There is a published method for calculating statistics for ratios: Bootstratio. Perhaps it helps you (https://www.researchgate.net/publication/221770913_BootstRatio_A_web-based_statistical_analysis_of_fold-change_in_qPCR_and_RT-qPCR_data_using_resampling_methods)

Article BootstRatio: A web-based statistical analysis of fold-change...

Please try micro array based software free fro bioconducter for fold change analysis

(I assume you are talking about gene or protein expression)

Just seconding Thomas' second paragraph:

1) The FC is not normal distributed, but pretty well log-normal distributed. Therefore, the log-transformed values (logFC) can be used with "standard" techniques based on the normal distribution.

2) You provide FC for two groups (control and test) what is non-sensical. The control is the control because it is used as the reference to what the FC of the test is calculated. There is no information in the FC of the control. The information about the expression in the control group is already included in the FC of the test group. Thus, testing FC is testing *only* one group, not two (the "change" in FC implies that this is already something relative to a reference!). Hence, a ONE-SAMPLE t-test on the logFC is ok. The base of the log is not important, but 2 has the advantage that the result can directly be interpreted as the "number of doublings".

Slightly opposing to Thomas' first paragraph (if I got it correctly):

I don't think that the expression values itself are normal distributed. To my experience, they already follow a log-normal, what can also be well explained by the mechanisms of gene regulation that is based on many multiplicatory working signals. Thus, one might test differences between the log expression values between test and control directly (TWO-SAMPLE t-test). This (using the logs) is equivalent to analyzing fold-changes (because logs transforms products to sums, so the analysis of the difference between logs is equivalent to analyzing fold-changes between the original data). If there is a considerable covariance between both groups, one can account for this by introducing additional exploratory factors to the model. The simplest case is to account of a pair-wise correlation, what can be done by a PAIRED-SAMPLE t-test (what again is equivalent to the one-sample test on the pair-wise differences, actually what I suggested above).

For small data sets, both non-parametric and bootstap methods are not working well.

Thomas, why should the variation of FCs depend on the method with what they are measured? Perturbations of phosphorylations, transcription factors, enhancer activities and so on all have a multiplicative effect on the gene transcription / mRNA concentration, further perturbations of mRNA stability, translation, and protein stability again have a multiplicative effect on the protein concentration. It should not matter much if the gene or protein concentration is measured with blots, arrays, qrtPCR, OD, ELISAs or whatever. Please correct me when I am wrong here.

Ok, so I do not get you... what values are derived from the FCs? And what is the relation to "testing the FC" then? Do you mean these "normalized FCs"? If so, then what is this normalization? I do not understand...Could you show an example?

Thank you all for your response. Here I would like to mention that above data is from luciferase experiment. But I have similar type of data from qRT experiment.

I have two more queries from your suggestion:

a) Jochen mentioned that "The FC is not normal distributed, but pretty well log-normal distributed". I did not get this completely. Why is it so?

b) Can we apply t-test on small sample size(as in my case it is 3)? What if I have only two test results? can I still use any statistical test?

Please pardon me if I sound very stupid.

Hi All,

Just looking at this data makes difficult to get some statistical inference since n=3 is really small. When sample size is really small such as n

dear Ramon, dear All,

I am also very often working with simple experiments, where I look for fold changes of my gene of interest in response to some treatments etc. Usually we repeat such experiments 3-6 times, depending how consistent are our observations. I have to questions, maybe using the example from Gunjan above:

1) Control = 1 , 1, 1

2) Test = 2.2, 2.4, 2.6 (fold change)

Could I then put simply the fold changes into the Bootstratio software, as samples already normalized? I tried, and what I get are only indications if I reach certain p values thresholds, but I cannot see the specific p value that I actually obtained. Or is it hidden somewhere?

Second question would be if alternatively I could transform my FC values into log2FC, and test with a one-sample t-test if the mean is significantly different from 0? I understood from above posts that this should be a correct approach. But then I have a very pragmatic question: could I still show my data in e.g. manuscript as a histogram, where my control samples are always set to 1 (without any error bars) and my experimental condition is shown as a FC value (mean of 2.2, 2.4 and 2.6), with let's say SD or SEM as an error bar. (although to test the significance I transformed my values into log2). I ask about this as showing simply FC is much more intuitive for the potential reader than e.g. showing the log2FC values.

many thanks for some additional feedback.

I can't help you with your 1st question since I don't know this software.

To your 2nd Question (use logFC for a t-test): Yes. You can do this.

To your "pragmatic" question:

There is no rule that the data must be shown axactly the way it was used for a test. There are many circumstances where this is not even possible. So, yes, you can do this. (Note: the plot you mean is called "bar plot" and not "histogram"! A histogram has another meaning and is a completely different type of plot).

I am only concerned with showing SD or SEM as error bars in such diagrams (sowing FC). They simply meake no sence there, and thus I find them clearly inappropriate. If error bars should be placed, then the only sensible choice is to use confidence intervals. They can be easily calculated on the log-scale and the limits can then be transformed to the scale of the fold-changes (by exponentiating).

Thank you for the answers.

Dr. Joshen is there any reference article where one can show that logFC can be used in a t-test to depict the significance of fold change?

Yes, you can take all papers using the ddct-method (real-time PCR) to investigate gene regulation, you can further use almost all papers about (DNA-)microarrays.

I attached two papers that may be of interest: one gives a theoretical background why gene expression leads to log-normal distributions of mRNA or proteins, the second is about the log-transformation and its use in biomedical science in general.

Another paper is

http://www.uh.edu/engines/generalizedgamma.pdf

but this is a bit difficult and you will need to understand that the log-normal ditribution is a special case of the generalized gamma.

Thank you once again Dr. Jochen. I am looking for a paper which has shown to incorporate one-sample t-test to estimate the significance of fold change?

Hi everyone,

I have read all of the thread but one point I want to make and will definitely seek guidance from you all.

Like as mentioned previously that one can do the t-test for the log-transformed fold change data from real time PCR... But I wonder that even by log transforming, the lost information in control (variability) is not coming back and statistical testing will not mean much...

What about this, if you have 3 experimental replicates (3 controls and 3 tests). After internally normalizing the gene of interest (gene X with internal control gene), you will get dCt value. Use this dCt value of all repeats of control and treatment to perform statistical testing...

I am doing real-time PCR quite alot and having same trouble. By thinking much on this and working around, I have come on above conclusion.

I shall request you all to please let me know, if it will work or not.

Regards

Dear Dr. Jochen Wilhelm,

Do we need log transformation of control group FC to perform one sample t test?

Dear Jochen,

Do you know any good reference for using one sample t-test for estimating statistical significance of FC (log transformed, or course)? I really liked this approach, in all cases where your control expression levels are variable between biological replicates and therefore doing direct statistics from Ct values is not working well..

Thanks!

Kasia