In case you are wondering why there is a lack of updates, I have moved the blog to my personal webpage and integrate statistical posts with general posts about cognitive psychology & academia.
The new blog is www.jimgrange.wordpress.com/blog
In a previous post, I extolled the virtues of Bayesian statistics. Bayesian statistics is often flouted as the superior alternative to p-values, and its popularity is growing rapidly in psychology. In that post I discussed how one of the advantages of Bayesian analysis is it allows us to assess evidence in support of a certain hypothesis, given the data (rather than what the p-value gives you, which is something quite different, as I discuss in this previous post).
However, that post neglected one critical aspect of Bayesian analysis: priors. In Bayesian statistics, you take your prior belief about something, then you view some data, and Bayes’ rule formally states how your beliefs should be updated in light of the new data; this updated belief is called the posterior. The prior not only states what your prior belief is, it also states how strongly you believe it. For example, if deciding whether a coin is fair or not, most people would have a strong prior that it is fair (i.e., that p(head) = 0.5), because most coins you encounter are fair. However, some people might believe in aliens, but the prior for this would be pretty weak. Strong priors require a lot of strong opposing evidence to overcome, whereas weak priors can be overcome with weaker evidence.
At this point, frequentist psychologists (i.e., those strongly committed to “traditional” statistics with p-values and null-hypothesis testing) will be shaking their head in fervent disgust. They will say “You shouldn’t allow your beliefs enter into scientific analysis; I want objective results, not subjective results! Do they have a point? I would argue NO!
We are all—yes, even you—Bayesian. We let our beliefs (priors) enter our assessment of “data” all the time. Let’s use an example. If I told you I had roast chicken for dinner this weekend, your prior would be likely in the form of believing me; therefore, you would not require much evidence from me (or any at all) to be convinced that I had roast chicken. However, if I told you I had roast chicken for dinner, and aliens joined me for desert, you would immediately demand photo evidence, video recordings, and a whole-host of scientific evidence to convince you it was true.
Priors should enter our decision-making process, and be paired with the data to reach rational conclusions. As another example, if I flipped a coin 10 times and it came up heads 8 times, you would likely not change your prior that much from believing it is a fair coin. However, if you knew the coin was from a magic shop, your prior would be that the coin is biased. These priors should be used for coming to conclusions.
Below is a screen-grab from a paper I have read recently. It shows a meta-analysis of a certain psychological phenomena of interest. For those unfamiliar, a meta-analysis is a statistical aggregation of all known published results about a particular effect of interest. It allows for a much more precise estimate of the effect size of the particular phenomena of interest. Meta analyses are important, because individual studies have error (sampling, experimental, and measurement, with residual errors also) associated with their reported results.
The figure below shows a recent meta-analysis. Each diamond reflects the estimated effect size for the phenomena under investigation in each study. The lines represent 95% confidence intervals around this effect size. Confidence intervals that overlap with zero do not provide convincing evidence that the effect size reported is different from zero.
The diamond at the bottom (circled in red) is the meta-analysis effect size; that is, the estimate of effect size when the data is aggregated across all studies (this is a very simplified, and ultimately incorrect, summary of what the meta-analytic effect size represents, but it serves our purposes). You will note that the 95% confidence intervals aren’t even close to zero, suggesting that the effect under investigation is “real”.
Now, to prove you are all Bayesian. Answer the following two questions:
1) How strongly do you believe in this final positive effect size if you knew the studies were investigating whether working memory capacity is related to intelligence?
2) How strongly do you believe in this final positive effect size if you knew the studies were investigating whether people are able to perceive future events? This is known as “psi”
Now, I believe that most—if not all—would take the first question at face value, and would have faith in the final effect size. You all either have no real prior for the relationship between working memory and intelligence, or you have a prior that there is indeed a relationship. Therefore, you aren’t really surprised (or perhaps interested) in the result of the meta-analysis.
However, I would be shocked if any had faith in the final effect size knowing that it provides “positive evidence” for people being able to see into the future. That is, you all have—and if you don’t, you really should have—a strong prior against believing in being able to perceive future events reliably.
The point is, in both cases you are viewing the same data. In the former, you would accept the data without much problem. In the latter case, you still think the results don’t provide convincing evidence. That is, you are Bayesian. Accept it. Embrace it. Move forward.
This meta-analysis is not made up; it’s from a paper by Cornell psychologist Daryl Bem. You can view the full (as yet, unpublished) manuscript here: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2423692
This paper has confirmed my prior belief (pun intended) that Bayesian analysis is the way forward in tackling how to interpret data. Priors entering the decision-making equation is not a weakness, but a strength. Despite Bem’s best efforts with this paper and his prior work, I am still not convinced that people can “feel the future”. To me, psi doesn’t exist. A frequentist should be convinced by the above meta-analysis—after all, the p-value for psi is very low!
A strong Bayesian approach to statistical decision-making, with well-defined (and defendable) priors, makes you robust against wacky claims, even if the data is “strong”.
“Extraordinary claims require extraordinary evidence”
—Carl Sagan (quite likely a Bayesian)
As a psychologist, the software I have used for much of my career when analysing statistics is, of course, SPSS. Its simplicity has meant SPSS has been quite a good friend over the years. However, the friendship is (almost) over. There’s a new “best friend” for this researcher: R.
R is a free (yep, FREE) software environment for statistical computing. Although that description might sound intimidating, do not let it put you off. I love R. No, I REALLY mean it. Since I’ve been using R, my statistical life has become a dream. The beauty of R is that you can do almost ANYTHING you want to with it (it doesn’t make coffee, although a package might be available for that someday; hey, I can dream, can’t I?). Continue reading
As the start of the research methods module I teach is fast approaching, I have been exploring new ways to try and get students more engaged with statistics. One aspect I believe increases student understanding—and dare I say it, enjoyment—of statistics is to get their hands dirty with lots of examples.
Although it would be great to get students to go off and collect lots of different data sets, each suitable for exploring a particular statistical test, a quicker alternative is perhaps to provide the data via computational simulation. Then, students can quickly explore new data sets in interesting ways. Continue reading
As I mentioned in an earlier post, p-values in psychological research are often misunderstood. Ask students (and academics!) what the definition of the p-value is and you will likely get many different responses. To jog your memory, the definition of the p-value is the probability of observing a test statistic as extreme—or more extreme—than the one you have observed, assuming the null is true. But, even with this definition in hand, many struggle to conceptualise what the p-value reflects. In this blog post, I take inspiration from a lovely paper I have recently read that advocates using computer simulation to understand the p-value. Continue reading
Students often wonder why we in Psychology ask them to learn statistics. There are many good reasons, but today I just want to focus on one; surprisingly, this reason has nothing to do with Psychology. Having even just a smattering of statistical savviness can protect you from a lot of…well…bullshit. And trust me, there is a lot out there. Continue reading
Wow – so, it’s been a LONG time since my last post. Apologies. I didn’t think anyone would miss it, but it seems that the blog has been receiving a modest amount of daily traffic despite the absence of posts.
A lot has happened in my professional life since the last post which has wrestled my attention away from the blog: Continue reading
Today I give my favourite lecture of the year. Second year students today will be introduced to some of the controversies surrounding the use of p-values. Although I don’t go into great depth, I highlight that p-values—among the many problems associated with them—don’t provide answers to the type of questions we—as researchers—are asking. Continue reading
As readers of contemporary psychology journals may well know, you can’t help but keep reading about something called “Bayesian Statistics”. Researchers “in the know” seem to extol Bayesian statistics as a superior method of making inferences from data compared to traditional, “frequentist”, methods (yes, Mr p-value, I’m looking at you!).
But what is it? What can it do that standard methods can’t? It turns out the answer is just about everything you’ve ever dreamed of (well, as a researcher, anyway!). Continue reading
Well, I haven’t posted for AGES! This is—in most part—due to the hectic return to teaching. Over the first two weeks of semester, I have noticed in lab classes that students struggle with formatting their results tables correctly in their reports. Thus, I decided to knock up a quick screen-cast showing you all how to do this in MS Word.
I shall add more “substantive” post over the next few days. Enjoy!