Steve Kass

In today’s number news (State-by-state cremation rates in U.S.), we learn that “slightly more than a third of all persons who died in 2006 were cremated, according to the Cremation Association of North America.” Happily, the article contained the raw data, but only as an alphabetical-by-state table of numbers.

Here’s an illumination, as MapPoint is my amanuensis. Click to embiggen.

Explanation: Pie areas are proportional to the number of deaths; the yellow slice is cremations, the red non-cremations.

Pies for our nation’s two newest states are not shown. Alaska’s looks like a two-thirds size Vermont pie; Hawai’i’s looks like a one-third size Oregon pie.

The Journal of the National Cancer Institute just published the results of a large study evaluating the survival benefit of contralateral prophylactic mastectomy in the surgical treatment of breast cancer. In some mastectomy patients, breast cancer will reoccur in the remaining breast, and that risk can be reduced (but not eliminated) by removing the non-cancerous breast .

Contralateral prophylactic mastectomy lowers post-mastectomy five-year death rate by 30%

The five-year death rate after mastectomy was 11.5% for women who had both breasts removed. It was 16.3% for those who only had the cancerous breast removed. Adding a contralateral prophylactic mastectomy to the original surgery therefore reduced the five-year death rate from 16.3% to 11.5%. Almost a third fewer mastectomy patients died within 5 years when the had chosen to remove the second (healthy) breast, compared to mastectomy patients who had not chosen to remove the second breast. The bilateral mastectomy decreased the 5-year death rate by 29.4%.

This strikes me as a significant benefit. Suppose I have breast cancer and need a mastectomy. I can choose a single mastectomy and have a one-in-6 chance of dying in five years, or I can choose a double mastectomy and have a one-in-9 chance of dying in five years. One-in-9 sounds quite a bit better to me. If 100,000 women with unilateral cancer need mastectomies, performing 100,000 double mastectomies instead of 100,000 unilateral mastectomies will reduce the number of deaths in the first five years from 16,300 to 11,500. About 4,800 fewer women will die within five years.

Contralateral prophylactic mastectomy benefits only 5% of mastectomy patients.

The reporting of this study takes a very different viewpoint. It compares the survival rate, not the death rate, and notes that the bilateral prophylactic mastectomy increases the survival rate from 83.7% to 88.5%, “a difference of less than 5%.” Five percent sounds like a small number, but 5,000 lives saved sounds like a large number.

Point of view

Both statements (lowers by 30%; benefits only 5%) are the same. Only the intent to communicate is different. Whether prophylactic mastectomy is good practice depends not only on the change in five-year survival rate, and I don’t have more information.. For example, how does a double mastectomy (which for 95,000 of the women will not change the five-year death rate outcome) affect a woman’s well-being and general health over time? What is the cost to save these 5,000 lives, and how will the disparity of death rates change more than five years after surgery?

Good news

There is a good piece of news in the study: The study data identified a subgroup of women for whom double mastectomy had an even greater benefit: women 50 and younger with early stage estrogen receptor negative cancer. Removing the second breast had no benefit for women 60 and older, and the benefit for women in their 50s was uncertain, presumably because of the small number of bilateral mastectomies in the sample.

The journalists writing about this study generally downplayed the benefits. It would have been better for them to downplay the benefits on most of the women, but hype the discovery that there is a subgroup of women who might get a substantial benefit from this procedure. If you read the study, or find better summaries of it, you may find that this study can help patients and doctors make wise treatment choices (contralateral mastectomy sometimes among them). With luck and more studies like this, prophylactic mastectomies might in the future go only to those women whom they might help. With even more luck, we’ll improve our diagnosis and prevention of breast cancer and the number of mastectomies will go down.

In July, I griped here about misleading news reports of a higher crash risk among texting truckers than among non-texting ones. I pointed out that the research quoted had not shown an increased crash risk, and in fact observed fewer crashes (zero, in fact) among the texting truckers. The data might suggest a decreased crash risk among texting truckers, I noted. The reason for the confusion was that crashes and near-crashes (which included sudden and possibly crash-preventing maneuvers) hadn’t been separated in the calculations. Nevertheless, the increased crash-or-near-crash risk was widely reported as an increased crash risk. I wondered whether more near crashes might in fact be a positive thing; those who occasionally swerve suddenly might be paying more attention to the road than those who rarely do.

Today, researchers and others are expressing surprise that just-in real crash data doesn’t support what they don’t realize the earlier research didn’t show in the first place. For example:

• Results of Study on Cellphone Use Surprise Researchers
• Cell phone bans do little to reduce crashes, study finds

If researchers (or journalists) are surprised by today’s news, they probably didn’t examine the research very closely. They may have believed the misleading headlines instead.

Disclaimers: I haven’t read all the research, and some studies might have in fact shown an increased crash risk, unlike the one I mentioned. That might be reason for real surprise. In addition, today’s data doesn’t specifically show that less phone use means no fewer accidents, because laws don’t always change behavior. But some of the reports I read did suggest that there was less phone use, yet no lower crash rate, in places that instituted bans.

Ironically, the crash data out today could make things worse. If drivers think phone use while driving is not as unsafe as previously thought, they might be less careful when using a phone while driving, and phone-related crashes might increase. Common sense suggests that multitasking requires greater concentration. If you do use a phone while driving, drive with even more care than usual. In other words, this might be one situation where being wary might have benefits, even when its not warranted by the facts – especially because being somewhat over-cautious while driving has no serious down side, it seems to me. (It’s not like it infringes upon hundreds of millions of people’s civil rights, like acting on other unwarranted fears can and does…)

A headline in today’s Washington Post is “Fewer Americans think Obama has advanced race relations, poll shows.” My statistics students know what I have to say about that: “fewer than what?” Yes, headlines are generally false, and the Post reports the statistics somewhat more carefully in the article. “On the eve of President Obama’s inauguration a year ago, nearly six in 10 Americans said his presidency would advance cross-racial ties. Now, about four in 10 say it has done so.”

According to the actual poll results, a year ago, 60 percent of Americans answered “help” to the question “Do you think Obama’s presidency will do more to (help) or more to (hurt) race relations in this country, or not make much of a difference?” Last week, 40 percent of Americans answered “help” TO A DIFFERENT QUESTION: “Do you think Obama’s presidency has done more to (help) or more to (hurt) race relations in this country, or has it not made much of a difference?”

Big deal. Not a story. Fewer Americans today also think they have died than thought last year that they would die.

Original title: Over 90% of Research Studies Make Me Want to Scream (P < 3E-12).

Shania Twain is in the news today. No, her new album still isn’t out, but her face is in the spotlight. It turns out someone “applied” the latest “research” to “determine” that she has the perfect face, “scientifically” speaking. The distance between her eyes and mouth are precisely 36% of the length of her face, and her interocular distance is exactly 46% of its width. These proportions, according to an article in press at Vision Research, are universally optimal (among low-resolution, mostly Photoshopped images of a few white women).

Garbage. Poppycock. Nonsense. Balderdash. Crap, crap, crap of a research paper, right from sentence 1: “Humans prefer attractive faces over unattractive ones.”

But you came here for the pictures. (more…)

Starting today, funny math is moving to a new and better home, www.lolmath.com. Vulpigeration and other serious number-related topics will stay here.

If you read my last post, you know I’ve been looking at some very fishy survey data from Strategic Visions, LLC. The data seems to stink no matter how anyone looks at it, and mathematicians, statisticians, and programmers have been looking hard and every which way. Instead of throwing yet another heavy mathematical brick at the poor numbers, Let’s see how it stands up to a feather.

You don’t have to read many poll or survey results to be familiar with the phrase “totals may not equal 100% because of rounding.”

So guess what? The numbers in Strategic Visions’ results all add to exactly 100. Oops. That’s a huge red flag. Huge, like big enough to wrap a planet in.

Ok, I didn’t check all their polls. Just the most recent 73, which is how many I checked before stopping. For the record, I didn’t stop because poll #74 added to 99 or 101. That would have been statistical misconduct on my part. I didn’t look at poll #74 or any others, because I wanted to write this up. (If anyone checks further, let me know and I’ll post an update.)

If Strategic Visions were not lying (or being extremely sloppy in a systematic way, which is their only hope of explaining this—see below), the chance the 73 most recent polls would add to 100 is less than 1 in 2,000. (That assumes intentional rounding to minimize 99s and 101s. For any pre-determined rounding rule, however, we’re talking 1 in 10,000,000 or worse. Maybe a real pollster can fill me in on what’s industry practice among those who aren’t lying.)

One-in-two-thousand-ish stuff happens all the time, but believe me, I didn’t find this on a data dredging excursion. I noticed that the first few poll results I saw added to exactly 100, I formulated a plausible hypothesis based on all the evidence at hand, I carried out an experiment, and I calculated the p-value.* Small enough to be incriminating in my home court, and I tend to be a benefit-of-the-doubt guy.

Just two more things. First, the possibility of systematic sloppiness:

The consistent adding-to-100 could be the result of a systematic error, as opposed to cheating. Of the possible excuses, this is the one I suggest SV choose if they decide not to come clean. Logically it can’t be distinguished from lying, and they can attribute it to a whipping boy like the web site designer. (This excuse doesn’t defend against the mountain of mathematical bricks I mentioned earlier, however.) They can say that they made a regrettable decision that to avoid the appearance of error, they calculated one percentage in each survey from the other percentages, not from the survey results. I won’t believe it for a minute (though if they show me their programs or confirm that some commercial product makes this error, I’ll reconsider), but it might get them out of hot water.

And second and last, an example and a bit of the math behind my calculations:

Most survey results are short lists of whole number percentages that express fractions to the nearest whole percentage point. Suppose 600 likely voters were polled in a tight race between Tintin and Dora the Explorer. Tintin had the support of 287 people, Dora was close behind with 286, and the rest of the 600 people surveyed (that would be 27 of them) said they weren’t sure. To the nearest whole percentages, that’s 48% for Tintin, 48% for Dora, and 5% undecided. The sum of the rounded percentages is 101%, and that’s due to honest mathematics, not fraud.

Let me skip some really fun mathematics and tell you that for survey questions that have three answers, the percentages add to 100 most, but by no means all of the time. Exactly how often they don’t add to 100 depends on several factors, two of which matter much at all in this case: the number of people surveyed, and how numbers ending in .5 are rounded to whole numbers. Strategic Visions, LLC’s usual survey sample size is 800, and even if ending-in-.5 numbers are rounded differently in each poll to avoid a sum of 99 or 101 when possible, three-choice result percentages should still add to 99 or 101 at least one time out of ten.

* Not to be taken as evidence of a non-Bayesian persuasion on my part. The frequentist approach seemed to me pretty straightforward and justifiable here, that’s all.

Today’s clicking (especially from fivethirtyeight.com) led me to two strikingly similar declamatory reports about high school student’s knowledge of civics, complete with chart-laden survey results.

“Arizona schools are failing at [a] core academic mission,” concludes this Goldwater Institute policy brief.

“Oklahoma schools are failing at a core academic mission,” announces this Oklahoma Council of Public Affairs article.

When asked to name the first president of the United States, only 26.5% of the Arizona high school students surveyed answered correctly. Only 49.6% could correctly name the two major political parties in the United States. An even smaller percentage of Oklahoma high school students gave correct answers to these and other questions from the U.S. citizenship test study guide. None of the thousands of students surveyed in either state answered all ten questions correctly.

The shocking thing is that these are garbage studies. Made-up numbers, probably. The acme of vulpigeration. Evil. Makes me sick. (Glad I coined the word, though.)

No way these are real studies. Danny Tarlow over at This Number Crunching Life has taken a mathematical hammer to the Oklahoma “study” quite effectively. (The blatant similarity of the Arizona “study” blows away any shred of possibility that the Oklahoma study is legit. I’d love to see Danny’s face when he sees the Arizona report.)

What’s frightening is that this kind of snake oil has far too good a chance of surviving as fact (which it isn’t) and influencing public policy.

The guilty parties? The Goldwater Institute, which as you might guess is a conservative “think” tank. The OCPA, which describes itself as “the flagship of the conservative movement in Oklahoma.” Matthew Ladner, the author of both reports, who is vice president of research for the Goldwater Institute. And last but not least, Strategic Vision, LLC, which Ladner says “conducted” the studies. In my opinion, the word is concocted. Read about them yourself.

[Updated with correct business name: Strategic Vision, LLC.]

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.

Kevin Underhill* emailed me today about the Schwarzenegger veto letter. Specifically, Kevin wondered whether “it might be possible to calculate the odds against this arrangement of letters being entirely random, as the Governor’s office has claimed.”

Kevin wasn’t the only one to wonder about the odds and contact a math professor.

Several hours after Kevin wrote me, SFweekly.com published “Odds Schwarzenegger’s ‘I F–k You’ Message Was Coincidental? About One in Two Billion, Says Math Prof.” SFweekly.com quoted Stephen Devlin, a mathematician at University of San Francisco, and Gregory McColm, a mathematician at the University of South Florida, and printed various tiny chances, including a) 1 in 10 million, b) 1 in 100 million, and c) 1 in 2 billion. These are (using rough estimates of initial-letter frequencies in English words) the chances a) that seven randomly selected lines of English text begin with f, u, c, k, y, o, and u, in that order, b) that eight randomly selected lines of English text begin with i, f, u, c, k, y, o, and u, in that order, and c) that the first letters of eight randomly selected lines of English text and two blank lines separating those lines into three groups appear in the sequence i, blank, f, u, c, k, blank, y, o, u. The chances of c) are 1 in 2.1 billion, by my calculation, but no matter—only the order of magnitude is of interest here.

Whether many visitors to SFweekly.com (well, male ones of any persuasion) read the piece is questionable. It appeared next to two photo links with more click appeal than “Blah Blah F–k Blah Blah Math Prof.”: Exotic Erotic Ball (image of exotically costumed ladies kissing) and San Franciso’s Hottest Chefs (image of boyish chef wearing five o’clock shadow and a uniform, smiling seductively).

Similar calculations appear with less distraction elsewhere.

For the record, even before making any calculations, I was convinced beyond a reasonable doubt that the governor meant full well to say “fuck you.” In a later blog post, however, I’ll try to explain that the unlikelihood of “fuck you” or “i fuck you” appearing in a letter isn’t in and of itself a smoking gun.

*I’ve e-known Kevin for a couple of months now, ever since I introduced myself with an unceremonious, if politely worded email to the effect of “your arithmetic is wrong.” More details here. It turns out to be a better way to meet people that you might think.

Kevin finds fortune at Shook, Hardy, and Bacon (shb.com, Alexa US traffic rank 292,302, with 132 sites linking in), and he finds fame at Lowering the Bar (www.loweringthebar.net, Alexa US traffic rank 59,588, with 145 sites linking in).