Boobies. There I had to say it. This is a post about boobies, and math, and consulting with experts before making too many claims.
In this click bait article that I found somehow searching on Google News for unrelated topics, I see that some "Medical Anthropologists" are claiming that Bras seem to cause breast cancer (not a new claim, their book came out in 1995, but their push against the scientific establishment is reignited I guess). At least part of this conclusion seems to be based on the observation from their PDF
Dressed To Kill described our 1991-93 Bra and Breast Cancer Study, examining the bra wearing habits and attitudes of about 4,700 American women, nearly half of whom had had breast cancer. The study results showed that wearing a bra over 12 hours daily dramatically increases breast cancer incidence. Bra-free women were shown to have about the same incidence of breast cancer as men, while those who wear a bra 18-24 hours daily have over 100 times greater incidence of breast cancer than do bra-free women. This link was 3-4 times greater than that between cigarettes and lung cancer!
They further claim "bras are the leading cause of breast cancer."
That's pretty shocking data! I mean really? Now, according to http://seer.cancer.gov/statfacts/html/breast.html there are about 2 Million women in the US living with breast cancer, and 12% overall will be diagnosed throughout their lives. There are around 150M women in the US overall. So
However, in our sample That's 50 times the background rate (ok 37.5 if you do the math precisely).
Doesn't it maybe seem plausible that in winnowing through the 1% of women living with breast cancer and are still alive, or even the 5 or 6 percent who have been diagnosed in the past but are still alive (figure half of women who are alive today who will at some point be diagnosed have already been diagnosed at this point) that maybe, just maybe they could have introduced a bias in whether or not their sample wears bras?
So "looking for cancer patients causes us to find bra wearing women" is actually maybe the more likely story here? Perhaps "cancer patients who were non bra wearers were overwhelmingly more likely to have died from their breast cancer, and so we couldn't find any of them?" That's somehow not as reassuring to the non-bra-wearers in the audience I think.
Symbolically: pretend BC and Bra are independent. We conclude or not wearing a bra reduces your chance of surviving by a factor of 10 or so if P(Bra) ~ 0.9? Put on those bras ladies! The exact opposite of their conclusion!
I personally suspect something else spurious in their research. But nothing in their PDF convinces me that they know what they are doing.
Note that wikipedia has some discussion of their book.
A friend of mine posted a link to news articles about a recent Surgeon General warning about sunscreen
Quoting from that article:
Skin cancer is on the rise, according to the American Cancer Society, with more cases diagnosed annually than breast, prostate, lung and colon cancer cases combined.
On Tuesday, the United States surgeon general issued a call to action to prevent the disease, calling it a major public health problem that requires immediate action. Nearly 5 million people are treated for skin cancer each year.
But let's dissect this a little bit. First of all, most skin cancer is unlikely to really hurt you. It's melanoma that is the real concern.
cancer.gov gives melanoma rates according to the following graph:
As for melanoma itself, clearly the diagnosed cases are on the rise, but look at the deaths per year. Totally flat. This is consistent with improved diagnosis procedures without any change in actual incidence in the population. Furthermore looking deeper on the melanoma site we see that 5 year survival rates have increased from 82% in 1975 to 93% in 2006, this is also consistent with earlier diagnosis (so that the 5 year rate measures from an earlier point relative to the initiation of the melanoma).
How about melanoma by state? Climate should be involved right? More sun exposure should mean more melanoma?
As you can see, states in the southwest have lower rates, and states in the northwest and northeast have higher rates. The cluster of southeastern states with high rates are interesting too.
Vitamin D is suspected to be involved in several health affects related to cancer, so overall exposure to sun may be beneficial. However, I think that the data is also clear that intense exposure to sun from tanning beds, intentional tanning, and repeated sunburn is bad for your skin.
Sun exposure, like other exposures such as alcohol, may have nonlinear risk effects. At moderate exposures you are better off than at zero exposure (in Oregon rain or Maine winters) or heavy exposure (leading to repeated sunburn and high melanoma risk).
So, is the advice to slather on sunscreen good? I don't think the conclusion is so clear cut, but I don't have any in-depth study data to go on either. All I can tell you is that I'll continue to avoid getting sunburn by covering up and using zinc based sunblock when I'm outside for long periods, but I'll continue to get regular sun exposure without sunblock in mornings, evenings, and mid day during non-summer months.
Last week I went swimming and met a couple who were very helpful and friendly. They were both professors at the Art Center College of Design here in Pasadena. Anyway, one of them pointed out that the pool does have a lap timer clock which I hadn't bothered to notice before, so I timed myself swimming 50 and 100 yard sets (two and four laps of the pool). My time for 50 was about 40 seconds (1.14m/s), and 100 was 1:40 (0.914 m/s) these were fairly consistent after the first set or two. I consider this not bad since I'm a father of two small boys who hasn't been exercising enough for the last 4 years or so. But another thing I discovered was that I know I swim 25 yards in about 13 or 14 strokes, sometimes 15. This, together with my lap times gives me a range of stroke rates which is around 36 or so strokes per minute. Now typical stroke rates are advocated to be around 60 to 70 per minute, which suggests that somehow I should increase my stroke rate. Intuitively, it seemed to me that this wouldn't work for me, but I thought I'd analyze the math of it. Thinking about how this would affect power output led me to the following analysis:
Let's add in the energy cost of stroking to the previous analysis. To get an energy cost, consider that your muscle "wastes" the kinetic energy of your arm at each end of the stroke. So the power you're putting out overcomes drag, and also accelerates and decelerates your arm every stroke.
Where is the length of the arm stroke, which is different from , so has the dimensions of power (kinetic energy per stroke time). Also, note that which I substitute for. We're going to try to control and so having things in terms of those variables is more helpful.
Another analysis I'm going to do here is to change from using and to and , where is still the height of the wake above the water, but is the full cross sectional profile of the body, so . I do this because for people who have a fairly flat profile in the water, is something fixed by their body shape, whereas can be controlled by things like head position. The previous analysis normalized by drag using but normalizing by full submersion drag using makes a little more sense because it's a single fixed point rather than something that changes when changes. Substituting , and normalizing by gives the following dimensionless power equation (note, as in the previous analysis, I've dropped the lateral viscous drag term as we've already seen it's multiplied by a trivially small constant).
This is a slightly different form than before, we see in this analysis that we have three dimensionless groups .
From this analysis, if we take a very low in the water body position, then , and we see that we can decrease our drag by riding up a little bit so that . Since the wave drag grows like when is small the decrease in piston drag exceeds the increase in wave drag. This probably accounts for the fact that elite swimmers often have a relatively forward looking head position. This lets them reduce their piston drag significantly while suffering only a small wave drag penalty. It also probably helps to produce the trough that allows for effective breathing.
Plugging in typical values for the mass of the arm and the density of the fluid and soforth, the order of magnitude estimates for the dimensionless groups are about 0.2 for the wave group, and for the arm group. So, although the equation predicts that if I double my stroke rate, I will waste 8 times more arm power, this still brings it into the same range as the wave and piston drag power. But why would I want to do that? The only reason I can see is that it would give me significantly more opportunities to breathe. One thing that I find with a 30 or so strokes per minute stroke rate is that I pretty much have to breathe one side only because otherwise I build up carbon dioxide. But doing so gives me an asymmetric stroke and might increase the wave and piston drag coefficients, and it makes it hard to empty my lungs fast enough. A faster stroke rate will have to be combined with a shorter stroke length, but will allow me to have more opportunities to breathe and may allow me to have a better sustained power output, actually making me faster even though my power output increases.
At least, that's how I interpret the math. I plan to get a little metronome that beeps regularly to help me time my stroke and see how it works for me.