I would like to know if anyone knows an easy way for calculating relative quantities under this delta Ct approach, without the use of qBase plus software.
Given the efficiencies of target and reference sequence are similar and known (a)), the normalized quantities are
a^(-ct[target]) / a^(-ct[ref]) = a^(ct[ref]-ct[target])
the relative quantities:
q[treated] = a^(ct[treated,ref]-ct[treated,target]) ("quanity of the target sequence in the treated sample, normalized to its reference")
q[control] = a^(ct[control,ref]-ct[control,target]) ("quanity of the target sequence in the control sample, normalized to its reference")
fc = q[treated]/q[control]
It is all easier if we stay on the log scale (ideally to base a). Simple arithmetics show that
log(fc) = (ct[treated,ref]-ct[treated,target]) - (ct[control,ref]-ct[control,target])
log(fc) = deltact[treated] - deltact[control]
Since deltact's have to be calculated per sample, one can only average the deltact's. In the above formula, "deltact" could replaced by the mean deltact values for the treated and the control samples. The "problem" then is that the log(fc) (or delta-delta-ct value) is the difference of just two values and there is no obvious variance estimate. However, the variance of the log(fc) can be determined by error-propagation vfrom the variances of the deltact's. If s[treated] and s[control] are the standard deviations (or standard errors or CI widths) of the two groups, then
s[log(fc)] = sqrt(s[treated]² + s[control]²)
with n[treated]+n[control]-2 degrees of freedom is the standard deviation (or standard error of CI width) of the log(fc).
From there one might un-log the limits (log(fc) +/- s[log(fc)]) to draw error error bars or give the confidence limits in the "fold-change" scale (rarther than in the "log fold-change sale")
If the efficiencies for target (a) and reference (b) are not similar, we have the normalized quantities
q[treated]: a^(-ct[treated, target]) / b^(-ct[treated,ref])
q[control]: a^(-ct[control, target]) / b^(-ct[control,ref])
what can not be further simplified. The distribution of q is not symmetric, so applying stanard t-tests is not appropriate. However, the distribution of the logs of these values is not much different to the normal distribution. Therefore, all statistics (summaries, tests) should be done on the logs.
What I have not figured out yet is how the uncertainty about the efficiencies a and b can be considered,
Given the efficiencies of target and reference sequence are similar and known (a)), the normalized quantities are
a^(-ct[target]) / a^(-ct[ref]) = a^(ct[ref]-ct[target])
the relative quantities:
q[treated] = a^(ct[treated,ref]-ct[treated,target]) ("quanity of the target sequence in the treated sample, normalized to its reference")
q[control] = a^(ct[control,ref]-ct[control,target]) ("quanity of the target sequence in the control sample, normalized to its reference")
fc = q[treated]/q[control]
It is all easier if we stay on the log scale (ideally to base a). Simple arithmetics show that
log(fc) = (ct[treated,ref]-ct[treated,target]) - (ct[control,ref]-ct[control,target])
log(fc) = deltact[treated] - deltact[control]
Since deltact's have to be calculated per sample, one can only average the deltact's. In the above formula, "deltact" could replaced by the mean deltact values for the treated and the control samples. The "problem" then is that the log(fc) (or delta-delta-ct value) is the difference of just two values and there is no obvious variance estimate. However, the variance of the log(fc) can be determined by error-propagation vfrom the variances of the deltact's. If s[treated] and s[control] are the standard deviations (or standard errors or CI widths) of the two groups, then
s[log(fc)] = sqrt(s[treated]² + s[control]²)
with n[treated]+n[control]-2 degrees of freedom is the standard deviation (or standard error of CI width) of the log(fc).
From there one might un-log the limits (log(fc) +/- s[log(fc)]) to draw error error bars or give the confidence limits in the "fold-change" scale (rarther than in the "log fold-change sale")
If the efficiencies for target (a) and reference (b) are not similar, we have the normalized quantities
q[treated]: a^(-ct[treated, target]) / b^(-ct[treated,ref])
q[control]: a^(-ct[control, target]) / b^(-ct[control,ref])
what can not be further simplified. The distribution of q is not symmetric, so applying stanard t-tests is not appropriate. However, the distribution of the logs of these values is not much different to the normal distribution. Therefore, all statistics (summaries, tests) should be done on the logs.
What I have not figured out yet is how the uncertainty about the efficiencies a and b can be considered,
Hi there,
@Jochen Wilhelm
Since data obtained from miRNA expression profiling experiments result in CT-values as well and one want to compare 2 different disease states for example diseased vs. healthy - there is no means to justify which positive sample is compared with which negative sample because there are different patient samples. So the deltaCt method should be appropriate for analysis (Livak and Schmittgen, 2008).
And as you indicated in your answer, deltact's have to be calculated per sample, so one can average the deltact's and the "deltact" could replaced by the mean deltact values for the treated and the control samples.
But if you can't detect any miRNAs in several samples within a group (e.g. healthy), averaging deltact's wouldn't be appropriate since it's not representative and the expression-ratio could be falsified in my opinion.
How can i overcome this problem?
Non-detection means non-quantification. You cannot quantify something that you don't detect. If you have frequent samples where the target is not detected, then you need a different approach. One way is to use the Poisson distribution and the numbers of positive and negative samples. However, to get acceptable precision, you will need quite a lot of samples. This is typical for quantifying rare things...
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