Bell Curve The Law Talking Guy Raised by Republicans U.S. West
Well, he's kind of had it in for me ever since I accidentally ran over his dog. Actually, replace "accidentally" with "repeatedly," and replace "dog" with "son."

Monday, August 17, 2009

Be Careful in Interpreting Experiments

This is from today's NYTimes: "Three recent experiments show that even the youngest children have sophisticated and powerful learning abilities. Last year, Fei Xu and Vashti Garcia at the University of British Columbia proved that babies could understand probabilities. Eight-month-old babies were shown a box full of mixed-up Ping-Pong balls: mostly white but with some red ones mixed in. The babies were more surprised, and looked longer and more intently at the experimenter when four red balls and one white ball were taken out of the box — a possible, yet improbable outcome — than when four white balls and a red one were produced. "

I find this a weird explanation. If I reached into a pile of mostly white balls and pulled out three red ones, my toddler would be pretty interested, but not because she understood this to be an improbable event - rather, because it would seem like magic. The experimenters assume that the child noticed the ratio of red/white in the box and compared it to the sample. Is that a fair assumption? My child notices red balls more than white ones, period. I also am not sure how this study can be double-blind: doesn't the experimenter who is inviting the baby's gaze know what she is doing, and isn't she looking for a reaction?

I wanted to blog about this because it's the kind of crappy science reporting I see in newspapers all the time. Either the study was much better than the reporter makes it out to be, or the reporter should have been more skeptical of the study, or (likely) both.


Anonymous said...

One obvious point left out of the experiment, red, in nature, signals is the color of is in that lizard part of our brain. Decorators use it when they want to stimulate the patrons of a night club, etc., so it would seem natural to me, that a baby would react to red in a different way than white, it is hard wired in us. WW

Anonymous said...

I call total bullpuckey. Babies like watching adults show them interesting things.

-Seventh Sister

Dr. Strangelove said...

I found the original article by Xu and Garcia. You will be happy to know that it was more sophisticated than reported in the short NY Times article.

First, they used red/white interchangeably, and found the same effect whether there was an anomalous number of white or red balls. They were well aware of that potential problem and I feel they actually controlled for color quite well. (And as a side note, it seems the folk wisdom of "red=danger" did not hold up in these trials.)

Second, they ran the experiment backward and forward. Sometimes they showed the contents of the box, closed it, and then drew a sample; other times they first drew a sample and then revealed the box. The results were the same either way. And in no case did the babies on average look longer or more intently when the sample actually matched what was in the box. There was clearly some awareness of a mismatch.

Third, the study authors were concerned that the infants might simply be reacting to seeing a mismatched set of colors, so they sometimes ran the experiment when the samples were not drawn from the box at all, but instead were clearly drawn from a hidden cache in the experimenter's pocket. In those cases the infants did not react to mismatches.

Finally, the experimenters closed their eyes when they drew and displayed the balls. It is also worth noting that the infants' parents were seated next to the infants, for the comfort of both, but the parents always faced in the opposite direction so they could not view the experiment.

I think it was a pretty good experiment. The summary of it was somewhat too hasty.

USwest said...

They should have used a different color.

The Law Talking Guy said...

It sounds okay, Dr.S., but does it prove that babies understand probabilities? Or isn't it just that they saw a mostly white box and out of it drawn something that didn't look like the box, and they were amazed. As if I showed you what looked like a basket of lemons but pulled out a couple of limes. I am sure the babies understood "mismatch" - I am totally doubtful that it is anything like understanding the "probability" of drawing randomly.

How did they control for the problem that a baby's gaze is not one-sided - it is MET with the interviewer's gaze - so the interviewer/experimenter might be looking more intently at the babies too.

The Law Talking Guy said...

Oh, I see, I missed the "closed eyes" thing at the end of your paragraph.

Dr. Strangelove said...

You are right, LTG. The babies saw a box containing "mostly" one color (they used a 70/30 mix) and the babies recognized when the samples drawn from it "didn't look like" what was in the box. (The sample was a 1/4 mix the other way.) The recognition of mismatch between drawn sample and the general population is a recognition of unlikelihood.

Note that they also ran the experiment backward, drawing the sample first from an opaque box and only later revealing the contents. The babies showed equal surprise when the box contained the "wrong" mix. But when the sample was drawn from a different source, they were not surprised.

USwest said...

I ain't buying it. Sorry.

Dr. Strangelove said...

Don't apologize, USWest. I have no stake or interest in this study :-)

The Law Talking Guy said...

"The recognition of mismatch between drawn sample and the general population is a recognition of unlikelihood."

I still think this is reading too much into the behavior. Not recognition of "unlikelihood," but of mismatch.

Or maybe the babies were intrigued by something else entirely.

For example, my toddler put a puzzle together yesterday (it was a wooden board with three pictures engraved into it and three wooden pieces matching those pictures, the idea was to fit the piece into the matching indent). After she succeeded with my prodding, I then turned the puzzle by 180 degrees. She attempted to put the pieces in the same place as she had before - only now they were 180 degrees off, so they didn't fit. This confused her. I realized she had not been matching up crab with crab and turtle with turtle - rather she had put the pieces in whatever the location that elicited praise from me, remembered their location relative to one another, and repeated the process even though the background "puzzle" had moved.

Another such incident: I observed my toddler frustrated that the milk cup she wanted was on the other side of the coffee table, 4 inches beyond her reach. She didn't seem to think to go around the table to get it. I got it for her; she was appeased. Later it occurred to me that her frustration was not that she couldn't figure out how to get it, but that she wanted ME to give it to HER. Since, in her primitive world, that's my function: to grant her wishes. The frustration was that the world wasn't working as it was supposed to, as she believed it did. She will soon learn not to wish, but to ask. It will then take another 30 years to learn that my function is not simply to do as she asks...

Dr. Strangelove said...

The infants only reacted to a mismatch between what was in the sample and what was in the box when the sample was actually drawn from the box. When the experiment was run in exactly the same way except the balls were drawn from somewhere else, a mismatch generated no reaction. The infants were therefore not reacting to the visual mismatch alone, but to the occurrence of a mismatch as the result of the drawing process.

And if you think about it, the ability to recognize that it is unusual for a drawing process to produce a visual mismatch is basically the definition of an intuitive grasp of probability.

Dr. Strangelove said...

As I mentioned in an earlier comment, I certainly have no stake in this study and frankly it actually does not interest me very much. But I have been somewhat surprised and disappointed to read the continuing comments of LTG and USWest. It looks to me like LTG and USWest both decided a priori that the study must be wrong, or at least wrongly interpreted. LTG has done nothing but try to pick apart the results, and USWest went so far as to flatly deny the results without bothering to explain why.

So I find myself wondering... How is it that two thoughtful, intelligent people--people whom I know have great respect for science in general--nevertheless feel perfectly comfortable disregarding a specific piece of scientific research simply because they do not agree with its conclusions? Neither LTG nor USWest appeared to consider the possibility that, despite their initial beliefs to the contrary, the study result might in fact be true--neither appeared willing challenge their own assumptions. While it is important to be skeptical of scientific research, it is also important to be open to it!

I apologize if I have misunderstood. I probably have. LTG picks at arguments for a living, so he might just have been having a little fun here, and USWest might just have felt my summary of the study was not clear enough to be convincing. I guess this discussion just really strikes a nerve in me, that's all...

Too many folks out there feel they can just pick and choose which science they want to believe and which they can ignore, especially when it comes to things like health care and the environment. Someone at my workplace has a sticker on their door that says, "The plural of 'anecdote' is not 'evidence.'" I wish more people grasped that. Scientific illiteracy is killing our country... quite literally.

Raised By Republicans said...

"It will then take another 30 years to learn that my function is not simply to do as she asks..."

You have plans to say "no?" ;-)

The Law Talking Guy said...

No, Dr.S, you have it wrong. You seem to be almost willfully blind to the distinction I am making.

"And if you think about it, the ability to recognize that it is unusual for a drawing process to produce a visual mismatch is basically the definition of an intuitive grasp of probability."

No, it's not! Of course it's not! That's reading way, way too much into what is observed. I don't see why this is so hard to figure out. Let me try it again.

The child isn't recognizing that it's "unusual" for there to be a visual mismatch. The child is probably just observing the visual mismatch, period. This is not an intuitive grasp of probability. The child isn't aware that there The issue is the principle of object permancence: that the objects IN the box should be the same as the objects when taken OUT of the box. The child isn't thinking it's unlikely (probability requires some idea of likely and unlikely) but that it's wrong. We know all about teaching "object permanence" to babies. That's nothing new.

This, of course, explains why there was no reaction when the balls were taken from somewhere else. With babies, we should always use a form of Occam's razor: the explanation that posits the least complicated understanding is likely correct.

Dr.,S's midnight lecture about scientific illiteracy is wrong. My point, which Dr.S. unwittingly demonstrates, is that some scientists will read too much into their experiments in order to fit the desired theory. And you should try to understand criticisms, not just reject them as the product of illiteracy.

I suppose I do "pick at arguments" for a living. Perhaps it's worth wondering if I have learned anything in the process about spotting logical or evidentiary flaws, and crediting me with some of it.

Dr. Strangelove said...

You raise a good point about object permanence. But this experiment probes something a bit more subtle. The box contained both red and white balls, and the experimenters were careful to draw attention to this: they even allowed the infants to handle several of the balls before the box was closed up. Likewise, the samples shown also contained both red and white balls.

In order to react as they did, the infants had to do more than just recognize that the objects drawn from the box were the not same as those contained in the box--they had to decide whether the sample was sufficiently similar. To me, the concept of similarity alone is halfway toward the essence of statistics, and the concept that a subset SHOULD be similar (but not identical!) to the larger set from which it is drawn is the other half. Of course no one is claiming that these infants actually made any computations. But that they could determine similarity between patterns and could tell when they should be similar and when they need not be similar is the essence of it. Maybe I just have a less formal notion of what "probability" means, and the rest is a distinction without a difference.

As to your final two paragraphs: (a) I specifically said you were thoughtful and respectful of science and I would never imply you were scientifically illiterate; (b) My extensive comments show that I did indeed try hard to understand and respond to your criticisms--a failure to agree is not a failure to understand; (c) I absolutely believe that you are very good at spotting logical and evidentiary flaws and I specifically credited you with that, although apparently you took the phrase "pick at arguments" to be an insult rather than a credit; (d) The only thing I ever faulted you for was for what I perceived as approaching the study with your mind already made up.

Pombat said...

As a mathematician, I'm with you Dr.S, especially the bit about the intuitive grasp of probability. Interesting study.

I also get very irritated at the kind of people who think that science is something you can choose to believe or otherwise - it's not, it's science, if there was a belief choice, it would be called faith. Saw a stand-up by the name of Dara O'Briain (yes, that's spelt correctly, it's an Irish name) on the box a week or so ago, and he made a comment along the lines of the people who question science, and say that it doesn't know everything - "well science knows it doesn't know everything, otherwise it would stop!" Look him up, you'd like him.

Oh, and I want one of those anecdote/evidence stickers!

Dr. Strangelove said...

LTG started the thread by complaining about shoddy reporting on science, and I do not want to ignore that important point. In truth most of the people I was bitching about are not actually in the position of picking and choosing which "science" they wish to believe because they never see any: they just read those godawful headlines/soundbites. They never see the reasoning. The critical reasoning skills LTG exercised here would be totally foreign to most people... And popular entertainment is no help.

If you watched the movies, you would think scientists have some magical talent that lets them think their way to the truth, instead of being just ordinary people who slowly pick their way to the truth by doing things-specifically by conducting laborious, careful, and usually unbearably boring experiments.

Dr. Strangelove said...

I re-read the study again. One thing I have to say is that I made a mistake in my summary: the boxes were not 70/30 mixes but 70/5 mixes. So they were actually much more biased toward one color than I had represented (and one needs to know the correct mix to understand the statement about which sample is more probable that I quote below). The other really important I realized was that the authors specifically raised LTG's objection and stated it it is a valid concern. They said more study is required to address it. Let me quote them here directly.

Several questions remain open. First, what is the computation that underlies the infants’ performance in these experiments? At least two possibilities are consistent with the current findings. One possibility is that infants had computed probabilities of the samples and used them to predict which of the outcomes was more probable. Another possibility is that infants had expected the sample to be related to the distribution of the population in a qualitative way... One way to distinguish these two alternatives is to contrast a 4:1 sample and a 5:0 sample; the former provides a better distributional match, whereas the latter is more probable. A recently published paper using a different methodology with 12-month-old infants suggests that by the end of the first year infants may be able to estimate probabilities in making predictions about future events. It remains to be seen whether younger infants are also able to do so.

So I think it is fair to say that the authors think either interpretation is acceptable.

Pombat said...

Dr.S: totally agree with your 9.27pm comment (the one immediately following my last one), and obviously also agree with LTG's complaint on shoddy reporting on science - a big hear hear here on that one!

My particular bugbear with that, other than the glaringly obvious fact that the reporters are all too often the kids who failed/skipped science in school, is the headlines that scream "activity [blah] TRIPLES your risk of disease [eek]!!!". They never, ever, then go on to explain that oh, the original risk is 0.001%, so the new risk is 0.003%, which even the least mathsy person would recognise as [insert favourite folksy expression to describe a pathetically small amount here]. Grumble.

I also get irritated by mentions of, e.g., "one in a million" chances. For example, "there's a one in a million chance that this DNA sample matches this person!". Most people respond with "ooooh, must be them then!". I respond with "hmmm, so there's nineteen other matches here in Aus then? Or another 300 or so in the US / 60 in the UK? So what you're really saying is that in Aus, there's a one in twenty chance this is the person?..."

And to be fair, the movies don't solely depict scientists as these amazing people who magically arrive at the answer - if you select the other 50% of movies that involve scientists, you'd conclude that science is an incredibly dangerous profession, and that wearing a lab coat virtually guarantees your death in a weird and/or wonderful way, in ninety minutes or less... Sigh!

Raised By Republicans said...

Not quite Pombat,

"So what you're really saying is that in Aus, there's a one in twenty chance this is the person?..."

Let's say there are 20 people in Australia who would be a "match" for some piece of DNA evidence in a crime in Sydney. First, the police would have to estimate the likelihood of all 20 of the Aussies in question living in Sydney (let's say .25). Then they'd have to estimate the probability that those 5 people would be near the scene of the crime. In that circumstance, they may have other evidence that suggests that one person had motive to be there.

So really the calculation is much more restrictive than 1 in 20. And could even be more restrictive than 1 in a million.

The Law Talking Guy said...

Doesn't "one in a million" actually mean a particularly low probability? It doesn't mean that every millionth is a match. It means that the likelihood that the DNA test is inaccurate is 1 in a million.

The reasoning that "oh, I have reasonable doubt about guilt because there could be 300 such people in America given a 1 in a million chance" - this is very dangerous.

Right now we are having a "CSI effect" problem in American criminal juries. Conditioned by TV to expect expert high-tech scientific evidence and 100% perfect matches, juries often have a hard time convicting on eyewitness testimony and low-tech science.

The Law Talking Guy said...

If the authors think either interpretation is acceptable - i.e., that the children are as likely observing mismatch as probability - then the article in the Times was completely off base.

I am surprised any mathematician on this blog would expect children to have an intuitive grasp of probability. Adults sure don't, beyond the very simplest items. It is well known through research on what is often now called mental accounting or behavioral economics (or some branch of psychology) that individuals tend, for example, to treat a 99,999 to 1 chance of a successful lifesaving operation differently from hearing about a 1 in 100,000 chance of death.

Similarly, adults will believe that the statement "4 out of 5 dentists recommend this toothpaste" and "1 in 5 dentists would not put that crap in theirm ouths" are contradictory statements.

The Law Talking Guy said...

Of course any time I talk about evidence, I feel obliged to recount Ambrose Bierce's description:

We have no admissible evidence that there ever was such a person as Caesar or Napoleon, since that is all hearsay. But the courts may take judicial notice of the records of other courts, so the evidence that there were witches is is indisputable.

Dr. Strangelove said...

"I am surprised any mathematician on this blog would expect children to have an intuitive grasp of probability. Adults sure don't, beyond the very simplest items."

Yes, but we are only talking about those very simplest terms here.

Pombat said...

Wow! So that's how the police do it! Thanks RbR!

I was talking about just the one in a million stat, and how easily people will believe it. And yep LTG, it could also be translated that way around - it depends on the context in which the one in a million is described/presented.

As Dr.S says - we're talking about the simplest terms here. The thing is, we all have an intuitive grasp of things like probability, and the ability to learn about them properly. Sadly though, a large number of people never do, because they're taught maths so badly when they're younger that they're switched off to it / anything that looks remotely mathsy once older (and by older I don't mean that old - 9, 10yrs old maybe), and simply refuse to grasp even the simplest concepts, on the basis that "I'm no good at maths" / "I don't like maths" / "maths is too hard". Which is so incredibly frustrating, because no matter how good a teacher you put in front of them, they just won't learn :-(

Dr. Strangelove said...

The English have an especially acute problem with this, Pombat, because they have many "maths" to learn. Fortunately we only have to learn one "math" over here :-)

Dr. Strangelove said...

That last comment was supposed to be a joke, by the way. (I am not actually suggesting the English have any worse time of it than anyone else.)

Pombat said...

No need for explanation Dr.S - I took it as the joke it was intended as (and smiled, for the record) - you know I'm not overly touchy about that kinda stuff :-) Or maybe it's just that I like language-related humour (all the grammar nerds say "wuh wuh". I watch too much Daily Show).

The Law Talking Guy said...

I really think the "one in a million stat" does NOT mean "every millionth should be a match" but rather "this test has a one in a million chance of being wrong."
Do it twice, and you might increase the certainty dramatically.

Dr. Strangelove said...

Here's how the test works. The lab compares short genetic sequences at about thirteen standard chromosomal sites in each sample. These thirteen or so genetic "markers" are used for DNA comparison because the sequences are statistically independent and easy to distinguish since only a handful of genetic sequences are possible at each of these sites for humans. The essence of the probability calculation is straightforward: If each genetic marker consisted of a single GATC base only, then the chance of a coincidental match would be 1 in 4^13 (67 million). There are usually more than four options for each genetic marker, however, so the actual chance that all 13 sequences would match is more like 1 in 100 billion.

Thus as a practical matter, DNA evidence is as close to certainty as one can get in this world. It may only refuted by showing that the incorrect sample was tested, or that the sample was contaminated, or that the laboratory test was botched... Or that one's identical twin did it ;-)

Dr. Strangelove said...

So just to sum up, DNA matching is not a single test but really two tests (one for each sample) whose results are then compared. Repeating the comparison gains you nothing. And once you see how the test works, it is clear that Pombat's interpretation of the "one-in-a-million" stat is correct.

It is worth emphasizing, perhaps, that laboratories compare only short sequences, instead of the entire DNA pattern, because that would be far too difficult. Indeed, the first time anyone sequenced the entire DNA sample of a person it took several years and we called it the "Human Genome Project."

The Law Talking Guy said...

Dr.S.- Let me see if I understand. IF every person's DNA is different, then the match is 1 in 100 billion. But what if some people (other than identical twins) have large segments of identical DNA. I'm not saying this is true, of course. But isn't it true that the test, then, depends on this not being the case?

Dr. Strangelove said...

These thirteen or so genetic markers have been shown to be almost completely statistically independent in humans. But as you rightly observe, the high confidence of the DNA match hinges completely on that statistical independence. It remains theoretically possible that these genetic markers might be more highly correlated in some small subgroup (say, the residents of some Romanian village).

Fortunately, using DNA to establish innocence does not suffer even from this potential minor flaw. If two samples are found not to match, and the laboratory work was not contaminated, then the statistics are irrelevant: there is simply no match.

Bob said...

Yow. I ignore the comments for weeks, and three or four interesting topics come up.

I wrote a comment, but it was too long, so I posted over here. Here is a dramatically abridged version, in an attempt to get you interested enough to click over:

A) you may be interested in the Iowa Gambling Task experiments, one observation of which is (I'm paraphrasing here, so read some real research if you want to see what the studies really claim) that people can develop a sense of a probability from a smallish sample, _before_ they consciously recognize that they have such a sense. (Paraphrase 2: drawing from a deck of possible rewards/penalties, someone will start reacting negatively to a "bad" deck before they consciously decide it's a "bad" deck.)

B) [not actually chronological] LTG says "I am surprised any mathematician on this blog would expect children to have an intuitive grasp of probability. Adults sure don't...It is well known...that individuals treat a 99,999 to 1 chance...differently from...a 1 in 100,000 chance..."

Mathematicians (especially in education) think about what is and isn't intuitive mathematics a great deal, because what _seems_ intuitive to us is clearly not to our students.
So we talk about "number sense" and "proportional reasoning" and "intuitive probability".

Thus, while mathematicians cannot help but see that most adults' sense of probability is _solely_ intuitive, and unconnected to any attempts at probabilistic abstraction or language, and (therefore?) usually lacks _nuance_, we cling to the belief (well justified, really) that there is in fact an intuition there.

C) I took Pombat's examples to just be illustrations of the way probabilities are bandied about in misleading ways, and not a claim about DNA evidence per se. For more examples of this and other grievous misquantification, John Allen Paulos' Innumeracy (and/or A Mathematician Reads the Newspaper) are excellent reading.

D) [cut for length]

E) If anyone wants to talk more about math, probability, and math education, I'm very keen on it -- you could say it's my job. Email me!

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