# Rationale about the difference between logarithms quocient and ratio? Fold-change?

Following this post I was wondering how to plot or even communicate gene expression using relative quantification, meaning through transformation of cycle threshold values. In my particular experiment, a longitudinal study made up by 3 groups with 2 measurements I performed the following calculations:

dCt_post = Ct_target_post - Ct_ref_post

dCt_pre = Ct_target_pre - Ct_ref_pre

ddCt = dCt_post - dCt_pre

To express the variation caused by treatment there are several possibilites reflected in bibliography:

• log2(dCt_post/dCt_pre)
• ratio: dCt_post/dCt_pre
• There is the possibility of adding a calibrator or even a pool of untreated sample in the different batches which will result in another ddCt value

dCt_test = Ct_target,test - Ctref_test dCt_calibrator = Ct_target,calib -Ct_ref, calib ddCt = dCt_test - dCt_calibrator. The non explicit part here is that Ct_target,test in longitudinal studies is made up by a pre- and post- values. This is just a note, not pertinent to the topic

I am a little bit confused between the ratio and the log2 of the dCt values. I have read something about variances in the post @EdM, but crudely speaking I dont know which contribution make each one statistically speaking. Is there one approach more appropiate in this context, from a mathematical aspect, coming from an inverse log space (sigmoid function) as RT-qPCR? Interaction and main effects are calculated with linear mixed-effects model

• @EdM here we are Mar 24, 2023 at 19:02

The quantitative real-time polymerase chain reaction (qPCR), the basis for this question, subjects a sample to a series of reaction cycles. Ideally, each cycle doubles the amount of a specific DNA sequence. A fluorescent dye reports the (amplified) amount after each cycle.

If the original amount of the DNA of interest is $$x$$, then the amount after $$C$$ cycles is ideally $$2^Cx$$. This makes it natural to work in the $$\log_2$$ scale, transforming that amplified amount to $$\log_2 x +C$$. This question is whether it's best to work in the log scale related to the number of cycles, or in some other scale.

The short answer: it's best to work in the log scale of $$x$$, using linear combinations of the associated cycle numbers to correct for differences in sample size or differences in pre-processing (e.g., reverse transcription of RNA to DNA). That's because linear modeling works best when error magnitudes are independent of the magnitudes of the observations, and in qPCR that assumption holds best in the scale of cycle numbers. See, for example, this answer.

Details

In practice, at low cycle numbers the specific fluorescence can't be distinguished from background fluorescence or instrumental noise, and at high cycles the consumption of reagents limits the fluorescence signal. That provides a sigmoid curve of fluorescence versus cycle number, as noted in the question.

Quantitation thus starts with the number of cycles that it takes for the fluorescence generated from a sample to pass a pre-defined threshold value that's higher than background but well below the limiting value. That number of cycles is called the Ct or Cq for the sample.

If you have two sets of qPCR reaction cycles with starting DNA amounts $$x_1$$ and $$x_2$$, then given their individual Ct values you can (ideally) compare their DNA amounts as follows:

$$\log_2 x_2 + \text{Ct}_2 = \log_2 x_1 + \text{Ct}_1$$ $$\log_2 \frac{x_2}{x_1} = \text{Ct}_1 -\text{Ct}_2 =\Delta \text{Ct} .$$

That can be used to correct for differences in sample amounts or in pre-processing by reverse transcription from RNA to DNA. Then $$x_2$$ is the DNA of specific interest in an experiment and $$x_1$$ is a reference DNA unaffected by the experimental manipulations, each determined from the same biological sample. (The dCt_post and dCt_pre in this question are written in the opposite direction, with the question's dCt the negative of $$\Delta \text{Ct}$$ as written above.)

The variance of a $$\Delta \text{Ct}$$ estimate, based on the formula for the variance of a weighted sum of correlated variables, is:

$$\text{Var} (\Delta \text{Ct}) = \text{Var} (\text{Ct}_1) + \text{Var} (\text{Ct}_2) - \text{Cov} (\text{Ct}_1, \text{Ct}_2),$$

where $$\text{Cov}$$ is the covariance (representing the shared errors between $$x_2$$ and $$x_1$$ due to their being analyzed in the same original sample). As noted above, these variances are typically constant over wide ranges of Ct values, with a normal error distribution often a reasonable approximation in the Ct scale. Thus linear modeling is best done via linear combinations of Ct values, as used to calculate $$\Delta \text{Ct}$$ values.

The proposed ratios of $$\Delta \text{Ct}$$ values, or the logs of such ratios, don't make a lot sense. They take you out of the Ct scale in which errors are well behaved, and they take you even farther away from the original DNA values $$x$$.

I have seen calculations based on things like $$2^{\Delta \text{Ct}}$$ to transform back to the $$x$$ scale of initial DNA amounts. But that also takes you out of the scale in which errors are well behaved. You might consider that type of transformation at the end of analysis, to express values and confidence intervals in terms of DNA amounts. But linear modeling, associated intermediate calculations, and statistical tests should be on the Ct scale where the underlying assumptions are best met.

Standard curves can be used to correct for inefficiency in PCR, less than the ideal doubling of DNA in each cycle. Standard curves can give estimates for $$x_1$$ and $$x_2$$ in original DNA amounts. But as those are still based on Ct values, it's best to work with those amounts in the logarithmic scales most directly associated with their Ct` values.