, 1.2" and so on (or worse, all positive values for one GOI, all negative for another), and this may confuse some less log-adept researchers (personal experience, here). If you're happy with logs, it's pretty trivial to recognise that a -2.3 vs 1.2 is simply 3.5 cycles of difference, or a smidge over ten-fold, and that the difference would be EXACTLY the same if it were 0.0 vs 3.5, or -7.5 vs -4, but if you're not happy with logs, this may be less obvious. Similarly, if a single sample has a Cq value five cycles higher than all other samples for both GOI and ref genes, this discrepancy is immediately lost upon calculating dCt, whereas converting to RQs before normalising reveals this sample to be around 32x lower than all the other samples in both GOI and refgenes, flagging it up as a very questionable sample that should probably be discarded (yes, a diligent researcher could recognise this at the Cq stage, but again, personal experience).

If you are dealing with a gene that changes expression by orders of magnitude, you can always log transform the values back again for statistical tests or graphical representation, but that's a choice you can make post-hoc on a case by case basis.

The numbers, in essence, are the numbers: they don't change (see excel sheet that obtains the exact same values via RQ or ddCt). I just find that people generally get a better handle on the numbers when they're not forced to mentally do inverse logs all the time.

0.5 vs 0.0005 is really easy for me to interpret: it's a big change, and I can see that it's three orders of magnitude.

5 vs -5 is....less easy. It's the same difference, but the magnitude isn't as obvious (and it's worse when you bring in decimal places).

Sure there are cases where log scales are implicit (pH is a great example), but people rarely are in a position where 'normalising the pH value' is a calculation they need to do. In most cases the limit of a molecular biologist's interaction with pH is "put pH electrode into solution, add HCl or NaOH until you get the right number".

Finally:

" *I personally don't get the point why 0.25 should be easier to be recognized as "less concentration" as -2. What is so difficult to understand the sign as the direction and the value as the amount? "

Like I said, some people are much more comfortable with logs than others. You are absolutely in this category, and I am...not. I mean, I can do it, but when I look at "-2", I am still mentally doing "1 divided by 22, oh, it's a quarter". I do not need to do the same for 0.25. :)

If you just don't want negative values and higher values should indicate higher expression, it would be way simpler to define Ct' = 45-Ct.

The layout of the Ct values in the Exel file indicates paired samples (Control and Treated are measured on the same subject), but the test is for independent samples.

The test is done on the normalized concentration values, testing the mean difference in these normalized concentrations. This is not ideal, because the assumption that these values are normally distributed is not reasonable. I also don't understand why you use a one-sided alternative.

The ratio of the mean values (1.009 / 0.363 = 2.781) is not the same like the mean of the ratios (4.539). This woul give very different results, depending on whether you make a paired analysis or not. Actually, you must use the geometric mean here, too!

To me, a -2 is a -2. Full stop. It's less than a -1, and way less then a +3. And a +1 is less then a +3, and -1 is less then 0. That's the math understanding of a 4-year-old (ok, in fact, "negative numbers" can be difficult for a 4-year old... but you get the point). At no point I feel the need to make any sophisticated calculations in my head. It's not difficult because there is nothing to calculate or imagine in your head. It's just the number. Put it on a real line and see if it is more on the left (lower) or more on the right (higher).

The 0.25 you would calculate needs exponentiation (mentally difficult), and it tells me... what? Is a quarter of ...what? And what is the biological meaning or relevance of having a quarter (of what?).

After all, we only want to know if a gene is induced or repressed (and we do a significance test to see if the data we have is sufficient to interpret this direction based on the data we observed). It is practically irrelevant to know if a gene is induced/repressed by 20% or by 500%. I am not aware of any instance where this information is really used or is of any practical relevance (for publication, for the interpretation of a treatment efficiacy or whatever). There might be real systems biology approaches where knowing the exact concentrations, changes and rates is relevant for the physiological models, but peopple doing such kind of research should really not have any problems with understanding Ct values (it would anyway require some kind of absolute quantification, and more sophisticated statistical models to estimate the biologically relevant parameters that is considerably beyon applying t-tests or doing ANOVAs).

"It is practically irrelevant to know if a gene is induced/repressed by 20% or by 500%"

Seriously, strenuously disagree. A 20% reduction in a transcription factor that is robustly expressed will make virtually no difference to downstream expression. Many transcription factors vary innately by more than this purely through stochastic noise. A 500% reduction is much more significant and likely to have biological consequence, and a 1000% reduction even more so.

A 20% reduction in something structural and essential like a myosin heavy chain gene may well be of biological consequence, however, while a 500% reduction in a myosin heavy chain suggests you're measuring the wrong muscle. The numbers absolutely mean something beyond mere "more or less", and this is much, much easier to interpret when you don't need to do log transforms.

I really don't think you need to be an expert at Ct values to understand the consequences of large or small biological changes.

It seems we are arguing statistical significance vs biological relevance: they are two different things, and you can't always get both: I've had data where a treatment absolutely, unarguably, increases expression of a gene by 5%. This is of essentially zero biological consequence, even though the stats show it is definitely significant.

We have no information for the OP beyond the question, so we don't know whether their scenario is "I just want to do stats" or "I want to know whether the effect is likely to be meaningful", but I would generally suggest linearising because it makes the latter call much, much easier to make.

Regarding the file:

One-tailed is because I did this at midnight: it should be two-tailed. It's significant either way (just double the p value), but the Ttest was there simply to show that it's a hell of a lot easier to do stats on nice linear values that make sense. The values are not supposed to be paired (hence not being assessed as such): see the other tab where paired/unpaired are assessed respectively. Again, copy paste, midnight, no decent sleep in months, pandemic, etc. Thanks for pointing out the errors. I've uploaded a corrected file, but frankly I think I should just stop talking because I clearly am making far, far too many simple mistakes.

I absolutely agree that there are in fact occasions where the size of the effect matters. I think you gave good examples. It's my own experience that researchers in the biomedical field usually don't have any idea about what effect to expect, about what effect would be relevant in their models, and about what effects might indicate problems. Most of the time, genes with no--so-well described functions are analyzed, and they are analyzed in complex cell mixtures (like organ homogenates) of unknown composition, and it is unknown if the target gene has some cell-type specific expression etc etc. All this makes is almost impossible to come up with a reasonable guess what change would make sense, and the question really reduces to the point whether the gene is induced or repressed.

This is just how you did your t-test, too. I know, it's just some silly example, but I am sure you would not do this any differently in a real research setting. Actually, you should test the minimally relevant change (not the "zero change" - with this you only confirm that your data allows you to interpret the sign). Alternatively, you should interpret the confidence interval w.r.t. to the minimally relevant change. And there is another typical problem (again, my own experience): the precision of effect size estimates are usually extremely bad, so you man come up with a confidence interval from 2-fold to 100-fold, or from 0.5-fold to 0.01-fold (the p-value for thesting 1-fold is < 0.001). Anything in this range is compatible with your data, so how can you judge if the effect you observed indicate any relevant induction, or a too high induction?

Thank you for the discussion. I am sure this will be helpful for many readers.