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.
I've been working on my front-crawl swim stroke as an effort to improve my fitness. I really enjoy the challenge of getting better at the technique, and I also enjoy the challenge of not over-thinking things while I'm swimming. Of course, when I'm not in the water it's another story. So here's a blog post about dimensionless analysis and the forward crawl.
First off, let's be clear what we mean. Some people call this stroke "freestyle", some people the "forward crawl" others the "australian crawl", you can read about this stroke at the wikipedia page but almost everyone who can swim has used some variant of this stroke, so I'm going to assume you know what I'm talking about.
Many people of course are interested in how to swim quickly. The front crawl is generally considered the fastest stroke, which is why people use it during the "freestyle" event (where you really could swim whatever you wanted). Let's think about how we get velocity:
In this equation, is the velocity of the swimmer through the water, and is the "stroke length" (the distance traveled per stroke) which we get by multiplying the swimmer's length L by a dimensionless factor (this is already anticipating later analysis of lengths), whereas is the stroke rate (the number of strokes per minute for example).
If we want to increase we can either make bigger, or bigger or both. Since we can't make the swimmer much longer without injury, to lengthen the stroke requires making bigger which is part of working on technique.
Now, let's think about energy consumption. If we want to go fast in a long distance swim, we will need to make large, which is the time averaged velocity, averaged over time . To make this make sense we had better be going long enough that we take a lot of strokes and can consider the time between strokes as "infinitesimal" compared to the overall time. A good stroking rate is on the order of 1 stroke per second, so we're interested in averages over minutes or even tens of minutes.
Now, when trying to go fast, what will happen is we have some ability to output power such that we can sustain it for the entire time . And we will go as fast as that power budget will allow, in such a way as the power from drag is equal to our sustained power output. So keeping drag low at a given velocity will let us go faster.
So let's try to model the power involved. The simplest model is a body averaged one:
where is the net drag force on your body.
To figure out what controls the drag consider that it has dimensions of force, and using dimensional analysis we need to generate a force term from the following variables the height of the bow wave the depth of your body in the water, the width of your body at the shoulders, the length of your body, the fluid density, the dynamic viscosity, and the velocity.
Our drag will be made up of several components. We can use dimensional analysis to get the functional form of these components. We need combinations of the above variables which have dimensions of force for use in our power equation.
A good candidate force term is proportional to . This is more or less related to the static pressure head generated by the bow wave in front of the swimmer. Another good candidate term is which has dimensions of force and relates to the difficulty of pushing the fluid out of the way in front of you (without lifting it). Finally, we expect drag to be related to viscosity as well, we're going to find that has dimensions of force, and is related to the drag along the length of the swimmer's body essentially caused by pulling the fluid along with us. Now we've identified several sources of drag. Let's write an equation for them:
In our overall problem, there are 2 equations and 9 variables () and 3 independent dimensions (length, time, mass) so we expect to have 6 independent dimensionless groups. One thing we can see though is that L,d,w,h are all lengths. Clearly we can make 3 dimensionless groups by doing which makes sense since is the largest dimension so all of these length ratios are less than 1.
Let's rewrite our power equation in dimensionless form. We could normalize the power equation by one of the types of drags, since we're interested in going fast, consider that the term multiplied by may dominate and normalize by that. Let's assume that over the range of speeds of interest to swimmers the dimensionless multiplier of this drag term is near constant and therefore the drag is equal to for some dimensionless constant as shown above. Dividing through we get:
In this equation, we've identified 3 important dimensionless groups , and , there is also another important dimensionless group, , this basically measures our velocity as a fraction of one body length per stroke interval, and is equal to the dimensionless stroke length. Finally, there is which are the natural measures of bow height, and depth in the water (we are essentially measuring all lengths as a fraction of the swimmer's full length). That is the full 6 dimensionless groups promised. If we write , , and we can rewrite our equations as (note, maxima computer algebra system is helpful in avoiding mistakes):
In other words, if we measure velocity in stroke lengths per stroke time, distance in units of your body length, and power in units of piston drag (the term proportional to ) we get things into a very simple form. Clearly we can learn a lot from this. It becomes obvious why tall swimmers are desirable, their wave drag decreases like and lateral drag like . Plugging the following approximate order of magnitude constants in we get order of magnitude estimates of how important our different drags are:
Now, we can't estimate or without data, but the fact that the typical size of the dimensionless group in front of the third term is orders of magnitude smaller than 1, suggests that lateral drag is really not very important in the real world (though we haven't accounted for kicking here, it's quite possible that excessive kick churn would enter into essentially lateral drag). To support this conclusion, consider that the Mythbusters tested out swimming in various viscosities and found that for non-elite swimmers the viscosity had essentially no important effect on speed over multiple orders of magnitude of viscosity.
Clearly though, with a dimensionless premultiplication factor of 3 or so, wave drag is important. One reason to keep your head low in the water ( small) and your stroke length long () is that small stroke lengths and high head positions will severely increase your wave drag. This is why swimmers try to stay under the water after a push off, so that waves aren't sapping their momentum.
So, to swim efficiently, we need small total power, which means a long stroke (large ), low in the water (small ), and we need to keep our body stretched out and lying flat in the water to maximize L and minimize (which produces a small dimensional piston drag ). Keep the stroke rate high, but keep it smooth so as to minimize other kinds of wave drag (choppiness) which is a kind of drag not included in this model. Since piston drag is proportional to depth in the water, and wave drag is proportional to height of the wave above the water squared, there is an optimum depth that involves not being too deep, but also not creating too big of a wave. It's probably a good idea to try to stay deeper in the water, with head down and see how well that works for you. Obviously if it interferes with breathing, or alters the constant in front of the piston drag due to strange turbulence around your head, it won't help. There is some optimum depth tradeoff for each swimmer.
What about the "glide"?? this is something that some people advocate, because it lengthens the stroke, and people have found that long strokes are efficient. However, it decreases the stroke rate as well. As long as increases, everything is good. So in other words, if you are "over paddling" with a short stroke length and a high stroke rate, slowing down your strokes and making them longer makes sense. But if you are "over-gliding" with a long stroke length (s) but a slow stroke rate (r) you should instead speed-up your stroking so that the overall combination increases. Note how the relative wave drag is inversely proportional to , which is related to the velocity. When we slow down, piston drag decreases, so wave drag becomes a bigger relative factor. Ideally, we're maximizing , which means that we're maximizing piston drag, and wave drag becomes relatively less important. Eventually we come to a steady state where the drag force and velocity are high enough that we can't go faster without exceeding our power budget and therefore getting tired out too soon. Since piston drag is related to which is what we're trying to maximize, we can't get rid of it entirely, but we can try to minimize the additional drag caused by waves.
I have a friend who is a Philosophy professor. He recently posted to his Facebook page that he was offering his students extra credit if they could come up with false positive and false negative rates for the drug tests that used to be required by Florida law in order to receive welfare benefits. His students were having a heck of a time finding anything on these rates.
The only data he had was that about 2.4% of people who took the test got positive tests (reported in the news I guess).
So, I did a little quick googling, and came up with a WebMD article that had an approximate range for these tests.
I wrote him the following email which describes how to use this range to construct a simple prior over the false positive and false negative frequencies for these tests, and then using a single 6 sided die to generate a sample from this prior and calculate the range of frequencies for a person who tests positive to be an actual drug user.
Since it's a Philosophy course, I put in some stuff at the end about the distinction between frequencies and probabilities within the Bayesian framework. If you're either a teacher, or a student just seeing these ideas for the first time, it might be helpful to read this description
If you follow my blog you probably know that I was a PhD student at USC. I graduated in December, so you can read my dissertation which discusses both the mechanics of liquefaction and the thermodynamics and mechanics of wave dissipation due to microscopic vibrations in a molecular solid, along with some Bayesian fitting of that ODE model.
The liquefaction research was recently published on Proceedings of the Royal Society A. My version of the preprint can be downloaded here: Grain Settlement and Fluid Flow Cause the Earthquake Liquefaction of Sand and my postprint version will be available after 12 months (though I understand the definitive version will become available from the journal after a year or so too). The supplemental materials are available as well.
The article in Proceedings A was picked up by two different news sources so far:
ABC News Australia interviewed me for their article from a week or two ago, and in the last few days ScienceNOW interviewed me and one of my advisers Amy Rechenmacher for an article that came out today.
I was particularly happy to see Ross Boulanger quoted with a favorable comment. He was a professor of mine when I did my second undergrad degree at Davis, and he definitely motivated me to think about the problems with standard liquefaction explanations.
In the mean time, I am now working on building up a consulting practice. My company is: Lakeland Applied Sciences LLC. My emphasis has been on building mathematical models of all sorts of phenomena, from things like liquefaction and wave vibration to scaling laws for the healing time of bones and cartilage, to economic decision making, to biochemical explanations for behavior. You can see examples of projects I've worked on in the past on the company web site.