If a gene is not expressed (or if expression is not detected), it is impossible to quantify its expression (there is nothing to be quantified). You donÄt have a Ct value in this case, so you cannot go along with dCt and ddCt, whatever you do. Inserting a constant value is not recommended, as this introduces bias in the mean (d)Ct-values and in the variance. This is not good.

If you have a large number / proportion of negative PCRs (no Ct), it may be better to analyze the odds of expression (rather than its quantity).

https://en.wikipedia.org/wiki/Logistic_regression

https://towardsdatascience.com/logistic-regression-detailed-overview-46c4da4303bc

If is happens occasionally that samples are negative (no Ct), you may consider a model with censoring (you should ideally censor at a cycle where you would expect less than a single molecule):

https://en.wikipedia.org/wiki/Censored_regression_model

https://www.jstor.org/stable/2335161?seq=1

https://stats.idre.ucla.edu/stata/dae/tobit-analysis/

Thank you very much for you well-written and well-thought answer Jochen Wilhelm and also the supplementary materials that you provided. It's a good idea to check the odds rather than the extent of expression. I'll have a look at the materials. But I really wanted to compare the expression of the gene of interest among various lines that were treated. Is there any way to do it by not considering the values of the control (since the controls did now show any expression at all)? :-(

There is no problem to compare the gene expression between different lines in which the gene is exressed. You can easily state that the gene is not expressed (orno expression is detected) in the control line, and that the gene is higher expressed in line X than in line Y, and lower in line Z etc.

If you just need tocheck that your data are sufficint to claim that the absence of expression in the controls is larger than in another cell line, you could use simple binomial or chi² test comparing the frequencies of "non-detects". Spoiler: p3 per group and only non-detects in the control and only detects in the other line.

Understand that ddCt methods give you a fold increase. If you control expression is "zero", then ANY expression will give you an infinite fold increase.

If your Cq values for actual expression are reasonable (

John Hildyard , I respectfully disagree with your statement that statistics there is "not much point for statistics (if...)". It is not such an uncommon problem that in one condition or cell line, there is just barely any expression above the detection level, or the gene is expressed so sporadically that most of the samples will be negative, -- and that in another condition or cell line the expression is less sproradical (or even well measureable). The problem is to show that the gene is expressed in one condition more likely (or more often, or less sporadically...) than in another. And for sure are there statistical tools to check if the observed data can be confidently interpreted that way. Tobit models are one way to use quantitative information where available and work with censoring; logistic models can be used if "positive" vs. "negative" is all information we can get (i.e. when the "positive" reactions are too noisy to be quantified with reasonable accuracy, like with Ct values above 34 or so).

Oh, I'd agree that "maybe no" vs "maybe yes" is totally statistically tractable, but for "literally zero" vs "something", stats typically fare poorly.

If I failed to detect anything, ever, in one cell line, but robustly in another, I would report it as such, and not attempt to apply statistics.

Possibly I've been lucky with reviewers thus far?

Ok, say you have lines A and B. You have measured n=1 values for each. A is negative, B is positive. Would you claim a difference between A and B? Certainly not. What about n=2, both A are negative and both B are positive? And n=3, 4,...? (surely, things become even more difficult if if there are positive among the A's and/or negatives among the B's, but you focussed on "robust results" only). How would you judge the observed difference? This is why we calculate the statistical significance. Depending on the sample size and the theory behind you might have been a bit lucky, yes:)

If I had an N=1 in each group, I wouldn't attempt statistics regardless of measured values. :-)

For zero values, though, especially with PCR (which often has a non-zero false positive rate), I don't think there's a huge degree of mileage to be had in measuring over and over again just to be extra super double sure there's nothing there. You just increase the chance of accidentally measuring noise. If you did this enough times you might generate a set of low but non-zero values to do stats on, but they would still probably just be "noise", and you also risk falling down a rabbit hole of confidence testing to decide whether something is real or not (the 'odds of expression' approach you noted above), all of which is increasingly divorced from biological context (if you have to measure something over and over again just to see if it's always not detected, then probably it is of little biological relevance even if sometimes there appears to be a single molecule or two).

Statistical tests are primarily about confidence, not magnitude, so "barely there, but detectable" vs "slightly there and detectable" is readily tractable, but "never detected" vs "slightly there and detectable" is less so (and of course, you cannot really calculate the distribution of a clutch of values if they are all "N/D")

In practice you are generally going to be measuring multiple genes at the same time (in the same panel of samples), and assuming you're doing this correctly, those will probably be sufficiently powered sample sets for you to make statistical inferences from the genes where metrics are more conventional "increase/decrease". Alongside these you could also report the "there/not there" data from the same groups, with "Not detected (N/D)" listed in place of the data points for the relevant samples.

This is less easy if you want to express everything as "fold change", but since you cannot calculate the fold change for "there/not there" data, I wouldn't take that approach in the first place.

Dear John, I think we essentially agree in practice. However, there are some aspects that don't seem so clear in theory and that may lead to overly uncritical interpretations or even to bad experimental designs. Therefor I want to give a short feedback, and you are heartly welcomed to disagree.

I think the line of arguments can not start with a non-zero false positive-rate of PCR. Having false-positives is clearly an error in the method (contamination or false interpretation of an amplification signal [determined, finite Ct-value]). I certainly agree that this happens in practice more often than not, but this is an experimental problem, that makes it impossible to measure occasional or extremely low expression. So if there is such thing like a non-zero false positive rate, the assay is not sufficiently well optimized or the experimental conditions don't allow to even think of such questions. If we want to talk about low or occasional expression, we must assume that there are no false-positive PCRs possible (what is in fact possible, in practice). It may still be that we won't be able to quantify small amounts accurately when considerably non-specific products are amplified. This just increases the LOQ, possibly also the LOD (if, due to competition, the few specific traget molecules are not amplified to detectable amounts). We may then not be able to distinguish 1 from 10 molecules and must use a higher censoring value.

I also see the danger of that rabbit hole of wanting a decision of what is true and what is false. But this is not what I meant with the "odds of expression". Thinking that a statistical analysis is done to proof that something is true or false is wrong anyway. This is not why statistics are done. They are done to check if the infromation in the observed data relevant to the scientific problem warrants a particular kind of expectation (e.g. the expression ratio is larger than 1, not smaller than 1) with a minimum (statistical) confidence. If we would not check that, we could just give purely annectotical reports. Note that the experiment itself is part of the statistical analysis. It makes no sense to report a p-value on non-random (e.g. purposefully selected) data. If the data are obtained from a proper independent measurements with no selection etc., knowing the data is sufficient to recognize the evidence it brings for or against some hypothesis, and the p-value is only a numerical summary that makes it easier to communicate (in more complicated models it can make this much, much easier). If you tell me that three non-detects are sufficint to expect that there is no expression is fine as long as I assume that these are unselected and complete results from independent experiments. This is already a statistical analysis.

You also say that looking for "almost not expressed" is "increasingly divorced from biological context". I don't think so. The occasional expression of a gene in a few cells of the tissue at a particular time can initiate biologically relevant processes, and it can be interesting to see if such gene expression events can be a possible reason. It further is sometimes important to show that there is no contamination, or that the contamination is below a limit close to the LOD, and it may be simpler to analyze a series of (small) samples instead of taking a large sample and make an extraction/concentration before the measurement. Generally, I don't think that we should be too fast in claiming that something is not biologically relevant because it is "too little". It can be a single mutation in a single cell in a multi-billion cell organism that leads to cancer and eventually to the death of this whole organism. Life is non-linear, and the context is important. (I think it is equally wrong to assume biological relevance just becuse something is "much". It can still be an irrelevant side-effect or counter-balanced/nihilated by other processes).

Mohammad Wasiful Gofur

It is not best to substitute non-detects with 40 (however, sometimes performed). I recommend an algorithm by McCall et al. (2014) Article On non-detects in qPCR data

Good luck!