Dear Arnoldo,

Q1) Why do you get RQs? You measure ct-values. If you stick to their scale a standard analysis is straight-foreward. Technical replicates can simply be averaged, leaving you some values per group (one value per biol.repl.), and these can be analyzed using any appropriate linear model. There is actally no need to think about SDs, and the linear models will internally take care of the calculation of the standard errors for the effect estimates (like, for instance, the expected difference between groups).

Q2) If you stick to ct values there is no need for a log transformation. And anyway, the normal distribution is an assumption (of an abstract mathematical property). You will never ever find that any real frequency distribution is identical to the normal distribution (so this assumption is always violated, anyway!). The relevant question is if the assumption is reasonable for the purpose. Actually, this should/must be decided on theoretical grounds. Data may give you information that your assumptions may not be reasonable, but this is hard to see when only little data is available. Generally, if you doubt that the assumptions are reasonable the only other choice you have to the information you wanted is bootstrapping, but this usually starts working well for sample sizes where the central limit theorem would also make strong violations of the assumptions irrelevant. 

Q3) Most post-hoc tests have nothing to do with ANOVA (or KW). You do such tests for two reasons; i) using a pooled variance estimate for the calculation of the standard errors (-> more robust) and ii) control the family-wise type-I error rate. For ii) it makes sense only when the experiment was planned accordingly (specification of the two alternative hypotheses, specification of alpha and beta, using the determined sample size). This would require to know a priori what comparisons you want to make and how you want to make your decisions between the specified alternatives. Since you are asking "a posteriori" (you have the data, now how to do the tests) all this does not make any sense. But given you want to plan a new experiment to test expected differences and you specified the effect size, alpha, beta, and the sample size, then you will have to find what the actual "family of tests" is that you like to preform. If it is "all possible pair-wise comparisons" then Tukey's HSD is a good choice, if you want to compare multiple treatment groups against one control group, Dunnett's MCP is an option. However, often in such experiments the tests do not constitute a family for wich the control of a family-wise error-rate would make any sense. The tests are sometimes sequential, or the tests are actually about a co-operative regulation of the investigated genes. I don't see how it would make sense to talk about reasonable error-rates in these cases. In such cases, significance tests might be more appropriate than hypothesis tests.

Q2+3) If you check your assumptions on a given set of data and decide for the test afterwards, the test on the same data will not keep the desired error-rates. You can use data (from a preliminary experiment, so to say) to check if the data may not too severely violate the assumption. Then you plan the experiment based on the appropriate test* and collect new data!

*Interestingly, there is little choice. You need to specify substantial hypotheses (or an substantial effect size), and this will determine what kind of statistical hypothesis you will test. Usually, the effect is an expected difference (ratios can be transformed to differences of the logs), and there is only the t-test testing this. There is no alternative (except bootstrapping, what comes with its own limitations). Rank-based tests test different hypotheses and might therefore require different specifications of alpha and beta to lead to reasonable decisions.

The main question is if we should do statistics on "egxpression values" or on "log expression values". It is well-known that expression values have an approximate log-normal distribution, so the log expression values have an approximate normal distribution. Therefore most standard analyses that assume normal distribution will neatly work with the log expression values.

Ct values are such log expression values. The difference betwenn Ct calues (dCt) are differences of such log values, what has the meaning of a log ratio. If dCt is calculated as Ct[ref] - Ct[goi], then the resulting value is the log of the expression of the goi, normalized by the expression of the ref (ref serves as a loading control). Hence, dCt values are "log normalized expression values" (normalized to a loading control).

Therefore, all the analyses should be done using these dCt values. A t-test, for instance, analyses the difference between dCt values (of different groups, like "treated" and "controls"). Such a difference between dCt values (= ddCt) is the log ratio of normalized expressions what is nothing else but a log fold-change.

Only to complement the commentary of Mr. Jochen Wilhelm, some time ago I found a good paper about this topic. It´s called: 

Quantitative PCR data analysis – unlocking the secret to successful results

2011

PCR Troubleshooting and Optimization: The Essential Guide, Publisher: Caister Academic Press, Editors: Suzanne Kennedy and Nick Oswald - MO BIO Laboratories, Publication date: January 2011, ISBN: 978-1-904455-72-1

Hellemans J, Vandesompele J

 Mr. Jochen Wilhelm, i have some questions for  you. I would appreciate your help please.

1Q. I'm working with the qBase method, but in general when you´r working in qrt-pcr assay, you should work with technical replicates (e.g 3 well of a same sample in a plate  ) and biological replicates (e.g 3 different samples of a same treatment). But in the end, you have your RQs values (or CNRQs,) for each sample with a standar desviation (SD), but when  performing a stastical analysis, Do you have to despise SD?

2Q. After of Log transformation, you're still violating the assumptions of normality or homoscedasticity, what do you do?. Obiously you can perform a non-parametric but, are there other ways to deal with that? (without repeat the assay).

3Q. What post-hoc test is recommended? after a ANOVA or Krus.W, for qrt-PCR.

Thanks in advance.

Dear Arnoldo,

Q1) Why do you get RQs? You measure ct-values. If you stick to their scale a standard analysis is straight-foreward. Technical replicates can simply be averaged, leaving you some values per group (one value per biol.repl.), and these can be analyzed using any appropriate linear model. There is actally no need to think about SDs, and the linear models will internally take care of the calculation of the standard errors for the effect estimates (like, for instance, the expected difference between groups).

Q2) If you stick to ct values there is no need for a log transformation. And anyway, the normal distribution is an assumption (of an abstract mathematical property). You will never ever find that any real frequency distribution is identical to the normal distribution (so this assumption is always violated, anyway!). The relevant question is if the assumption is reasonable for the purpose. Actually, this should/must be decided on theoretical grounds. Data may give you information that your assumptions may not be reasonable, but this is hard to see when only little data is available. Generally, if you doubt that the assumptions are reasonable the only other choice you have to the information you wanted is bootstrapping, but this usually starts working well for sample sizes where the central limit theorem would also make strong violations of the assumptions irrelevant. 

Q3) Most post-hoc tests have nothing to do with ANOVA (or KW). You do such tests for two reasons; i) using a pooled variance estimate for the calculation of the standard errors (-> more robust) and ii) control the family-wise type-I error rate. For ii) it makes sense only when the experiment was planned accordingly (specification of the two alternative hypotheses, specification of alpha and beta, using the determined sample size). This would require to know a priori what comparisons you want to make and how you want to make your decisions between the specified alternatives. Since you are asking "a posteriori" (you have the data, now how to do the tests) all this does not make any sense. But given you want to plan a new experiment to test expected differences and you specified the effect size, alpha, beta, and the sample size, then you will have to find what the actual "family of tests" is that you like to preform. If it is "all possible pair-wise comparisons" then Tukey's HSD is a good choice, if you want to compare multiple treatment groups against one control group, Dunnett's MCP is an option. However, often in such experiments the tests do not constitute a family for wich the control of a family-wise error-rate would make any sense. The tests are sometimes sequential, or the tests are actually about a co-operative regulation of the investigated genes. I don't see how it would make sense to talk about reasonable error-rates in these cases. In such cases, significance tests might be more appropriate than hypothesis tests.

Q2+3) If you check your assumptions on a given set of data and decide for the test afterwards, the test on the same data will not keep the desired error-rates. You can use data (from a preliminary experiment, so to say) to check if the data may not too severely violate the assumption. Then you plan the experiment based on the appropriate test* and collect new data!

*Interestingly, there is little choice. You need to specify substantial hypotheses (or an substantial effect size), and this will determine what kind of statistical hypothesis you will test. Usually, the effect is an expected difference (ratios can be transformed to differences of the logs), and there is only the t-test testing this. There is no alternative (except bootstrapping, what comes with its own limitations). Rank-based tests test different hypotheses and might therefore require different specifications of alpha and beta to lead to reasonable decisions.

Mr. Jochen Wilhelm 

Thank you very much for all your answers and comments.

I have the concepts much clearer now. You have helped me so much. 

My best regards.

Thanks, you are wellcome :)

Dr Wilhelm,

I calculated my dCts according to the order you suggested (first the reference, then the GOI)

dCt = Ct[ref] - Ct[goi]

I ended up with negative values for my "healthy" group, and positive values for my "sick" group. 

Now, when planning to calculate my ddCts (or log-fold changes)

ddCt = dCt[sick] - dCt[ctl],

I wonder:

1) should calculate the arithmetic or geometric mean for my "healthy samples " and "sick samples"-dCts?

2) If so, how do I handle the standard deviation or error of these means

3) could i run a t-test between the healthy and sick dCts?

4) should i lose the "minus" in the 2^(minus)ddCt calculation, since i substracted the reference to the goi in the dCt calculations?

PS: Thank you SO MUCH for the explanations already given in the forums. 

"I ended up with negative values for my "healthy" group, and positive values for my "sick" group."

This already tells you that the expression in the "sick" group is larger than in the "healthy" group (the sign does not matter; it's just that the dCt values in "sick" are larger than in "healthy"). If you calculate the ddCt (as you did: "sick" vs. "healthy") the result in your case is positive, again telling you that the expression in the "sick" is larger than in "health".

1) No. It is simpler: it's the arithmeic mean!

2) It's the usual standard statistics. You can calculate the means (as in 1), the SDs, SEs and CIs directly from the dCt values. There is nothing special. dCt values can be assumed normal distributed.

3) Yes.

4) Yes. The 2^(minus)ddCt  was used by Livak and Schmittgen because they calculated the dCt values "the wrong way", and this is the place where they had to correct this mistake to get the correct fold-change values.

Thank you so much for your quick reply.

Sorry to bother you again, but something you said (about the expression in the sick group being lower than the healthy) confused me.

If my values were:

dCT [sick]= 3 , dCT [healthy]= -2

Then calculating ddCT

ddCt [sick-healthy]= 3-(-2)= 5

Wouldn't this mean the expression in the sick group is higher than in the healthy group?

Oh, yes, sorry. I corrected my post above! I confused the groups. My mistake!

You are absolutely correct.

My apologies, again.