Finishing your vegies may still not get you dessert

Wish I had captured this on video. It is absolutely priceless!

Mother: Jeff, if you don't finish your food you won't go with us. [pause] Jeff Anderson! Did you hear what I said?!

My 4-yr old nephew finally takes his eyes off of the TV and turns toward his mom: Yes mommy. If I don't finish my food I can't go with you. If I finish my food I can go.

Classic logical fallacy from the mouth of babes! And let's not pat ourselves on the back because we all make this mistake just about every time.

Let
A = you don't do 100 push-ups
B = you won't be allowed to take your meal

The drill sergeant yells, If A then B. Does it follow then that if not A then not B? That if you complete the number of push-ups you'll finally be able to fill your empty belly?

No, of course not. To say so would be to commit the fallacy of inversion.

Here's Deborah Bennett in her Logic Made Easy giving an example which played right before my eyes this morning. Talk about textbook case! :)

As a part of natural language and conversation, 'if p then q' conditional statements that are promises and threats commonly invite the inference, 'if not p then not q.' Take, for example, the promise, 'If you eat your dinner, you may have dessert.' We would probably agree that this promise invites the threat of the inverse, 'If you don't eat your dinner, you won't have dessert.' But the statement says no such thing.... It is curious that even though the common interpretation of this parental warning has no basis in logic, both the parent and the child (and probably all of us) understand the intention of the statement.

I love the example given by Jonathan Baron [who the heck is he?!]. Presented with the threat, 'If you don't shut us I'll scream,' we would all be surprised if the speaker screamed anyway after you shut up. The speaker probably intends that your interpretation of this conditional to include its inverse, 'If you don't shut up, I'll scream and if you do shut up, I won't scream.'" This interpretation may be illogical but it isn't unreasonable; it makes perfect sense. In ordinary discourse, we make practical assumptions about what a person likely means.

---

Bennett, Deborah J.. Logic Made Easy: How to Know When Language Deceives You. New York: W.W. Norton & Co., 2004. p.115-116

Screening mammography is not recommended

A relative recently had a screening mammography. She was asymptomatic, therefore, this was screening not diagnostic. Moreover, she's 45 years old, thus age-wise, not (yet) in a high risk group. I sugested she not to go for mammography again unless she starts noticing symptoms, not least because being irradiated poses risks and can, ironically, induce mutations that can lead to cancers.

A simple computation provides a perspective on the unreliability of screening mammography. At any given time around 100 out of every 10,000 women have breast cancer, while 10,000 - 100 = 9,900 don't. Out of 100 who have cancer and who undergo mammography, the test correctly detects 95 of them and will miss 5, incorrectly showing those women as tumor-free. Out of the 9,900 who don't have cancer and who undergo mammography 9,603 will correctly test negative, but 297 will erroneously be interpreted as having a tumor.

So if we screen 10,000 women, how many will test positive? 95 + 297 = 392. And what proportion of those women would falsely be shown to have cancer? 297 / 392 = 75.8%! That's right. 3 out of every 4 women who test positive actually don't have cancer.

Now imagine the anxiety upon getting the results and being told that there is a chance you have cancer, and the need for further testing to rule out (or in) that possibility.

But what if the test comes out negative? What's the likelihood the test failed to detect the tumor? The total number of women who test negative is 5 + 9,603 = 9,608. The proportion of missed tumors = 5 / 9,608 = 0.05%, or close to 1 out of 2,000 women screened. Mammography, therefore, can very accurately rule out breast cancer (which, however, does not indicate continued absence).

----

In the latest Cochrane systematic review of the harms and benefits of screening mammography the authors conclude:

"[S]creening will result in some women getting a cancer diagnosis even though their cancer would not have led to death or sickness. Currently, it is not possible to tell which women these are, and they are therefore likely to have breasts or lumps removed and to receive radiotherapy unnecessarily. If we assume that screening reduces breast cancer mortality by 15% after 13 years of follow-up and that overdiagnosis and overtreatment is at 30%, it means that for every 2000 women invited for screening throughout 10 years, one will avoid dying of breast cancer and 10 healthy women, who would not have been diagnosed if there had not been screening, will be treated unnecessarily. Furthermore, more than 200 women will experience important psychological distress including anxiety and uncertainty for years because of false positive findings."

----

Notes:

Prevalence rate for breast cancer is difficult to come by. Estimate for the UK a decade ago was 1%. It must be noted that the frequency of cancers is higher in older people. 

I've taken the most optimistic sensitivity and specificity figures for mammography: 95% and 97%, respectively. Therefore, real false positive rate and false discovery rate are likely to be worse. For instance, specificity of mammograms can be as low as 90%. With sensitivity unchanged, If specificity were 95% the false discovery rate climbs to 83.9%, and with a specificity of 90% FDR shoots up to 91.2%.