What's wrong with Bonferroni adjustments? I read the following paper: Perneger (1998) What's wrong with Bonferroni adjustments.
The author summarized by  saying that Bonferroni adjustment have, at best, limited applications in biomedical research and should not be used when assessing evidence about specific hypothesis:

Summary points:
  
  
*
  
*Adjusting statistical significance for the number of tests that have been performed on study data—the Bonferroni method—creates more problems than it solves
  
*The Bonferroni method is concerned with the general null hypothesis (that all null hypotheses are true simultaneously), which is rarely of interest or use to researchers
  
*The main weakness is that the interpretation of a finding depends on the number of other tests performed
  
*The likelihood of type II errors is also increased, so that truly important differences are deemed non-significant
  
*Simply describing what tests of significance have been performed, and why, is generally the best way of dealing with multiple comparisons
  


I have the following data set and I want to do multiple testing correction BUT I am unable to decide for the best method in this case.

I want to know if it is imperative to do this kind of correction for all the data sets that contain lists of means and what is the best method for the correction in this case?
 A: Thomas Perneger is not a statistician and his paper is full of mistakes. So I wouldn't take it too seriously. It's actually been heavily criticized by others.
For example, Aickin said Perneger's paper "consists almost entirely of errors": Aickin, "Other method for adjustment of multiple testing exists", BMJ. 1999 Jan 9; 318(7176): 127.
Also, none of the p-values in the original question are < .05 anyway, even without multiplicity adjustment. So it probably doesn't matter what adjustment (if any) is used.
A: A nice discussion of Bonferroni correction and effect size http://beheco.oxfordjournals.org/content/15/6/1044.full.pdf+html
Also, Dunn-Sidak correction and Fisher's combined probabilities approach are worth considering as alternatives. Regardless of the approach, it is worth reporting both adjusted and raw p-values plus effect size, so that the reader can have the freedom of interpreting them.
A: Maybe it's good to explain the  ''reasoning behind'' multiple testing corrections like the one of Bonferroni.  If that is clear then you will be able to judge yourself whether you should apply them or not. 
In a hypothesis test one tries to find evidence for some known or assumed fact about the real world.  It is similar to ''proof by contradiction'' in mathematics, i.e. if one wants to prove that e.g. a parameter $\mu$ is non-zero, then one will assume that the opposite is true, i.e. one assumes that $H_0: \mu=0$ and one tries to find something that is impossible under that assumption.  In statistics things are rarely impossible, but they can be very improbable.  
So if we want to show that $H_1: \mu \ne 0$ then we assume the opposite namely $H_0: \mu = 0$ and we try to find something very improbable.  Very improbable is defined in terms of a probability lower than an a priori fixed significance level $\alpha$. Note that, because of the analogy I will use terms such as ''statistically proven'' or ''statistical evidence'', these terms aree just used for didactical reasons and are not used in general. 
In order to find that ''low probability'' we draw a random sample form a distribution that is known when $H_0$ (our assumption of the ''opposite'' of what we want to prove) is true.  As we assumted $H_0$ te be true we can compute the probability of this outcome (more precise something that is at least as extreme as this outcome). 
As the sample is a random draw from a distribution, it may be that we obtain a low probability just by ''bad luck with the sample'' and then we reject $H_0$ just because we had bad luck with the sample.  Rejecting $H_0$ means that we consider to have found evidence for $H_1$ but it is false evidence in these cases where we have bad luck with the sample.  
False evidence is a bad thing in science because we believe to have gained true knowledge about the world,  but in fact we may have had bad luck with the sample.  This kinds of errors should consequently be controled.  Therefore one should put an upper limit on the probability of this kind of evidence, or one should control the type I error.  This is done by fixing an acceptable significance level in advance.  
So if we fix our significance level at $5\%$ then we are saying that we are ready to reject $H_0$ when it is true (because of bad luck with the sample) with a chance of $5\%$.  As (see supra) rejecting $H_0$ is ''statistical evidence'' for $H_1$ this means that we falsely consider $H_1$ as ''statistically proven''.
Assume now that we have two parameters, and we want to show that that at least one is different from zero.  Follwing the logic of ''proof by contradiction'' we will assume $H_0: \mu_1=0 \& \mu_2=0$ versus $H_1: \mu1 \ne 0 | \mu_2 \ne 0$ and that we use a signficance level $\alpha=0.05$. 
One possibility to do this is to split this hypothesis test and to test $H_0^{(1)}: \mu_1=0$ versus $H_0^{(1)}: \mu_1 \ne 0$ and to test $H_1^{(2)}: \mu_2=0$ versus $H_1^{(2)}: \mu_2 \ne 0$ both at the significance level $\alpha=0.05$. 
To do both tests we draw one sample , so we use one and the same sample to do both of these tests.  I may have bad luck with that one sample and erroneously reject $H_0^{(1)}$ but with that same sample I may also have bad luck with the sample for the second test and erroneously reject $H_0^{(1)}$
Therefore, the chance that at least one of the two is an erroneous rejection is 1 minus the probability that both are not rejected, i.e. $1-(1-0.05)^2=0.0975$, where it was assumed that both tests are independent.  In other words, the type I error has ''inflated'' to 0.0975 which is almost double $\alpha$. 
The important fact here is that the two tests are based on one and the sampe sample !
Note that we have assumed independence.  If you can not assume independence then you can show, using the Bonferroni inequality$ that the type I error can inflate up to 0.1. 
Note that Bonferroni is conservative and that Holm's stepwise procedure holds under the same assumptions as for Bonferroni, but Holm's procedure has more power. 
When the variables are discrete it's better to use test statistics based on the minimum p-value and if you are ready to abandon type I error control when doing a massive number of tests then False Discovery Rate procedures may be more powerful. 
EDIT : 
If e.g. (see the example in the answer by @Frank Harrell) 
$H_0^{(1)}: \mu_1=0$ versus $H_1^{(1)}: \mu_1 \ne 0$ is the a test for the effect of a chemotherapy and 
$H_0^{(2)}: \mu_1=0$ versus $H_1^{(2)}: \mu_2 \ne 0$ is the test for the effect on tumor shrinkage, 
then, in order to control the type I error at 5% for the hypothesis $H_0^{(12)}: \mu_1=0 \& \mu_2 = 0$ versus $H_1^{(12)}: \mu_1 \ne 0 | \mu_2 \ne 0$ (i.e. the test that at least one of them has an effect) can be carried out by testing (on the same sample)
$H_0^{(1)}$ versus $H_1^{(1)}$ at the 2.5% level and also $H_0^{(2)}$ versus $H_1^{(2)}$ at the 2.5% level.
A: What is wrong with the Bonferroni correction besides the conservatism mentioned by others is what's wrong with all multiplicity corrections.  They do not follow from basic statistical principles and are arbitrary; there is no unique solution to the multiplicity problem in the frequentist world.  Secondly, multiplicity adjustments are based on the underlying philosophy that the veracity of one statement depends on which other hypotheses are entertained.  This is equivalent to a Bayesian setup where the prior distribution for a parameter of interest keeps getting more conservative as other parameters are considered.  This does not seem to be coherent.  One could say that this approach comes from researchers having been "burned" by a history of false positive experiments and now they want to make up for their misdeeds.
To expand a bit, consider the following situation.  An oncology researcher has made a career of studying efficacy of chemotherapies of a certain class.  All previous 20 of her randomized trials have resulted in statistically insignificant efficacy.  Now she is testing a new chemotherapy in the same class.  The survival benefit is significant with $P=0.04$.  A colleague points out that there was a second endpoint studied (tumor shrinkage) and that a multiplicity adjustment needs to be applied to the survival result, making for an insignificant survival benefit.  How is it that the colleague emphasized the second endpoint but couldn't care less about adjusting for the 20 previous failed attempts to find an effective drug?  And how would you take into account prior knowledge about the 20 previous studies if you weren't Bayesian?  What if there had been no second endpoint.  Would the colleague believe that a survival benefit had been demonstrated, ignoring all previous knowledge?
A: For one, it's extremely conservative. The Holm-Bonferroni method accomplishes what the Bonferonni method accomplishes (controlling the Family Wise Error Rate) while also being uniformly more powerful. 
A: One should look at the "False Discovery Rate" methods as a less conservative alternative to Bonferroni.  See
John  D. Storey, "THE POSITIVE FALSE DISCOVERY RATE: A BAYESIAN
INTERPRETATION AND THE q-VALUE,"
The Annals of Statistics
2003, Vol. 31, No. 6, 2013–2035.
A: 
He summarized saying that Bonferroni adjustment have, at best, limited applications in biomedical research and should not be used when assessing evidence about specific hypothesis.

The Bonferroni correction is one of the simplest and most conservative multiple comparisons technique. It is also one of the oldest and has been improved upon greatly over time. It is fair to say that the Bonferroni adjustments have limited application in almost all situations. There is almost certainly a better approach. That is to say, you will need to correct for multiple comparisons but you can choose a method that is less conservative and more powerful.
Less Conservative
Multiple comparisons methods protect against getting at least one false positive in a family of tests. If you perform one test at the $\alpha$ level then you are allowing a 5% chance of getting a false positive. In other words, you reject your null hypothesis erroneously. If you perform 10 tests at the $\alpha = 0.05$ level then this increases to $1-(1-0.05)^{10}$  = ~40% chance of getting a false positive 
With the Bonferroni method you use an $\alpha_b$ at the lowest end of the scale  (i.e. $\alpha_b  = \alpha/n$) to protect your family of $n$ tests at the $\alpha$ level. In other words, it is the most conservative. Now, you can increase $\alpha_b$ above the lower limit set by Bonferroni (i.e. make your test less conservative) and still protect your family of tests at the $\alpha$ level. There are many ways to do this, the Holm-Bonferroni method for example or better still False Discovery Rate
More Powerful
A good point brought up in the paper referenced is that the likelihood of type II errors is also increased so that truly important differences are deemed non-significant.
This is very important. A powerful test is one that finds significant results if they exist. By using the Bonferroni correction you end up with a less powerful test. As Bonferroni is conservative, the power is likely to be considerable reduced. Again, one of the alternative methods eg False Discovery Rate, will increase the power of the test. In other words, not only do you protect against false positives, you also improve your ability to find truly significant results.
So yes, you should apply some correction technique when you have multiple comparisons. And yes, Bonferroni should probably be avoided in favour of a less conservative and more powerful method
A: Suppose we have 20 null hypotheses, all of which happen to be true. Consider two cases:

*

*If 20 scientists each independently pick and test one of the null hypotheses at p=0.05, on average they will correctly accept 19 of the hypotheses and incorrectly reject 1.


*If a single scientist does one big study testing all 20 null hypotheses at p=0.05 each without the Bonferroni correction or another multiplicity correction, on average they will correctly accept 19 of the hypotheses and incorrectly reject 1.
Then the scientists in case 1 are each Bonferroni correction-compliant, but the scientist in case 2 is not. Why should the second case be considered any different from the first?
I suspect that studies testing many hypotheses are more likely to speculatively include hypotheses for which the relevant data happens to be available, e.g. "I know how strongly each person in my sample agrees with each of 20 opinions, guess I might as well check for correlation between every possible pair of opinions". So the average prior plausibility of the alternative hypotheses would be smaller. Ideally this would be handled by choosing a custom significance level to test each hypothesis at based on this prior plausibility. The Bonferroni correction could be seen as a step towards this.
