(This article was originally published at Statistical Modeling, Causal Inference, and Social Science, and syndicated at StatsBlogs.)
Stephen Martin started off with a question:
I’ve been thinking about this thought experiment:
—
Imagine you’re given two papers.
Both papers explore the same topic and use the same methodology. Both were preregistered.
Paper A has a novel study (n1=300) with confirmed hypotheses, followed by two successful direct replications (n2=300, n3=300).
Paper B has a novel study with confirmed hypotheses (n=900).
*Intuitively*, which paper would you think has the most evidence? (Be honest, what is your gut reaction?)—
I’m reasonably certain the answer is that both papers provide the same amount of evidence, by essentially the likelihood principle, and if anything, one should trust the estimates of paper B more (unless you meta-analyzed paper A, which should give you the same answer as paper B, more or less).
However, my intuition was correct that most people in this group would choose paper A (See https://www.facebook.com/groups/853552931365745/permalink/1343285629059137/ for poll results).
My reasoning is that if you are observing data from the same DGP, then where you cut the data off is arbitrary; why would flipping a coin 10x, 10x, 10x, 10x, 10x provide more evidence than flipping the coin 50x? The method in paper A essentially just collected 300, drew a line, collected 300, drew a line, then collected 300 more, and called them three studies; this has no more information in sum (in a fisherian sense, the information would just add together) than if you didn’t arbitrarily cut the data into sections.
If you read in the comments of this group (which has researchers predominantly of the NHST world), one sees this fallacy that merely by passing a threshold more times means you have more evidence. They use p*p*p to justify it (even though that doesn’t make sense, because one could partition the data into 10 n=90 sets and get ‘more evidence’ by this logic; in fact, you could have 90 p-values of ~.967262 and get a p-value of .05). They use fisher’s method to say the p-value could be low (~.006), even though when combined, the p-value would actually be even lower (~.0007). One employs only Neyman-Pearson logic, and this results in a t1 error probability of .05^3.
I replied:
What do you mean by “confirmed hypotheses,” and what do you mean by a replication being “successful”? And are you assuming that the data are identical in the two scenarios?
To which Martin answered:
I [Martin], in a sense, left it ambiguous because I suspected that knowing nothing else, people would put paper A, even though asymptotically it should provide the same information as paper B.
I also left ‘confirmed hypothesis’ vague, because I didn’t want to say one must use one given framework. Basically, the hypotheses were supported by whatever method one uses to judge support (whether it be p-values, posteriors, bayes factors, whatever).
Successful replication as in, the hypotheses were supported again in the replication studies.
Finally, my motivating intuition was that paper A could basically be considered paper B if you sliced the data into thirds, or paper B could be written had you just combined the three n=300 samples.
That said, if you are experimenter A gaining three n=300 samples, your data should asymptotically (or, over infinite datasets) equal that of experimenter B gaining one n=900 sample (over infinite datasets), in the sense that the expected total information is equal, and the accumulated evidence should be equal. Therefore, even if any given two papers have different datasets, asymptotically they should provide equal information, and there’s not a good reason to prefer three smaller studies over 1 larger one.
Yet, knowing nothing else, people assumed paper A, I think, because three studies is more intuitively appealing than one large study, even if the two could be interchangeable had you divided the larger sample into three, or combined the smaller samples into 1.
From my perspective, Martin’s question can’t really be answered because I don’t know what’s in papers A and B, and I don’t know what is meant by a replication being “successful.” I think the answer depends a lot on these pieces of information, and I’m still not quite sure what Martin’s getting at here. But maybe some of you have thoughts on this one.
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