Thursday, February 25, 2010

Disease Screening & Base Rate Fallacy

Definition
The base rate fallacy refers to the neglect of prior probability of the evidence that supports the conditional probability of a hypothesis.
(Based on Wikipedia)


(Photo: mayaevening at Flickr)
Commentary
A recent example is the controversy about breast cancer screenings.

Imagine that about 1% of women (1 in 100) have breast cancer. You have a diagnostic test that correctly detects cancer 85% of the time (i.e. if the test is given to 100 women with cancer, it catches 85, but misses 15 of them).

Also, the test sometimes incorrectly detects cancer (when none is present) about 10% of the time (i.e. if the test is given to 100 women without cancer, it accidentally tells 10 of them they have cancer, but correctly tells the other 90 they don't have cancer).

Now the tricky bit: Imagine we give the test to 1,000 women in the population. If the test says a women has cancer, what is the probability she actually has cancer?

This question is hard for many people (including doctors!) because it's hard to make the trade-offs in our head about whether or not the test is accurate for this particular woman. Here's how you would do the calculation correctly:
  1. Based on the rate of cancer in the population (1%), how many of the 1,000 women tested do we expect to have cancer?
    Answer: About 10.

  2. Of those 10 who have cancer, about 9 will be told they have cancer, and 1 will missed. (Recall, the test only catches 85% of cancers.)

  3. Now of the remaining 990 women who don't have cancer, about 99 of them will be told they have cancer (10% false-alarm) while the rest (891) will be correctly told they don't have cancer.

  4. So how many women are told they have cancer?
    Answer: 9 + 99 = 108.

  5. How many of those women actually have cancer?
    Answer: Just the 9.

  6. So if you're told you have cancer, what's the chance you actually have cancer?
    Answer: 9 / 108 = 8.3%
Pretty strange, right? What about the people who are told they don't have cancer? What's the probability you actually do have cancer?
  1. How many women are told they don't have cancer?
    Answer: 1 + 891 = 892.

  2. How many of those actually have cancer?
    Answer: Just the 1.

  3. So if you're told you don't have cancer, what's the chance that you actually do have cancer?
    Answer: 1 / 892 = 0.1%
That means that it's pretty unlikely for you to have cancer if the test says you don't.

The reason this occurs is because the number of women who have breast cancer to begin with is not that high (10 of 1,000). Therefore, the mistakes the test makes start to matter when applied to the entire population.

Naturally, this has policy implications: if you test more and more people, a large percentage of people will be told they have cancer when they don't-- leading to more invasive testing that has other real side-effects. The trick is either to try to test a high-risk subpopulation (where the prevalence rate is higher) or to improve the test by reducing its false-positive rate.

See Also

Wednesday, February 24, 2010

Review: The Autonomy of Mathematical Knowledge

(Image: Amazon)
Review
Curtis Franks' (full disclosure: he's is a friend of mine) PhD-turned-book Autonomy of Mathematics: Hilbert's Program Revisited is an exciting new look at an overlooked aspect of early twentieth century mathematics. Franks' writing is crisp and engaging, as he paints the picture of a man and his philosophy that so many have spurned.


The face of a brilliant mathematician or of a sun-hat enthusiast.
(Photo: Wikimedia)


Commentary
In the 1920's, Hilbert launched a program that was ostensibly aimed at solving the foundational crisis of mathematics-- the issues of paradoxes (e.g., Russell's paradox). The traditional understanding is that Hilbert's program failed because Gödel's incompleteness theorems threw a monkey wrench into any sufficiently sophisticated system that tried to prove itself.

Franks' thesis is that this is a narrow understanding of Hilbert's goals. While Gödel's results did complicate certain endeavors, Franks' suggests that Hilbert was really trying to take back mathematics. That is, certain other endeavors were trying to resolve the foundational crisis by rooting mathematics in some other discipline (e.g., philosophy). Hilbert's goal was to keep mathematics strictly within the realm of mathematics-- a unique feature of the discipline.

The Autonomy of Mathematical Knowledge is admittedly not for everyone (perhaps not even for me)-- about 20% of the book involves theorems I faithfully assume describe what the surrounding text tells me they do. Yet, about 80% of the book is eminently accessible-- the historical context, the epistemic issues, and the attempt to reconstruct a neglected approach combine for a great read.

See Also

Tuesday, February 23, 2010

The Arnell Group & PepsiCo

Summary
The Arnell Group is a design company famous for its work with DKNY and Banana Republic (among many others). In the last several years, several of its high-profile projects with PepsiCo have met with derision and consumer outrage.


Blank paper does not count towards the page count of a report.
(Photo: jacqueline-w at Flickr)


Commentary
In 2008, an Arnell Group design document called "Breathtaking" was leaked to the public. The 27-page document was the point of much ridicule as its contents attempt to explain how the proposed Pepsi logo was the ultimate realization of thousands of years of good design. Gems include:
  • conformance with the golden ratio (frequently used in the world of design) [p. 18] and detailed instructions on how to construct the logo using the golden ratio spiral [p. 19]
  • comparison of the logo to the Earth's geodynamo and magnetic fields [p. 21]
  • a 5000+ year timeline of all the "authentic Constitution of Design" (culminating with the Pepsi "Breathtaking" logo) [p. 6]
Apparently, the new logo fulfills the destiny of good design in addition to making Pepsi's logo look similar to other modern logos.

In 2009, the Arnell Group orchestrated the Tropicana orange juice package redesign. (Tropicana is owned by PepsiCo). The design met with amazing backlash from consumers (supposedly, there was a 20% drop in sales) and the existing iconic straw-in-orange design was reinstated.

Meta
What was your reaction when Pepsi changed its logo? What about Tropicana? Other companies?

See Also

Sunday, February 21, 2010

Reader Question: What does the furious brushing in curling accomplish?

This post is a response to a reader question. Please email your questions to metaist.blog@gmail.com. Not all questions will be answered.

Question
An anonymous reader asks: What does the furious brushing in curling accomplish?


From the little I've read, curling is a combination of darts and marbles. Sort of.
(Photo: andreassolberg on Flickr)


Summary
Curling is a 400+ year-old sport about which I know very little. Wikipedia saves the day again.

Explanation
Curling is played on level ice. However, droplets of water called pebbles are added which cause the stone (pictured above) to curl when it travels across the ice. According to Wikipedia, sweeping accomplishes two goals: to increase the distance the stone travels, and to change the amount of curl. The strategy of sweeping seems to involve figuring out when to sweep and how hard to sweep (apparently that has an effect on the amount of moisture created).

Overall, the point of sweeping seems to be to influence the curl of the stone by wearing down the pebbles.

Meta
Do you know more about curling than I do? If so, please post a comment so that I can update this post appropriately.

See Also

Sunday, February 14, 2010

Shoes & Gait

Summary
A study in the journal The Foot showed that wearing shoes isn't that good for your feet.


(Video: YouTube)
Meta
Suppose you were shown the overwhelming evidence of the harm of shoes on gait-- how would you respond?

See Also

Zotero

Summary
Zotero is an excellent way to organize, save, and cite your research.

In other words, magic.


(Photo: libraryriot at Flickr)

Commentary
Zotero is plugin for Firefox that provides inline citation and bibliography for hundreds of different styles including MLA, APA, IEEE and many more. Zotero also provides plugins for OpenOffice.org and Microsoft Word. One thing I particularly like is its ability to store PDFs and extract metadata; it saves so much time and lets you focus on continuing to do your research.

Meta
What tools do you use to organize your research?

See Also (Updated 2010-02-24)
  • Zotero for the plugin and related resources.

Friday, February 12, 2010

Feedback Friday: February 2010

Summary
Feedback Friday is a week late, and there hasn't been a post in two weeks; The Metaist is in ad-hoc mode (but not for long).


Things got kinda slow, didn't they. No worries, we'll bounce back.
(Photo: streamishmc on Flickr)

Feedback
As always, this is your place to voice your opinion on things you'd like discussed in future posts.