qPCR ΔΔCt Fold Change Calculator
qPCR ΔΔCt Fold Change Calculator
Relative expression by the Livak method: two ΔCt values, one ΔΔCt, and 2 raised to its negative. The arithmetic is trivial; the assumption underneath it is that both assays amplify at 100% efficiency.
qPCR ΔΔCt Fold Change
Four Ct values → fold changeTest sample: target Ct 22.4, reference Ct 18.2. Calibrator: target Ct 25.1, reference Ct 18.0
The Livak (2⁻ΔΔCt) method
ΔCt (calibrator) = Ct target − Ct reference, in the control
ΔΔCt = ΔCt (sample) − ΔCt (calibrator)
fold change = 2^(−ΔΔCt)
- ΔCt
- normalisation. Subtracting the reference gene’s Ct from the target’s removes differences in how much template went into each well, which is the whole purpose of a housekeeping gene
- ΔΔCt
- calibration. Subtracting the calibrator’s ΔCt expresses the sample relative to a chosen baseline — untreated, time zero, normal tissue — so the answer is always a ratio and never an absolute quantity
- the 2
- THE ASSUMPTION OF THE WHOLE METHOD: that one PCR cycle doubles the product, for BOTH the target and the reference. That is exactly true only at 100% amplification efficiency. If either assay is at 90%, the base is 1.9, not 2, and the error compounds with every cycle of ΔΔCt
- the minus sign
- a higher Ct means less template, so the sign is inverted to make an increase in expression come out above 1
- if efficiencies differ
- use the Pfaffl method, which takes each assay’s measured efficiency as an input instead of assuming both are 2. Measure efficiency from a standard curve before deciding which method applies
Worked example
Test sample: target Ct 22.4, reference Ct 18.2. Calibrator: target Ct 25.1, reference Ct 18.0
ΔCt (sample) = 22.4 − 18.2 = 4.2
ΔCt (calibrator) = 25.1 − 18.0 = 7.1
ΔΔCt = 4.2 − 7.1 = −2.9
Fold change = 2^(2.9) = 7.46 — expression is about 7.5 times higher in the test sample
Sanity check on the normalisation: the reference gene came up at 18.2 and 18.0, a difference of 0.2 cycles, so the two wells received closely similar amounts of template and the normalisation had little work to do
Now suppose the target assay runs at 90% efficiency rather than 100%. The true base is 1.9 rather than 2, and 1.9^2.9 is about 6.4 — a 14% overstatement from a discrepancy most people would not notice on a standard curve
ΔΔCt and the fold change it corresponds to
| ΔΔCt (cycles) | Fold change | Read as |
|---|---|---|
| −3.32 | 9.99 | Very nearly ten-fold — exactly ten-fold at −3.3219 |
| −2.00 | 4.0 | Four-fold increase |
| −1.00 | 2.0 | Doubled |
| 0.00 | 1.0 | No change |
| +1.00 | 0.5 | Halved — report as a two-fold decrease |
| +2.00 | 0.25 | Four-fold decrease |
| +3.32 | 0.10 | Very nearly a ten-fold decrease |
Livak or Pfaffl?
| Situation | Method |
|---|---|
| Target and reference efficiencies both close to 100% and within about 5% of each other | Livak (2⁻ΔΔCt) — this page |
| Efficiencies differ meaningfully between target and reference | Pfaffl, which takes each measured efficiency as an input |
| Efficiency unknown | Measure it from a standard curve first. Neither method is safe without that figure |
| Absolute quantity required rather than a ratio | Neither. Use a standard curve of known copy number, or digital PCR |
The assumption hidden inside the number 2
The Livak method answers a narrow question well: how much more, or less, of a transcript is present in one sample than in another, once differences in input have been normalised away. It does it in two subtractions and one exponentiation. Subtract the reference gene’s Ct from the target’s in each sample to normalise, subtract the calibrator’s normalised value from the test sample’s to calibrate, and raise 2 to the negative of the result. Because a higher Ct means less template, the sign is inverted so that an increase in expression appears as a number above 1.
Everything rests on that 2. It says that one cycle of PCR doubles the amount of product, for the target assay and for the reference assay alike, which is true only at 100% amplification efficiency. Real assays are not all at 100%. An assay at 90% efficiency multiplies by 1.9 each cycle, not 2, and because the term is exponentiated the discrepancy compounds: over a ΔΔCt of 2.9 cycles, using 2 where 1.9 is correct overstates the fold change by about 14%. Over larger ΔΔCt values the error grows further, and it does not average out across replicates because it is systematic.
So efficiency has to be measured, not assumed, and measured for both assays from a standard curve before any relative quantification is reported. Where both come out near 100% and close to each other, Livak is appropriate and its simplicity is a genuine advantage. Where they do not, the Pfaffl method takes each measured efficiency as an explicit input and gives the right answer with the assays you actually have. The MIQE guidelines ask for the efficiency figures to be published for exactly this reason: without them a fold change cannot be evaluated by anyone else.
Two points about reporting. First, a result below 1 is a decrease, and convention is to state it as a reciprocal with the direction named — ‘reduced four-fold’ rather than a bare 0.25 — because fractions below 1 compress the apparent size of a change while values above 1 do not. Second, statistics belong on the ΔCt values and not on the fold changes. Ct values are logarithmic, so ΔCt values are approximately normally distributed while fold changes are not; testing the exponentiated figures inflates the apparent significance of large increases and buries decreases. Report the fold change, test the ΔCt.
Frequently asked questions
How do you calculate fold change from Ct values?
Subtract the reference gene Ct from the target gene Ct in the test sample and again in the calibrator, subtract the second result from the first to get ΔΔCt, then raise 2 to the negative of that. A ΔΔCt of −2.9 gives 2^2.9, about a 7.5-fold increase.
What does the 2 in 2^-ΔΔCt assume?
That every PCR cycle exactly doubles the product, for both the target and the reference assay — that is, that both amplify at 100% efficiency. If an assay is at 90%, its true base is 1.9, and because the term is exponentiated the error compounds with the size of the ΔΔCt.
When should I use the Pfaffl method instead?
Whenever the target and reference assays do not amplify at similar, near-100% efficiencies. Pfaffl takes each measured efficiency as an input rather than assuming 2, so it gives the correct ratio with real assays. Measure both efficiencies from a standard curve before choosing.
How should a fold change below 1 be reported?
As a decrease, with the direction stated and usually as the reciprocal — a value of 0.25 is a four-fold decrease. Bare fractions below 1 make a change look smaller than the equivalent increase would, which is a well-known source of misreading.
Should statistics be done on fold changes or on ΔCt values?
On the ΔCt values. Ct is a logarithmic quantity, so ΔCt values are roughly normally distributed and suitable for a t-test or ANOVA, while fold changes are not. Run the test on ΔCt and convert the result for presentation.
Related calculators
References
- Livak KJ, Schmittgen TD. Analysis of relative gene expression data using real-time quantitative PCR and the 2(−ΔΔC(T)) method. Methods. 2001;25(4):402–408.
- Pfaffl MW. A new mathematical model for relative quantification in real-time RT-PCR. Nucleic Acids Res. 2001;29(9):e45.
- Bustin SA, Benes V, Garson JA, et al. The MIQE guidelines: minimum information for publication of quantitative real-time PCR experiments. Clin Chem. 2009;55(4):611–622.
Medical Disclaimer: The tools and content provided here are for educational and reference purposes only. They are not intended to substitute for professional medical advice, diagnosis, or treatment. Clinical decisions should always be based on the comprehensive assessment of a qualified healthcare professional.