# Test(s) For Comparing qPCR results / ΔΔCт values

To be clear, I'm talking about real-time polmerase chain reaction (qPCR), not Principle Component Regression. Though from a company selling qPCR reagents, here is a good walk-through of the procedure.

This is something I've seen lots of arguments over, and generally believe biologist should ask for statistical help more often. Often reliance is given to a computer program like RealTime StatMiner, which in addition to putting the stats on the back-end, could easily be misused by a user.

To quote Livak and Schmittgen (because I can't get it better):

The endpoint of real-time PCR analysis is the threshold cycle or Cт. The Cт is determined from a log–linear plot of the PCT signal versus the cycle number. Thus, Cт is an exponential and not a linear term.

A commonly used method for reporting qPCR results is the ΔΔCт method, the "normalized" cycle point the concentration of a target cross a threshold. The threshold is normally set visually against the data set. The value is normalized by adjusting for the concentration of an abundant known house keeping gene, like GAPDH, and the ΔCт values of naive/vehicle group.

If I understand correctly, taking: $$2^{-ΔΔCт}$$ will give one a liner form representing the factor change in the gene expression. Data sets from such results can often be zero weighted, and transformed further depending on the results.

It would seem most appropriate to apply statistical tests to the liner form, but it's been so heavily modified from the actual read out (normalized and hopefully linearized), I am not certain that is appropriate.

Regression models often would be difficult because different dilutions of your target may be impossible, undetectable, and unknown. Are the standard ANOVA considerations (follows a normal curve, equal population, and possibly different population averages) sufficient for deciding if an ANOVA is appropriate for the liner form?

Often I find when comparing between drug effects over virus strains or animals the liner form is not normally distributed. Would be most appropriate to apply a transformation (log, arc-sine, or what ever seems applicable) to the linear form, or would an earlier point be more appropriate?

For some reason my gut twists every time I take $2^{-ΔΔCт}$ and just use them in an ANOVA/MANOVA.

It is truly a mystery to me why rtPCR data are so often analyzed on the multiplicative scale.

My experience is that those data and their noise variance are much more naturally represented on the $C_t$ scale. That includes analyses performed on preserved samples.

Meaning, that I think it is really best to analyze the data on the $C_t$ scale, not after transforming to the multiplicative scale.

And, I know that the $\Delta\Delta C_t$ method is widely used, but I think that it is pointless other than its prosody and mathematical appearance.

In practice, I recommend the following procedure:

1. Normalize each technical repeat's gene-specific $C_t$ value by subtracting from it the housekeeping gene $C_t$ value.

2. Then, analyze the data using analysis of variance (ANOVA) on the repeat-normalized $C_t$ values, including the control group.

3. If desired, translate effects estimates from the ANOVA onto the multiplicative scale.

Here are some notes on each point:

1. Sometimes there is only one technical repeat per subject but often there are more. If you have multiple technical repeats per subject or per subject/time-point, then you could average them after normalizing each repeat separately.

2. Alternatively, if you have a more complicated design, you may want to use some sort of linear mixed effects model. That would allow you to model hierarchical variation --- for example, variation due to technical repeats.

As a side note, I have seen some terrible approaches to "normalizing" for controls, including randomly selecting individual measurements from the control group to subtract from treatment group measurements.

There is no point in subtracting the mean of the control group from all the values, as that simply amounts to analyzing only the treatment groups, but shifted along the scale.

This recommended procedure has several advantages:

• It normalizes for differential expression appropriately, accounting for repeat-to-repeat and/or subject-to-subject differences in rtPCR analysis.

• It allows use of all data in estimating the noise variance.

• Typically, rtPCR data are well-behaved with respect to the ANOVA assumptions when analyzed on the $C_t$ scale rather than the multiplicative scale.

• All of the tools of linear model theory are available including linear mixed models, means separation methods, weighted least squares, and fairly faithful representation of experimental design.

• It still allows comparisons of treatment groups with each other, but also allows comparisons with the control group.

These comparisons are implicitly performed on a relative scale. It is also possible to calibrate expression.

Resources:

Here are some resources on normalization with respect to housekeeping genes:

For more about the $\Delta\Delta C_t$ methodology:

• Section VII of Applied Biosystems guide Guide to Performing Relative Quantitation of Gene Expression Using Real-Time Quantitative PCR gives all the calculations.

Discussion about the statistical aspects of analyzing rtPCR data:

References:

Vandesompele, J., De Preter, K., Pattyn, F., Poppe, B., Van Roy, N., De Paepe, A., and Speleman, F. (2002). Accurate normalization of real-time quantitative RT-PCR data by geometric averaging of multiple internal control genes. Genome biology, 3(7), research0034.

Yuan, J.S., Reed, A., Chen, F., and Stewart, C. N. (2006). Statistical analysis of real-time PCR data. BMC bioinformatics, 7(1), 85.

• Thank you for the answer. Do you by any chance have a reference to support doing the ANOVA directly on the normalized Ct values? It's not that I disagree, but as it's not my field it's better to have backup. Also, I would never dream of any normalization other than sample internal comparisons. I've never seen real data sets were subtracting the mean of the control group looked like a reasonable idea. Commented Dec 3, 2014 at 14:39
• @Atl LED Only my experience. It is pretty clear when you look at the data, though. Someone has to have written a polemic at this point! My guess about why it works is that the analytic machinery has been optimized to have linear response on the $C_t$ scale. Commented Dec 3, 2014 at 15:30
• The Yuan et al paper was exactly what I was looking for. Thanks. Commented Dec 4, 2014 at 14:24