The rulings of the US Supreme Court aren't the place most people would start looking for a discussion of what the phrase "statistically significant" might mean. But in March 2011, America's highest court of justice made a decision that involved exactly that.
Contemplation of Justice: A figure outside the US Supreme Court building
The case involved a case originally brought by investors in a company called Matrixx Initiatives Inc., who manufacture a range of over-the-counter medicaments including one called Zicam Cold Remedy.
The investors alleged that Matrixx had failed to disclose material information to them, when it had received reports that people who had taken Zicam Cold Remedy had suffered a loss of sense of smell. When these reports did become public, the share price fell, thus (allegedly) losing the investors money.
People do lose their sense of smell sometimes, for reasons that have nothing to do with a drug. Matrixx argued that the number of cases of loss of sense of smell in people who had taken Zicam was not significantly higher, in the statistical sense, than the number one would expect to see if the drug had no effect on sense of smell.
For that and other reasons, Matrixx argued that they had no obligation to disclose the reports to their investors. The case depended to a major extent, therefore, on the idea of statistical significance testing.
What's all that about? I've given a brief explanation in another OpenLearn article, Confusing terms in statistics, so I won't repeat all that, but briefly the point of doing a significance test is to see whether it is plausible that some feature in data can be explained by chance alone.
In the Matrixx case, perhaps they did get more cases of loss of sense of smell than one might have expected if the remedy had no effect on sense of smell, but maybe this could have been due entirely to chance – people who were going to lose their sense of smell anyway might just have happened to take the cold remedy.
If, after performing a statistical significance test, the conclusion was that the data are consistent with this "it's all just chance" explanation, one would report that the results were "not statistically significant".
Matrixx did claim this, but went further and argued that, because they were not statistically significant, they did not have to be disclosed.
An interesting feature of the Matrixx case is that arguments to the court were not submitted only by lawyers, investment experts, and so on, but also by Deirdre McCloskey and Steve Ziliak. They are economists and statisticians who are well known for publishing a book called The Cult of Statistical Significance, which, as you might guess from the title, is not complimentary about the way that significance testing has been used.
McCloskey and Ziliak submitted an amicus brief, or more precisely an amicus curiae brief (from Latin words meaning "friend of the court"). This is a submission to the court from a person or organisation that is not one of the parties involved in the case, but which is intended in some way to throw light on the case.
They did not argue wholesale against the use of the ideas of statistical significance. Instead, they argued as follows:
If a feature in data is not statistically significant, it's reasonable to take that to mean that the feature might plausibly be due solely to the workings of chance.
But just because it might be due to chance, that doesn't mean it is due to chance. That's only one possible explanation.
There will be others, and thus a decision on whether some piece of information is relevant can't be based on statistical significance alone. The court agreed with this line in their judgment. They said quite clearly that "Matrixx's premise that statistical significance is the only reliable indication of causation is flawed."
Most, if not all, statisticians would agree with them. Statistics students often get annoyed with what seems like excessive pedantry from their teachers. The statisticians will insist that one does not accept the hypothesis that some data feature is due to chance alone.
One merely fails to reject such a hypothesis, if the results are not statistically significant.
The idea here is that "accept" sounds too much as if one is saying, "Yes, we know it's just chance". "Fail to reject" allows the possibility that it might just be chance, but allows other possibilities too.
There's reason in our pedantry. To quote another cliché, absence of proof isn't the same as proof of absence, and a result that's not statistically significant is absence of proof that something is happening that goes beyond mere chance.
A couple of aspects of this discussion that might interest you, depending on your background:
First, what is really going on with statistical experts like Ziliak and McCloskey rubbishing significance tests? Surely they are a key and fundamental aspect of the way statistics is done?
Well, not everyone would be quite as critical as Ziliak and McCloskey, but actually, many statisticians would be very critical of the unthinking and routine use of significance testing that goes on in many areas of enquiry. (If you understand a little of the technical ideas behind significance testing, you might enjoy reading another criticism from a slightly different angle, Mindless Statistics, by Gerd Gigerenzer, the well-known author on risk. [The article opens as a pdf.])
Second, some people find it surprising at first sight that the law should be concerned with ideas and methods of statistics. But if you think about it, it's not really surprising at all.
Both statistics and the law involve dealing with uncertainty – admittedly often with different goals in sight, but uncertainty is there as an inescapable aspect in both. Gary Slapper, Professor of Law at the Open University, has written about statistics and the law.
Statisticians and lawyers would not always agree on what should be done about the uncertainty, but perhaps that's inevitable – the law has to come to a clear decision on every case, however much uncertainty there is, whereas we statisticians can sometimes have the luxury of saying that there's simply not enough evidence to decide.
If you're interested in the legal details, the briefs submitted to the Supreme Court in the Matrixx case (including the McCloskey and Ziliak brief) are online, as are a pdf transcript of the oral argument, and the "opinion" (i.e. judgment), again as a pdf.