11 August 2010

Devaluation of Significance

I referred in my last post to a lost writing of mine on the subject of abuse of statistics in economics. I've sort of found it - I sent it as an email in response to this blog post by Noel Campbell at Division of Labour. (Read it - it's short).

He quoted from my response, but I can't find the actual email I sent him. I do have a draft of it, so it would have been very much like this:



That's a superb question, and I think the answer will surprise (and disturb) many.

Your paper will include a calculation of significance. This is essentially an estimate of the probability that a correlation as strong as the one you found would exist purely as a result of randomness in the data, even if your theory is false.

This calculation assumes the "proper" sequence of events. You have a theory, and you test the data for a correlation. Since you in fact poked around for correlations, then came up with a theory, the significance calculation is not valid. The true significance depends on the probability that, having found a randomly-caused correlation somewhere, you can then invent a theory to explain it. That probability is very difficult to estimate, but is probably much greater - meaning that the significance of the correlation is much smaller.

It is very counterintuitive that the order of your actions affects the validity of your findings, and indeed it is a close relative of the famous Monty Hall problem - the poster child for counterintuitive probability. When you reveal the correlation that you already knew of, you are revealing no information about the chance of your theory being correct, much as when the quizmaster opens the door that he already knows doesn't have the car, he reveals no information about the chance that the door you first picked has the car. Conversely if you pick a door and find that it doesn't have the car, that does change the probability that the first door had it, and if you had no prior knowledge of the data, the correlation does change the probability of your theory being true.

Back to science. As you say, theories aren't formed in a vacuum, and so there is not such an clear division between the "right" way of doing it and the "wrong" way of doing it. Nobody is completely ignorant of the data when they start to theorize. That is a real problem with nearly all statistics-based results that are published today. They are all presented with significance calculations based on the assumption that the forming of the theory was independent of the data - an assumption that is very unlikely to be completely true. Therefore nearly every significance published is an overestimate.

This was much less of a problem when collecting data and analysing it was difficult and laborious. Now that large data sets fly around the internet, and every researcher has the capability of running analyses at the click of a mouse, it is a problem that has already got out of hand.


I didn't want to be rude at the time, but I found Campbell's response shocking. He seemed to fully accept my argument, but wasn't bothered by the implication that pretty much all published research relying on analysing pre-existing statistics was wrong. Rather, his conclusion was that since everybody else was doing what he was doing, nobody should complain and demand "purity" (his scare quotes). That came to mind particularly reading Bruce Charlton's discussion of the state of honesty in science.

2 comments:

Anomaly UK said...

Division of Labour no longer exists: This is the original post referred to:

November 23, 2009
Random Observations
1. Suppose I run a regression out of curiosity, just to see what I find. 2. The results puzzle me mightily. 3. I get an idea, and tinker some more, pretty much confirming the idea. 4. Now I write it up. Here's where it gets interesting.

A. I write up my explanation as if I was never confused and as if I expected to find what I found. Reviewers consider this to be good theorizing and the paper is accepted.

B. I write up my explanation totally honestly, and in chronological order, as above. Reviewers dismiss my work as ad hoc theorizing and the paper is rejected.

Agreed, if I behave as outlined in the opening comments, I would not conform to the ideal scientific method. Then again, scientific theorizing does not occur in a vacuum. I must know there's a research question out there before I try to answer it. If I consider the scientific method to begin when I formulate my research question, then I start at #2, and my behavior is ethical and consistent with the ideal scientific method.

Does the problem lie with me or the review process?

Posted by Noel Campbell at 10:58 AM

Anomaly UK said...

..And this is the response quoting my comment:

November 25, 2009
The Ghastly Specter of Monty Hall!
Anomaly UK responds to my Random Observations . He was heading not entirely down my path, but he makes a good point that’s worth repeating. Andrew references the Monty Hall problem .

Your paper will include a calculation of significance. This is essentially an estimate of the probability that a correlation as strong as the one you found would exist purely as a result of randomness in the data, even if your theory is false.
This calculation assumes the "proper" sequence of events. You have a theory, and you test the data for a correlation. Since you in fact poked around for correlations, then came up with a theory, the significance calculation is not valid. The true significance depends on the probability that, having found a randomly-caused correlation somewhere, you can then invent a theory to explain it. That probability is very difficult to estimate, but is probably much greater - meaning that the significance of the correlation is much smaller.

As you say, theories aren't formed in a vacuum, and so there is not such a clear division between the "right" way of doing it and the "wrong" way of doing it. Nobody is completely ignorant of the data when they start to theorize. That is a real problem with nearly all statistics-based results that are published today. They are all presented with significance calculations based on the assumption that the forming of the theory was independent of the data - an assumption that is very unlikely to be completely true. Therefore nearly every significance estimate published is an overestimate.

I believe the issue Anomaly mentions to be very nearly universal. In order to publish, I must read and reference the related literature. That action, alone, nearly guarantees that I am NOT ignorant of the data.

In light of this fact, I am occasionally dismayed when reviewers demand a particular type of “purity” in the scientific process, which, I suppose, was the point of my original post.

I also find it blackly amusing when we bring such arguments to bear against research that contradicts our preferences, but tend not to use such arguments against research the conforms to out preferences. Yes, I’m guilty, too.
Posted by Noel Campbell at 10:38 AM