I have a question to Jochen Wilhelm and other experts in the field. I am a physician scientist currently working in basic science. We work on rodent experimental models and use qPCR a lot. We follow very stringent qPCR protocols with various reference genes for different tissues and cell lines. I believe our technical standards are good and our results are controlled for as many confounders as possible.
Reading high impact publications in our field I am often struck how little information is provided as to how qPCR data was analyzed. In expensive animal models or rare human tissue samples I understand the limitations on sample size and I acknowledge that whichever statistical test is applied, any conclusion on few samples should be taken with a pinch of salt. Again, statistical standards differ greatly between an exploratory n=5 rodent study or a n=5000 human biomarker validation for clinical application and I can live with this difference in statistical quality.
Returning to my question: Dr. Wilhelm is a strong advocate of working with log values in qPCR data and I completely agree with this. It is a common practice to perform log transformation to reduce skweness or variation in data, so why transform log data to increase variation before analysis? Yet, it is common practice to show fold changes on a linear scale in the world of physicians and I have been criticized by reviewers for showing dCt values. I know it is correct and mathematically makes sense, but it is the reviewer and to an extent the (bad) convention that decides what and where data is published. So I am wondering what a good compromise between mathematical sense and convention could be.
My current approach: Perform statistics on dCt or ddCt values and mention this in the methods section accordingly. Then show 2^-dCt (or 2^-ddCt) in the figures. Mean and error are calculated from dCt values and then I use exponentiates of these values for my graph (because I am not interested in the mean of relative expression!). On the y-axis I use a log-scale in order not to underrepresent downregulated GOIs. So it leaves me with a graph of relative expression (or fold changes) with asterisks to indicate levels of significance from the log values. Is this a valid approach? I feel this might be a compromise between convention and actual mathematical sense. What do you think?
Hi Hannes,
I face the same problem very often! I first try hard to convince my co-authors to use (IMHO) better graphical representations of their results. And if I succeed there, manuscripts often come back with complaints from reviewers who don't seem to understand the plots or who are very strict about what they want to see. A controversy with the reviewer will delay publication, and there is a risk that the manuscript will be rejected. So there is a huge pressure to follow the reviewer's suggestions, and in most cases we (my co-authors and I) do give into that pressure.
At least I tried to improve old habits a bit. This is by far not always successful, but that's life. Fall down and stand up again. I try to educate students and postdocs to think about these issues, to get a better understanding of data analysis and visualization. My hope is that they will once become better reviewers.
To your analysis: I think I don't need to confirm that I find your strategy correct (doing stats on dCt, and do any transformation with the mean and confidence limits if desired).
You say that you show the exponentiated values in a diagram, but use a log-scale for the y-axis in order to give the same emphasis on up- and down-regulations. This approach is surely valid. I just want to note that this is the same like showing the -dCt or -ddCt values right away.* The diagram will look the same, only the numbers at the y-aaxis will be different (giving either y or log2(y), where y is either 2-dCt or 2-ddCt).
I am only a bit concerned about showing 2-dCt. I am not sure what this value should represent. Many people (like my co-authors, for instance) are of the opinion that this is a "relative expression" (the expression of the GOI relative to the reference/loading control). But this is not correct, because the dCt value includes an assay-specific unknown constant:
-dCt = log2( Ngoi/Nref ) + u
This unknown (u) depends on the probes, the length and sequence of the amplified PCR products and on the threshold used to get the Ct values. The dCt value is related to the "log relative expression", but it is not equal to it. The 2-dCt are thus
2-dCt = Ngoi/Nref * 2^u
what is the "relative expression" multiplied by some unknown, assay-specific factor. What should be interesting in knowing the value of ^-dCt, when it will depend on the assay? How would this help us to interpret the data?
Th dCt values are not directly "log relative expressions". They are only proxies that can be compared between different groups/samples. Their actual value is (almost) completely irrelevant. A dCt value can only be interpreted w.r.t another dCt. A dCt of 0 has no particular meaning (in particular it does not indicate that the expression of the goi would be as high as the expression of the ref-gene!).
The ddCt is the numerical result of a comparison of dCt values. In this difference the unknown constant (u) cancels out and the result is really a log2-ratio of relative expressions, a "log2 fold-change". Here the value 0 means "no change".
I therefore find it strange to show 2-dCt, because this is a weird quantity that does not give us any useful information on its own and what makes it more difficult to compare them between groups. Plotting 2-dCt on a log-scaled axis will produce the same result as plotting -dCt directly, so I wonder what the fun of this double forth-and-back transformation game should be. Choosing a correct axis-title is also difficult, because "relative expression" is actually wrong. "Arbitrary units" would be correct but likely upsetting the reviewer. I therefore suggest simply showing (-)dCt values and write this in the axis title. This forces a reader to read these values as proxies without any particular quantitative meaning in their values (but in their differences!) and not to overinterpret the values.
This is slightly different for ddCt values, because they do have an interpretable quantitative meaning. A ddCt value of -1.5 for instance gives me a very concrete information that does not depend on the assay. This does not change by any monotone transformation like exponentiation: 2^1.5 = 2.8 also has a very concrete meaning. Showing 2-dCt on a log-sclaed axis or showing -dCt directly is equivalent. Showing 2-dCt on an untransformed ("linear") axis is inapt because it visually over-emphasises up-regulations (as you already said).
Good luck!
---
*From your post I infer that you calculate dCt = Ctgoi - Ctref. This is a bit inconvenient because then higher (larger) dCt values would correspond to a lower expression, what I find counter-intuitive. If you'd calculate dCt = Ctref - Ctgoi right away, then higher values would indicate higher expression, and the ddCt would be the log-ratio of "treated versus control" (in your way the ddCt is the log-ratio of "control vs. treated"). No need to change the sign afterwards.
Hi Hannes,
I face the same problem very often! I first try hard to convince my co-authors to use (IMHO) better graphical representations of their results. And if I succeed there, manuscripts often come back with complaints from reviewers who don't seem to understand the plots or who are very strict about what they want to see. A controversy with the reviewer will delay publication, and there is a risk that the manuscript will be rejected. So there is a huge pressure to follow the reviewer's suggestions, and in most cases we (my co-authors and I) do give into that pressure.
At least I tried to improve old habits a bit. This is by far not always successful, but that's life. Fall down and stand up again. I try to educate students and postdocs to think about these issues, to get a better understanding of data analysis and visualization. My hope is that they will once become better reviewers.
To your analysis: I think I don't need to confirm that I find your strategy correct (doing stats on dCt, and do any transformation with the mean and confidence limits if desired).
You say that you show the exponentiated values in a diagram, but use a log-scale for the y-axis in order to give the same emphasis on up- and down-regulations. This approach is surely valid. I just want to note that this is the same like showing the -dCt or -ddCt values right away.* The diagram will look the same, only the numbers at the y-aaxis will be different (giving either y or log2(y), where y is either 2-dCt or 2-ddCt).
I am only a bit concerned about showing 2-dCt. I am not sure what this value should represent. Many people (like my co-authors, for instance) are of the opinion that this is a "relative expression" (the expression of the GOI relative to the reference/loading control). But this is not correct, because the dCt value includes an assay-specific unknown constant:
-dCt = log2( Ngoi/Nref ) + u
This unknown (u) depends on the probes, the length and sequence of the amplified PCR products and on the threshold used to get the Ct values. The dCt value is related to the "log relative expression", but it is not equal to it. The 2-dCt are thus
2-dCt = Ngoi/Nref * 2^u
what is the "relative expression" multiplied by some unknown, assay-specific factor. What should be interesting in knowing the value of ^-dCt, when it will depend on the assay? How would this help us to interpret the data?
Th dCt values are not directly "log relative expressions". They are only proxies that can be compared between different groups/samples. Their actual value is (almost) completely irrelevant. A dCt value can only be interpreted w.r.t another dCt. A dCt of 0 has no particular meaning (in particular it does not indicate that the expression of the goi would be as high as the expression of the ref-gene!).
The ddCt is the numerical result of a comparison of dCt values. In this difference the unknown constant (u) cancels out and the result is really a log2-ratio of relative expressions, a "log2 fold-change". Here the value 0 means "no change".
I therefore find it strange to show 2-dCt, because this is a weird quantity that does not give us any useful information on its own and what makes it more difficult to compare them between groups. Plotting 2-dCt on a log-scaled axis will produce the same result as plotting -dCt directly, so I wonder what the fun of this double forth-and-back transformation game should be. Choosing a correct axis-title is also difficult, because "relative expression" is actually wrong. "Arbitrary units" would be correct but likely upsetting the reviewer. I therefore suggest simply showing (-)dCt values and write this in the axis title. This forces a reader to read these values as proxies without any particular quantitative meaning in their values (but in their differences!) and not to overinterpret the values.
This is slightly different for ddCt values, because they do have an interpretable quantitative meaning. A ddCt value of -1.5 for instance gives me a very concrete information that does not depend on the assay. This does not change by any monotone transformation like exponentiation: 2^1.5 = 2.8 also has a very concrete meaning. Showing 2-dCt on a log-sclaed axis or showing -dCt directly is equivalent. Showing 2-dCt on an untransformed ("linear") axis is inapt because it visually over-emphasises up-regulations (as you already said).
Good luck!
---
*From your post I infer that you calculate dCt = Ctgoi - Ctref. This is a bit inconvenient because then higher (larger) dCt values would correspond to a lower expression, what I find counter-intuitive. If you'd calculate dCt = Ctref - Ctgoi right away, then higher values would indicate higher expression, and the ddCt would be the log-ratio of "treated versus control" (in your way the ddCt is the log-ratio of "control vs. treated"). No need to change the sign afterwards.
Thank you Jochen! This is extremely helpful and I totally agree with every point. And yes, 2-dCt on a log scale is equivalent to -dCt... I will try to explain, why I do this, please let me know if my reasoning is wrong.
I do not like the 2-ddCt expression, neither the -ddCt, because I prefer to show individual data points. I know a box-plot contains similar information, however, I think showing the entire cluster of data provides more information. Our brain does "statistics" pretty well on its own (if data is plotted correctly). Also, sometimes there is not control group (e.g. comparing two interventions to one-another).
I would argue that the term "relative expression" is right, however it does not imply the relative expression to a reference gene (which has a different unknown "u" as you mentioned) but the relative expression to another group (of the same gene) or even the relative expression of each data point to one-another in the same group. The units are indeed arbitrary which is why I use 2-dCt values and divide every individual value by the mean expression of my control group (for example untreated). I then get individual data points which each represent a "fold change" from the mean control group expression.
Then I plot everything on a log2scale and use antilog number format. You then get even representation of upregulation and downregulation and the mean of each group is the fold difference in expression to the control group (which has a mean of 1). I use the nature protocol publication from Schmittgen and Livak (2008) and basically followed example 4 and 5.
I have two questions:
1) I prefer doing statistics on dCt values because the variance is less and I do not mind showing dCt values to begin with. However many people prefer to see 2^dCt presentation.Would it be valid to run statistics on dCt values and show the data in a graph as I described above? If reviewers complain about log scales (which has happened) would it still be possible to display statistics from dCt values on a linear scale (obviously if mentioned in the methods/ statistics)?
2) What would the y-axis labelign look like in my scenario? I am inclined to call it relative expression (without units) and explain in the legends or methods that individual data points are normalized to a mean control group expression. Or could I just call it "fold change to ctrl"?
Thanks again for the input - I am still trying to find the most convincing way to show my data.
Best, Hannes
1) Absolutely. There is no law saying that a particular visualization must go with a particular statistical analysis. It is only important that the methods section clearly and correctly describes how the data was analyzed.
2) I think that's pretty much a matter of taste. To my humble opinion, "fold change to control" would be a bit more on the point and less umbigious.
Best, Jochen
I think it is asked for 2^dCt, where dCt = Ct[ref]-Ct[goi].
I would just plot the dCt values as you did, but label the Y-axis with the values corresponding to 2^dCt instead (you can even show two Y-axes, one labelled with dCt and the other with the corresponding 2^dCt). The person then finds the desired 2^dCt values, the plot (the distribution and relative positions of the dCt values) is not distorted and you can use standard errors to indicate the precision of the mean dCts.
Yes, you understood correctly.
The signs of the dCt values are irelevant. -3 is higher than -5, just as -27 is higher than -29. Note that the 2^dCt values shown at the axis will all be positive!
All statistical analyses, including the calculation of means, standard deviation, t-test etc. should be done using the dCt values. This is because the dCt values (but not the 2^dCt values!) can be assumed normal distributed, so that all these calculations make sense and results are interpretable in the usual way.
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