Steve Kass

Early this morning, Wikileaks began posting alphanumeric pager messages from four carriers (Arch, Metrocall, Skytel, and Weblink_B) that were intercepted during a 24-hour period beginning early on September 11, 2001. Alphanumeric pager messages are unencrypted, and, like communications over a public 802.11 wireless network, they’re skimmable with the right (and not exotic) software and hardware.

• “Due to today’s tragic events, it makes sense to cut back wherever feasible on payroll. Expect a very light business day. Please call all stores and review payroll issues”
• “RING ALL CHICAGO AIPORTS AND EVERY MAJOR BUILDING DOWNTOWN. BUSH IS DOING A SPEECH.  THIS IS SERIOUS POOH..”
• “Holy crap, are you watching the news.”
• “I hope you have gone home by now. The BoA tower and space needle here are closed. I suspect tall buildings across the country will be closed. Take care my love.-cb”

This might be the most interesting public data mine since the AOL breach. The total volume is far less, but unlike the AOL data, this data hasn’t been anonymized. There are full names, phone numbers, and other identifying information in the mix.

Need to run a one-sample t-test or z-test? Here’s a little calculator written in Excel to help you out.

Need to calculate z-scores, percentiles, or scores based on a normal distribution? Here’s little calculator for that, too.

The following subheadline on the Scientific American website caught my eye today (and not only because of the missing period):

New research makes the case for hard tests, and suggests an unusual technique that anyone can use to learn

I may be a bit thick, because neither the article nor the research paper it mentioned suggested any unusual technique to me. But this was better than my last wild goose chase reading episode, when I vainly sought a footnote on a cereal box (there was a dagger: †, but no footnote. Can you believe that?).

Henry Roediger and Bridgid Finn, the Scientific American article’s authors, write that researchers Kornell, Hays, and Bjork found that “learning becomes better if conditions are arranged so that students make errors.” There’s that pesky word “better.” Better than what? The eternal unanswered question. My guess is that Scientific American is reporting that Kornell et al. have found that learning under a) conditions arranged so that students make errors is better than learning under b) conditions arranged so that students do not make errors. In other words, that the researchers found errorful learning to be better than errorless learning. Not that it’s a bad article, but it would be nice if Roediger and Finn had stated what they’re reporting a bit more clearly. (This is why I give writing assignments to my statistics students. By the end of the semester, they better learn not to use adjectives like better without answering “Better than what?”.)

Anyway, Kornell et al. do mention errorless learning in their paper, recently published in the Journal of Experimental Psychology: Learning, Memory and Cognition® (yes, the name of the journal is a registered trademark), but they don’t study it. The abstract notes that they examine the question of “what happens when one cannot answer a test question—does an unsuccessful retrieval attempt impede future learning or enhance it?” Kornell et al. didn’t exactly examine this question either, because they didn’t (and possibly couldn’t) isolate what part of the learning in their scenario was “future” learning. In addition, they only studied learning after wrong answers, so one must be careful not to assume their research sheds light on getting test questions wrong vs. getting them right. (Suppose a researcher reported that “Student learning among African-Americans is enhanced when they are given test questions they cannot answer.” If the researcher only studied African-Americans and made no comparison to other populations, the reported finding might easily be misinterpreted.)

What Kornell et al. did was compare two scenarios for learning previously unknown information. One scenario was unsuccessful retrieval attempts (the students were asked to provide the not-yet-learned information as answers to test questions, and they answered incorrectly). In this scenario, the retrieval attempt was followed by feedback that included a brief presentation of the new information (i.e., the correct test question answer). The second scenario was a longer-lasting presentation of the new information with no retrieval attempt (the students were not asked to answer a test question, and it’s unclear in some of the experiments whether the students knew what kind of test question they would later be asked). Not surprisingly, unsuccessful retrieval attempts enhanced learning (as measured by scores on a test containing questions like those in the retrieval attempt), when compared to presentation of new information with no retrieval attempts. Despite the Scientific American article’s subheadline, this research makes no case that “hard tests” are better for learning than non-hard tests. They may be, but this research doesn’t help us figure it out. The research does support the value of tests, hard or not-hard, so long as there’s feedback with the right answer.

Almost every semester, I use the AOL Breach data as a point of departure for something in at least one of my classes. The data is fascinating. Most data is fascinating, but this data is particularly so: at once shocking, funny, creepy, poignant, sad, frightening, noble, ignoble, shrewd, and lewd. It’s also rich in the way data can be rich. It’s completeness—for a sample of several thousand AOL accounts, it includes the complete account search history during March, April, and May of 2006—which includes timestamped search strings and the result rank and destination of clicks-through, makes it ripe for discovering all sorts of patterns of human thought and behavior.

It’s AOL data week in one of my classes now. This morning, I proposed several nontrivial questions about the data that could be answered with SQL queries. We looked at the results and discussed what they might say about the unwitting study subjects. Then I asked my students to suggest some questions of their own. What are the typical time-of-day and day-of-week patterns of an individual AOL customer’s searches? Are there identifiable differences in the patterns (and by extension in the sleep, social, and perhaps employment or school behavior) of people whose searches included, say, “britney”? For what kinds of searches do users most often click through several pages of results? And so on.

One of my students suggested an excellent simple question. What are the most common searches of the form “how to …”? Out of millions of queries in the AOL data, there were many thousands of “how to … ?” searches. The most frequent was “how to tie a tie,” requested 92 times by a total of 47 distinct users. The rest of the top ten (in terms of most distinct users asking the question) were how to write a resume, gain weight, have sex, get pregnant, write a book, write a bibliography, start a business, lose weight, and make money, each sought by a dozen or more different people. AOL converted the queries to lower case and removed much of the punctuation, but they didn’t correct spelling. Consequently, how to masterbate and how to masturbate appear separately at ranks 49 and 51 respectively. The question would have nearly hit the top 10 without the misspellings.

Here’s a PDF file of the top 1000 “how to” queries submitted through AOL explorer by a sample of AOL users in the spring of 2006. You can probably guess that it’s not safe for work. Although there are no pictures, plenty of sex, drugs, and gambling is spelled out, and there are more than a few questions likely to offend in one way or another. Have a look.

A colleague recently drew my attention to this CNN.com article, titled “Confident students do worse in math; bad news for U.S.”

It’s interesting, I guess, but the analysis is flawed.

The article reports the results of a Brookings Institute study based on the 2003 Trends in International Science and Mathematics Study (TIMSS).

It’s the culture, stupid.

Where to start? First off, the countries with the best average math scores were East Asian countries. That confirms other studies and general perceptions, and it didn’t surprise anyone. Obviously, then, anything that correlates with East Asianness (straightness of hair, size of epicanthal folds, or facility in Chinese or Japanese) will also correlate with math scores when East Asian country-wide averages are compared with other country averages. One trait that correlates with East Asianness is self-effacement. Self-effacement tends to be highly valued in countries like Taiwan and Japan, but praising one’s self is not, and there was a survey question that measured self-effacement: “How strongly do you agree with the statement ‘I usually do well in mathematics.’”

It shouldn’t have surprised anyone that students in East Asian countries were less likely to answer “strongly agree” when asked how much they agree with the statement “I usually do well in mathematics,” so it shouldn’t have surprised anyone that country-wide averages on this survey question correlated (inversely) with mathematical ability. The correlation is simply explained by culture and the known difference between countries in mathematical ability.

Hurting the chances for a fair airing in the press (if there is ever any chance), the Brookings analysts repeatedly confuse expressed self-confidence with true self-confidence, too: “In the TIMSS data, when one looks at the math scores of students within each country, those who express confidence in their own math abilities do indeed score higher than those lacking in confidence,” and “The world’s most confident eighth graders are found…”, and “students in [East Asian] countries do not believe that they do very well in math.” Combine this with the fact that while the Brookings folks do notice the culture question, they dance around it enough to let reporters to come away with conclusions that are a good six fallacious leaps away from the data and statistics, like “Happy, confident students do worse in math” headlining an article by Association Press education writer Ben Feller. Nothing in the data suggests that confidence in mathematical ability is inversely related to actual mathematical ability at the level of individuals, but the headline gives that impression, strongly.

Another problem with the study is that it commits the ecological fallacy. It speculates about how confidence and ability are related in individual students from country-wide aggregate data, and the press drag these wrong conclusions in the wrong direction. The aggregate data is screwy to begin with, given the very strange list of countries surveyed. Many have small populations: Jordan, Israel, Bahrain, Cyprus, Latvia, the Palestinian Authority, Moldova, Lithuania, Scotland, Norway, Flemish Belgium, Botswana, Macedonia, and Serbia, and a few have large populations: Russian Federation, Japan, Indonesia, Malaysia, the United States, and the Philippines, but each country’s average mathematics score counted the same in the statistical analysis to which the press is paying the most attention. The country list was hardly comprehensive, either. Chile was the only American country on the list besides the U.S.

A regression weighted by population would have been a little better, but drawing any conclusions about individual students from country-wide averages is invalid. Here are a couple of good articles about this fallacy: (link) (link).

The TIMSS data contains plenty of useful information to support a conclusion opposite to the one reported. The data showed that all other things being equal (that is, among students within any one individual country) higher student confidence (which, ceteris paribus, should now correlate with expressed confidence) in math ability tended to be associated with higher math ability. The Brookings folks observed this, and they called it paradoxical: “So an interesting paradox emerges from the international data on student confidence and achievement. The relationships are the opposite depending on whether within-nation or between-nation data are examined.”

When Brookings writes “The international evidence makes at least a prima facie case that self-confidence, liking the subject, and relevance are not essential for mastering mathematics at high levels,” it’s easy to think they are suggesting that the ecological fallacy analysis is telling us something, and they are diminishing the different conclusions supported by the ceteris paribus analysis. And the reporters take the hook. I wonder how good the press corps and the Brookings researchers think they are at statistics.