I have heard that it is better to show PCR gene expression results via log 2 fold change rather than just fold change, but how does one calculate the confidence intervals (error bars) for that?
For log 2 fold change is it just delta delta Ct +/- 1.96*SD?
And the standard deviation for delta delta Ct = standard deviation for delta Ct??
For the regular fold change is the confidence interval just 2^-(ddCt +/- sd)? Would that mean it is a 68% confidence interval?
I agree that results of gene(/protein) expression experiments should be shown as log fold changes. This is particularly simple for rtqPCR, where the measured variable is "dCt", which is already proportional to the logarithm of the initial concentration of the analyte (the nucleic acid molecules with the amplicon sequence). There are a couple of statistical reasons (approximate log-normal distribution + typically large dynamic ranges of concentrations of biological/biochemical/chemical compounds), but, particularly for the presentation of the results, there are good "psychological" reasons to use log fold-changes, e.g. the fact that the sign directly indicates the direction of the change, that equal relative concentration differences are represented by equal differences on the log scale, what typically maps to the biological relevance of the change, and that the log changes are symmetric (if you compare male vs female you get the same numeric result as if you compare female vs male - only with the opposite sign; nature does not care which group you chose as "control" or "reference"!).
Using dCt values, you calculate the mean and its confidence interval following the standard procedure shown in any basic stats course. Nothing special, nothing new. I'd let a stats software do the calculations, because you'd need the pooled standard error, which is slightly complicated to calculate. If you have paired samples, the calculation is a bit simpler, but using a software is still simpler and possibly less error prone.
The problem becomes a little larger when you have more than two groups, like a couple of different groups, or a time-course, or more than one experimental factor (like genotype and treatment etc.). Data (that means: dCt values) from such experiments is best analyzed using appropriate linear models (regression, ANOVA, ANCOVA), and the stats software will provide the estimates and the corresponding confidence intervals of the relevant model coefficients.
The formula you gave for the confidence interval of the "regular fold change" (as you called it) is not correct. The confidence interval is based in the standard error (SE) of the estimate (ddCt), not the standard deviation (SD). In fact, in an unpaired design there is no SD of the ddCt (you may calculate something like the SD of a ddCt only for paired samples). The stats software will give you a SE, and the formula for a 68% confidence interval is estimate ± SE, for a 95% confidence interval it is estimate ± 2*SE.
Jochen Wilhelm Thanks for your detailed answer.
Is it possible for the error bars of the fold change to involve standard error of the mean (S.E.M.)? If so, how does one calculate it? Is it the standard error divided by square root of the number of technical replicates? And then can we input that value into the confidence interval (± 2*SEM)?
I know you just explained its better to use log 2 fold change, but supervisor insists.
If you have the SE of the mean logFC "m", you can calculate the 95%CI of the logFC as m ± SE, so the lower limit is lwr = m-2SE and the upper limit is upr = m+2SE. You can back-transform the mean and these limits to the scale of the FC with 2^m, 2^(m-SE) and 2^(m+SE). Note that the mean FC is not in the middle of the CI (it is not "symmetric").
Suggest your supervisor to improve his/her data-analytic literacy and statistical skills ;)
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