This one is wrong: ddct = dCt [target]/dCt[calibrator] it should be

ddct = dCt [target] - dCt[calibrator]

Althogh it is not very clear to me what "target" "reference" and "calibrator" in your examples stand for.

Let us consider two genes. One is the "gene of interest" of which you like to quantify a differential expression. Let's denote it "goi". The other gene is a loading control, sometimes termed the reference gene. Let's denote this by "ref".

You want to know the change in expression of the goi (normalized to ref) when you change the condition. The conditions could be "treatment" and "control", or "diseased" and "heathy". Something like this. Let us consider that we want to know the induction (or repression) of the goi by a treatment, relative to a control condition. Let us denote there conditions as "trt" and "ctl".

Usually, the relative normalized expression of the goi in a given condition is expressed as

dCt = Ct[ref] - Ct[goi]

You have dCt values for both conditions:

dCt[ctl] = Ct[ref,ctl] - Ct[goi,ctl]

The log fold-change between the conditions is then

ddCt = dCt[trt] - dCt[ctl]

Nice to know that I already could help you a bit. However, I actually wanted you to find following:

(1) the amount of amplicon, N, increases exponentially with the cycle number c, with the amplification efficiency E:

N(c) = No*E^c

(No is the starting amount we actually want to quantify)

(2) this amount is measured by fluorescence being proportional to N:

F(c) = p*No*E^c

(p is the sequence and probe-specific, generally unknown proportionality factor)

(3) the principle of measurement is to detemine the cycle number at which some arbitrary thershold (F=Ft) is crossed (c=Ct):

Ft =p*No*E^Ct

This is the basic formula. From this you should go on and find out, how the normalized log-ratio can be determined from the measured Ct values.

Honestly, take a week off and develop the equations (starting from the ones above) until you finally get to the ddCt. Try it by your own. It's simple math, nothing complicated. You only need patience, a clear mind, and some endurance. You are able to do it.

Finally you asked why dCt is often calculated as Ct[goi]-Ct[ref]. The reason, I think, is that Livak et al, who first describes the "2^(-ddCt) method" made it like this. They havent given any explanation for this. I think they did not recognize this as counter-intuitive. The result was correct, so the method "works", but the logig is one step harder to understand due to this unfortunate fact. Many many (most?) people doing real-time qPCR blindly follow "instructions" without really understanding the method or the calculations. A scientist should not do be "blind" in methods he/she is applying. The math is not sophisticated. It is helpful to know how to deal with logarithms, but this is a topic of 16-year old pupils. I see that scientists have a perferred topic, where they are experts, but in nature sciences some fundamental math-literacy is as important as being able to understand English (or any other language). Math is the language of science, it is the grammar of logic. There is no way to hide away from math when doing any nature science (or any empirical science). No chance. People who effectively do this (and there are many) are actually bad scientists. They are condemned to scratch at surfaces and they are lost when things do not work as expected.

At a slighliy different level, I think that PhD students who use a method must know the method deeply and thoroghly. This is (a very important) part of beeing a PhD student. I see many students applying hundreds of methods but who do not really understand a fraction of these methods. Having used a particular method in a PhD means that you are *expert* in this method, not just a "blind, helpless user". Methods are the fundament of empirical sciences - just using them is not and should not be enough for a scientist.This is my opinion, and this is why I always try to push people to think harder. Not an easy way, and not short-term rewarding...

In order to be able to do this, you need to think about the measurement uncertainty of your experiments. Straight up statistical comparisons only look at the values that you have and do not take into account the precision of the methods and so can make things look significant when they really are not or look insignificant when they actually are.

Have a read of the paper I linked and let me know if you have any questions.

Article Quantitative polymerase chain reaction: A framework for impr...

You said: "I have seen that others log their ddCt values and then perform anova or t-test." - is this really true?... I can't believe this!

What exactely is the meaning of a ddCt value? If you have found out and you can explain it, you will almost have the answer to your question.

So my advice is that you start understanding what real-time PCR measures and what the measured and calculated values really mean (Ct, dCt, ddCt). WIthout a sound understanding, any further analysis is no more than a game of chance with little hope of success. But, on the other hand, when you really understand the method you are using, then the analysis will be rather clear and simple.

Ct is the cycle number at which the fluorescent signal passes the threshold ; which inversely correlates to the amount of template added.

dCt is the ct value of a treated or untreated sample normalised against the endogenous control gene of the same sample. So after subtracting the ct value of the reference from the treated or untreated the value you have is the delta-ct for either treated or untreated. Given a reaction with 100% amplification efficiency then 1 ct difference between dct corresponds to twice the amount of RNA copies.

When you then subtract the dct of the untreated from the dct of the treated you have the ddct. A large negative ddct value corresponds to high expression fold change when 2^-ddct. When I get a fold change that is 0.6 for instance i have been taking the negative reciprocal of this value to give a final fold change of -1.7. Maybe this is wrong, but it seems to make sense to me.

I know i am probably getting confused with terminology and find all this fold change stuff tricky to explain, so forgive me. If i were to use the ddct (not fold change) values then I have a range of values that are not normally distributed. Or use the fold change values then I have negative and positive or whole numbers and decimals. I dont know enough about how these values would effect the statistical test and ultimately the significance. I'm very confused :(

Please take a look at my workbook and see if you can tell me the error of my ways.

"Ct [...] inversely correlates to the amount of template added." -> it correlated to the log amount. Can you write the pysical relation as a formula? You will then see that there needs to be some proprotionality factor that translates "amount" into "fluorescence". This factor is obviousely probe- and/or sequence-specific, and also depending on the selected threshold (that may be different for different genes! [it is sometimes not even possible to select a reasonable threshold that is similar for both genes]).

"[...] 1 ct difference between dct corresponds to twice the amount of RNA copies." -> not sure what exactly you mean here: a dCt of 1 (so a difference between Ct values) or a ddCt of 1 (difference between two dCt values)? Considering your next paragraph I suppose that you mean the former, what would be wrong. You neglected the proportionality factor I mentioned above. Exand the formula for the Ct to the dCt, and you will see what I mean. The dCt is some "relative normalized expression measure" that contains unknown proportionality factors, so dCt values have no particular, interpretable meaning in themselves - but they have a relative meaning, i.e. they can be compared between conditions.

"A large negative ddct value corresponds to high expression fold change when 2^-ddct" -> the ddCt is the log ratio of the normalized expressions under two conditions. You can again exapand the formula for the dCt to see that. You will then find that now the proportionality factors and thresholds cancel out and the result is really interpretable as a (negative)(*) log-ratio. The log, as you can see from the formulas, if you carefully develop them, is to the base E, where E is the efficiency. To get the fold-change from the log fold-change, you have to anti-log the ddCt by E^(-ddCt). Hence for E=2.0, a ddCt of 0.6 would translate to a fold-change of 2^(-0.6) = 0.66.

(*) the negation (finally leading to the "minus" in 2^(-ddCt)) comes from the fact that the dCt valu was calculated as Ct[target_gene] - Ct[reference_gene] following Livak et al. I find this conuter-intuitive and I really wonder why this is repeated by so many. When the difference is calculated this way, it means that the amount of the reference gene is normalized to the amount of the target gene (what I say is counter intuitive!). Hence, the *smaller* (or more negative) such dCt values are, the *higher* is the relative normalized expression (RNE). This makes little or no sense to me. I would always prefer to calculate the dCt as any normal thinking person would calculate a normalized measure: as ratio of target divided by the loading control (and not as the reciprocal!). The ddCt is then, funnily, calculated "correctly", dividing the RNE of the treatment condition through the NRE of the control condition. This eventually leads to the neccessity to once more calculate a reciprocal, what is achieved by the "minus" in 2^(-ddCt). If the dCt values would be calculated correctly as dCt = Ct[reference] - Ct[target], then this minus would not be required.

"If i were to use the ddct (not fold change) values then I have a range of values that are not normally distributed." -> Why? To all my experience, the distribution of ddCt values is in good accordance with the normal distribution, whereas this is clearly not the case for fold-changes.

It will be very, very helpful and enlightning to really develop the formulas from the physical basis of fluorescence signal generation and exponential amplification step by step, on your own. It may take a week of work, but it is really worth it. You will gain a deep understanding and skies will clear up.

Jochen Wilhelm, You are clearly very knowledgeable on this subject area and I am truly truly grateful that you have taken the time to push me towards a better understanding. Even though I can see I am testing your patience and kind of annoying you. These things are sometimes very plain to see when you have taught them for many years and have a real aptitude for maths and patterns. For others, these things can be a real challenge and in reality are things we only encounter from time to time and not our chosen field. I can however see the errors in my initial question and you have given me a better understanding from the few paragraphs you have written to help me.

I have just repeated an example using the formula you describe and have received a very different answer (a better answer, if there is such a thing?)

I calculated,

dCt = Ct[reference] - Ct[target] then,

ddct = dCt [target]/dCt[calibrator]

I then performed 2^(ddCt) "without the minus".

I now have a positive fold change for that gene! (initial 0.7, now 2.1) clearly a large difference!!

Tell me then, why do the vast majority perform the calculation the way I initially did if it is wrong? There must be a logical reasoning for this. The examples I followed (possibly blindly) were those provided by the company who supplied the instrument. They are clearly set out and I imagine designed to be easily followed by those who need the help. Why would they be telling people to calculate their data in this way?

This one is wrong: ddct = dCt [target]/dCt[calibrator] it should be

ddct = dCt [target] - dCt[calibrator]

Althogh it is not very clear to me what "target" "reference" and "calibrator" in your examples stand for.

Let us consider two genes. One is the "gene of interest" of which you like to quantify a differential expression. Let's denote it "goi". The other gene is a loading control, sometimes termed the reference gene. Let's denote this by "ref".

You want to know the change in expression of the goi (normalized to ref) when you change the condition. The conditions could be "treatment" and "control", or "diseased" and "heathy". Something like this. Let us consider that we want to know the induction (or repression) of the goi by a treatment, relative to a control condition. Let us denote there conditions as "trt" and "ctl".

Usually, the relative normalized expression of the goi in a given condition is expressed as

dCt = Ct[ref] - Ct[goi]

You have dCt values for both conditions:

dCt[ctl] = Ct[ref,ctl] - Ct[goi,ctl]

The log fold-change between the conditions is then

ddCt = dCt[trt] - dCt[ctl]

Nice to know that I already could help you a bit. However, I actually wanted you to find following:

(1) the amount of amplicon, N, increases exponentially with the cycle number c, with the amplification efficiency E:

N(c) = No*E^c

(No is the starting amount we actually want to quantify)

(2) this amount is measured by fluorescence being proportional to N:

F(c) = p*No*E^c

(p is the sequence and probe-specific, generally unknown proportionality factor)

(3) the principle of measurement is to detemine the cycle number at which some arbitrary thershold (F=Ft) is crossed (c=Ct):

Ft =p*No*E^Ct

This is the basic formula. From this you should go on and find out, how the normalized log-ratio can be determined from the measured Ct values.

Honestly, take a week off and develop the equations (starting from the ones above) until you finally get to the ddCt. Try it by your own. It's simple math, nothing complicated. You only need patience, a clear mind, and some endurance. You are able to do it.

Finally you asked why dCt is often calculated as Ct[goi]-Ct[ref]. The reason, I think, is that Livak et al, who first describes the "2^(-ddCt) method" made it like this. They havent given any explanation for this. I think they did not recognize this as counter-intuitive. The result was correct, so the method "works", but the logig is one step harder to understand due to this unfortunate fact. Many many (most?) people doing real-time qPCR blindly follow "instructions" without really understanding the method or the calculations. A scientist should not do be "blind" in methods he/she is applying. The math is not sophisticated. It is helpful to know how to deal with logarithms, but this is a topic of 16-year old pupils. I see that scientists have a perferred topic, where they are experts, but in nature sciences some fundamental math-literacy is as important as being able to understand English (or any other language). Math is the language of science, it is the grammar of logic. There is no way to hide away from math when doing any nature science (or any empirical science). No chance. People who effectively do this (and there are many) are actually bad scientists. They are condemned to scratch at surfaces and they are lost when things do not work as expected.

At a slighliy different level, I think that PhD students who use a method must know the method deeply and thoroghly. This is (a very important) part of beeing a PhD student. I see many students applying hundreds of methods but who do not really understand a fraction of these methods. Having used a particular method in a PhD means that you are *expert* in this method, not just a "blind, helpless user". Methods are the fundament of empirical sciences - just using them is not and should not be enough for a scientist.This is my opinion, and this is why I always try to push people to think harder. Not an easy way, and not short-term rewarding...

Note also that if you don't have paired samples (that is, the two conditions are done on the same cell culture/cell cultures from the same individual), the « second » d [dCT[trt] - dCT[ref]] is just a waste of time and computation power, a direct test between dCT[trt] and dCT[ref] is enough...

ddCT is needed only for paired samples!

@Emmanuel Curis

Why is it a waste of time and power to do ddCT on unpaired data and why is it useful on paired data?