Release the data
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That being said, calls of "release the data" are sometimes disingenuous. When a scientific publication conflicts with someone's religious or political ideology, the latter is likely to slip into denialism; they may in turn use this phrase to concern troll and gain the moral high ground with claims that, because research is often publicly funded, scientists owe it to the taxpayers to share their work in an unmediated fashion. This was particularly (and comedically) abused in the Lenski affair.
The data that people are asking to be released, as such, are typically of little use to anyone who is not trained in the scientific field in question. Those with the right amount of training or expertise probably have their own research group (and their own data to worry about) or already have the data, because they read the current state-of-the-art literature on the subject. Therefore, the actual point of the strategy is less to get the data into the public eye than to put scientists on the defensive. Usually, as most data is released anyway, scientists actually don't release any new data and the public watching any furor tend to come away with the message "scientists are hiding data," even though this is total nonsense.
Particularly following Climategate, demands for data were accompanied by Freedom of Information requests.
Getting it wrong
Usually, the data is mined for instances where it doesn't match the general conclusion. It is a given, especially when statistics needs to be applied, that one or two pieces of data will not match the conclusion. The trick with statistics, indeed the entire point of statistical analysis, is to determine whether such things are relevant or significant, and dozens of tests and equations designed to determine these factors are available for use. Those who demand that data be released often bypass this process, instead preferring to look at whatever data they can and cherry-pick it.
A recent example involved tabloid newspapers in the UK declaring that cocaine was flooding school playgrounds after its use doubled in a year. This headline was only plausible because the reporters examined the data but failed to do the relevant statistical analysis to show that the change from 1% use to 2% use wasn't significant (indeed, these figures were rounded off so the increase wasn't even that bad).
Getting it right
In one particular example, releasing the data did show that people were trying to hide something. This was the case of the Bush administration in the US "hiding" satellite photographs showing a reduction in sea-ice that was evidence of climate change. In another scenario, Gilles-Eric Séralini was part of a very poor study that purportedly shows higher tumor rates in rats when fed GMO corn. The study received intense criticism from scientists and researchers not only from its poor design, but also because Séralini refused to release the raw data and did not answer many major questions pressed against him.