We are all familiar with the idea that we can estimate height in male adults from their weight. … But not one of us believes that adding 20 pounds by eating and minimizing exercise will add an inch to our height.

The problem is not simply that the direction of causality backward, it’s that we cannot use a static description to predict what will happen if we change something.

Although regression situations may give one the illusion of finding out what would happen if we changed something, in the absence of an experiment they offer merely offer guesses.

He summarizes his point by quoting George Box:

To find out what happens to a system when you interfere with it, you have to interfere with it (and not just passively observe it).

Remember this next time you hear claims such as every dollar spent on X saves so many dollars spent on Y. Or every minute spent exercising increases your life expectancy by so many minutes. Or every time you do some activity you increase or decrease your risk of cancer by so much. First of all, these kinds of statements are linear extrapolations on situations that are not linear. Second, they may be observations that do not describe what will happen when you change something. They may be no more true than the idea that gaining weight makes you taller.

Here’s an example of how observation and intervention differ. Lottery winners often go bankrupt within a couple years of receiving their prize. If you suddenly make someone a millionaire, they’re not a typical millionaire.

Related posts:

Numerator-only data
Randomized trials of parachute use

If hot things cool, and cool things warm up, could something hot cool down and warm back up?

The people I asked didn’t understand my question and just laughed. I have no idea how old I was, but I wasn’t old enough to articulate what I was thinking.

Here’s what I had in mind. I knew that hot things like a cup of coffee grew cold. And I knew that cold things, say a glass of milk, get warm. Well, could the coffee get so cold that it becomes a cold thing and start to warm back up?

Could the coffee become as cold as the glass of milk? Common sense suggests that can’t happen. When we say coffee grows cold, we mean that it becomes relatively colder, closer to room temperature. And when we say the milk is getting warm, we also mean it is getting closer to room temperature. We’ve never left a hot cup of coffee on a table and come back later to find that it has cooled off so much that it is colder than room temperature. But could there be small fluctuations?

As the coffee and milk head toward room temperature, could they overshoot the target, just by a little bit? Say room temperature is 70 °F, the coffee starts out at 150 °F, and the milk starts out at 40 °F. We don’t expect the coffee to cool down to 40 °F or the milk to warm up to 150 °F. But could the coffee cool down to 69.5 °F and then go back up to 70 °F? Could the milk warm up to 70.5 °F and then cool back down to 70 °F?

I didn’t get a satisfactory answer to my childhood question until I was in college. Then I found out about Newton’s law of cooling. It says that the rate at which a warm body cools is proportional to the difference between its current temperature and the ambient temperature. This law can be written as a differential equation whose solution shows that the temperature of a warm body decreases exponentially to the ambient temperature. The temperature curve always slopes downward. It doesn’t wiggle even a little on its journey to room temperature. Cold bodies warm up the opposite way, exponentially approaching room temperature but never exceeding it.

In case it this seems obvious, think about thermostats. They don’t work this way. Say the temperature in a room is 85 °F and you’d like it to be 72 °F, so you turn on the air conditioning. Will the temperature steadily lower to 72 °F? Not exactly. If you were to plot the temperature in the room over time and look at the graph from far enough away, it would look like it is steadily going down to the desired temperature. But if you look at the graph more closely, you’ll see wiggles. The AC may cool the room to a little below 72 °F, maybe to 70 °F. The AC would cut off and the temperature would rise to 72 °F. Unlike the cup of hot coffee, the AC will often overshoot its target, though not by too much. The temperature may feel constant, but it is not. It oscillates around the desired temperature.

I want to focus on one line from the video. After showing simulated lightning strikes that hit a wooden rod five times and a copper rod five times, the narrator says

It’s five all, proof that metal does not attract lightning.

No, such an experiment would prove no such thing. I imagine the researchers conducted a much larger experiment and selected a representative sample. And I’m willing to accept their conclusion that metal does not attract lightning. But I would not accept such a conclusion from an experiment with 10 samples. What the experiment proves is that, under their experimental conditions, lightning will sometimes strike wood even a metal rod is nearby.

I have two complementary criticisms of this made-for-video science.

1. The results could easily happen if their conclusion were not true.
2. The results could easily not have happened if there conclusion were true.

Suppose in reality, lightning will not always strike the metal rod, but will prefer the metal. Suppose in the long run, lightning will strike the metal rod 60% of the time. It would not be unusual in that case to do an experiment with 10 strikes and find that half or more of the strikes hit wood.

Now suppose the researchers are exactly correct. In the long run, lightning has no preference for one rod or the other. What would viewers have thought if they showed a clip of 10 strikes, of which 6 hit metal and 4 hit wood? Many would have howled in protest. If lightning really had no preference for metal, the result should have been an even split, right? This is an example of the Law of Small Numbers. People underestimate the variability of small samples.

If the probability of lightning striking each rod is 50%, then in a sequence of experiments each containing 10 strikes, most will not have an exact 5-5 split. If you flip 10 fair coins, the most likely outcome is a 5-5 split, but this will happen only about 1/4 of the time. It’s more likely that you’ll get near a 5-5 split, sometimes with more heads and sometimes with more tails.

The exact 5-5 split in the video is good showmanship, but it’s misleading science.

Related posts:

Law of small numbers
Example of the law of small numbers
Law of medium numbers

The law of large numbers is a mathematical theorem. It describes what happens as you average more and more random variables.

The law of small numbers is a semi-serious statement about about how people underestimate the variability of the average of a small number of random variables.

The law of medium numbers is a term coined by Gerald Weinberg in his book An Introduction to General Systems Thinking. He states the law as follows.

For medium number systems, we can expect that large fluctuations, irregularities, and discrepancy with any theory will occur more or less regularly.

The law of medium numbers applies to systems too large to study exactly and too small to study statistically. For example, it may be easier to understand the behavior of an individual or a nation than the dynamics of a small community. Atoms are simple, and so are stars, but medium-sized things like birds are complicated. Medium-sized systems are where you see chaos.

Weinberg warns that medium-sized systems challenge science because scientific disciplines define their boundaries by the set of problems they can handle. He says, for example, that

Mechanics, then, is the study of those systems for which the approximations of mechanics work successfully.

He warns that we should not be mislead by a discipline’s “success with systems of its own choosing.”

Weinberg’s book was written in 1975. Since that time there has been much more interest in the emergent properties of medium-sized systems that are not explained by more basic sciences. We may not understand these systems well, but we may appreciate the limits of our understanding better than we did a few decades ago.

Related posts:

Laws of large numbers and small numbers
Gerald Weinberg’s law of twins
Subnatural and supernatural

John Ioannidis suggested popular areas of research publish a greater proportion of false results in his paper Why most published research findings are false. Of course popular areas produce more results, and so they will naturally produce more false results. But Ioannidis is saying that they also produce a greater proportion of false results.

Now Thomas Pfeiffer and Robert Hoffmann have produced empirical support for Ioannidis’s theory in the paper Large-Scale Assessment of the Effect of Popularity on the Reliability of Research. Pfeiffer and Hoffmann review two reasons why popular areas have more false results.

First, in highly competitive fields there might be stronger incentives to ‘‘manufacture’’ positive results by, for example, modifying data or statistical tests until formal statistical significance is obtained. This leads to inflated error rates for individual findings: actual error probabilities are larger than those given in the publications. … The second effect results from multiple independent testing of the same hypotheses by competing research groups. The more often a hypothesis is tested, the more likely a positive result is obtained and published even if the hypothesis is false.

In other words,

1. In a popular area there’s more temptation to fiddle with the data or analysis until you get what you expect.
2. The more people who test an idea, the more likely someone is going to find data in support of it by chance.

The authors produce evidence of the two effects above in the context of papers written about protein interactions in yeast. They conclude that “The second effect is about 10 times larger than the first one.”

Related posts:

Why microarray conclusions are so often wrong
Using Photoshop on experimental results
Irreproducible analysis
Make up your own rules of probability

The family had settled near a creek that was infested with mosquitoes. All the settlers around the creek bottoms came down with malaria, though at the time (circa 1870) they did not know the disease was transmitted by mosquitoes. One of the settlers, Mrs. Scott, believed that malaria was caused by eating the watermelons that grew in the creek bottoms. She had empirical evidence: everyone who had eaten the melons contracted malaria. Charles Ingalls thought that was ridiculous. After he recovered from his attack of malaria, he went down to the creek and brought back a huge watermelon and ate it. His reasoning was that “Everybody knows that fever ‘n’ ague comes from breathing the night air.”

It’s easy to laugh at Mrs. Scott and Mr. Ingalls. What ignorant, superstitious people. But they were no more ignorant than their contemporaries, and both had good reasons for their beliefs. Mrs. Scott had observational data on her side. Ingalls was relying on the accepted wisdom of his day. (After all, “malaria” means “bad air.”)

People used to believe all kinds of things that are absurd now, particularly in regard to medicine. But they were also right about many things that are hard to enumerate now because we take them for granted. Stories of conventional wisdom being correct are not interesting, unless there was some challenge to that wisdom. The easiest examples of folk wisdom to recall may be the instances in which science initially contradicted folk wisdom but later confirmed it. For example, we have come back to believing that breast milk is best for babies and that a moderate amount of sunshine is good for you.

Related posts:

A little coffee on the prairie
Galen and clinical trials
Randomized trials of parachute use

… scientists at Johns Hopkins … have shown that a protein made by a gene called “Twist” may be the proverbial red flag that can accurately distinguish stem cells that drive aggressive, metastatic breast cancer from other breast cancer cells.

Related posts:

Detecting breast cancer from a hair sample
Visualizing cancer DNA scrambling
Killing too much of a tumor

The older scientists believed that the smallest particles of matter moved according to strict laws: in other words, that the movements of each particle were “interlocked” with the total system of Nature. Some modern scientists seem to think — if I understand them — that this is not so. They seem to think that the individual unit of matter … moves in an indeterminate or random fashion; moves, in fact, “on its own” or “of its own accord.”

He goes on to explain that the macroscopic behavior of matter appears deterministic because the average behavior of billions of particles is very regular. His explanation is remarkably cogent for a professor of medieval literature writing in the 1940’s. He then discusses the philosophical consequences of quantum mechanics.

Now it will be noticed that if this theory is true we have really admitted something other than Nature. If the movements of the individual units is “on their own,” … then those movements are not part of Nature. It would be, indeed, too great a shock to our habits to describe them as super-natural. I think we should call them sub-natural. But all our confidence that Nature has no doors, and no reality outside herself for doors to open on, would have disappeared. There is something outside her, the Subnatural. … And clearly if she thus has a back door opening on the Subnatural, it is quite on the cards that she may also have a front door opening on the Supernatural …

From Miracles by C. S. Lewis, chapter 3.

Related post:

The world looks more mathematical than it is

Grad:

Div:

Curl:

Related link:

Div, Grad, Curl and All That

One theme that emerges is that the most common errors are simple (e.g. row or column offsets); conversely, it is our experience that the most simple errors are common.

The full title of the article by Keith Baggerly and Kevin Coombes is “Deriving chemosensitivity from cell lines: forensic bioinformatics and reproducible research in high-throughput biology.” The article will appear in the next issue of Annals of Applied Statistics and is available here. The key phrase in the title is forensic bioinformatics: reverse engineering statistical analysis of bioinformatics data. The authors give five case studies of data analyses that cannot be reproduced and infer what analysis actually was carried out.

One of the more egregious errors came from the creative application of probability. One paper uses innovative probability results such as

P(ABCD) = P(A) + P(B) + P(C) + P(D) – P(A) P(B) P(C) P(D)

and

P(AB) = max( P(A), P(B) ).

Baggerly and Coombes were remarkably understated in their criticism: “None of these rules are standard.” In less diplomatic language, the rules are wrong.

To be fair, Baggerly and Coombes point out

These rules are not explicitly stated in the methods; we inferred them either from formulae embedded in Excel files … or from exploratory data analysis …

So, the authors didn’t state false theorems; they just used them. And nobody would have noticed if Baggerly and Coombes had not tried to reproduce their results.

Related posts:

Irreproducible analysis
Highlights from Reproducible Ideas
Reproducible Ideas blog winding down

Something similar is true of software projects: the big, visible projects, the ones people write about, are a small fraction of the total. Certainly there are more small projects in the world than large projects. And I imagine more programmers in total work on small projects than on large projects. I don’t have any hard numbers on this, and I doubt anyone else does. Most hard numbers come from large, visible projects! Who is going to do a census of all the little one-man projects that go unnoticed?

This post is a continuation of a comment I made as part of the discussion following my blog post on medieval software project management. My contention there was that most projects involve one developer, have no written requirements, and no external testing. That may not be correct, but I imagine it’s closer to the truth than assuming everyone works on projects with a dozen developers, formal requirements documents, and a staff of testers.

The first books on the “right” way to develop software codified the experience gained from working on enormous federally funded software projects. For example, the recommended practice was to spend huge proportion of the total effort in up-front planning. While that made sense when coordinating the efforts of thousands of contractors in the days of punch cards, it doesn’t make as much sense now. The agile software development movement began when people realized that the world had changed and the “best practices” of a previous generation were not optimal for smaller projects and vastly superior hardware.

Agile software development has replaced the best practices of the 1960’s in many organizations. However, there is still a strong tendency to think that small projects should use the same tools and techniques as large, enterprise projects. Most books are written about medium to large projects and many developers worry unnecessarily about scaling up their projects. (”What if I get a million visitors an hour to my web site?” You should be so lucky. Worry about that after it becomes a remote possibility.) Few pundits give advice that scales down, that is, advice appropriate for small projects. I wrote about one exception in a previous post in which Rob Page suggests different methods for projects with a budget of less than $1M and projects with a larger budget.

Related posts:

Million dollar cutoff for software technique
Enterprising software
Medieval software project management

The Alternative Blog has a post this morning entitled Hawthorne effect debunked. The original Hawthorne effect was apparently due to a flaw in the study design; correcting for that flaw eliminates the effect.

The term “debunked” in the post title may imply too much. The effect in the original studies may have been debunked, but that does not necessarily mean there is no Hawthorne effect. Perhaps there are good examples of the Hawthorne effect elsewhere. On the other hand, I expect closer examination of the data could debunk other reported instances of the Hawthorne effect as well.

The Hawthorne effect makes sense. It has been ingrained in pop culture. I heard a reference to it on a podcast just this morning before reading the blog post mentioned above. Everyone knows it’s true. And maybe it is. But at a minimum, there is at least one example suggesting the effect is not as wide-spread as previously thought.

It would be interesting to track the popularity of the Hawthorne effect in scholarly literature and in pop culture. If the effect becomes less credible in scholarly circles, will it also become less credible in pop culture? And if so, how quickly will pop culture respond?

Sometimes one treatment will shrink a tumor as much as possible as a prelude to another treatment, such as shrinking a tumor with chemotherapy prior to surgery. But if only one treatment is being used, the situation may be like the old saying that you don’t want to wound the king. If you’re going try to kill the king, you’d better succeed.

In a recent interview on the Nature podcast, Robert Gatenby of Moffitt Cancer Center advocates an alternative approach, treating cancer as a chronic disease. Instead of killing as much of a tumor as possible, it may be better to kill as little of tumor as necessary to keep it under control. Patients would continue to take anti-cancer treatments for the rest of their lives, just as patients with heart disease or diabetes take medication indefinitely.

Related post:
Repairing tumors

If I have seen farther than others it is because I have stood on the shoulders of giants.

Later Mathematician R. W. Hamming added

Mathematicians stand on each other’s shoulders while computer scientists stand on each other’s toes.

Finally, computer scientist Hal Abelson quipped

If I have not seen farther, it is because giants were standing on my shoulders.

(Thanks to Mark Reid for the Hamming quote.)

Here is the abstract of the medical paper discussed on the podcast.

Related post: Cartoon guide to cancer research

I’m using the word “program” as in an “exercise program” or a “research program.” (I could think of my exercise program as a project, but it’s one I hope not to complete for a few more decades.) Sometimes there is a neat hierarchy where programs spawn off projects that can be divided into tasks. But sometimes you just have programs and tasks.

One of my frustrations with managing software development in an academic environment was the large number of programs disguised as projects. (Sorry, I know it’s confusing to talk about “programs” in the context of software development and not mean computer instructions.) You can’t manage programs as if they were projects. For example, you can’t talk about “after” project is done if it’s not really a project but a never-ending program. You have to either acknowledge that a program is really a program, or you have to have some way to make it into a finite project.

Three reasons to distrust microarray results
Provenance in art and science
Forensic bioinformatics (continued)
Preserving (the memory of) documents
Programming is understanding
Musical chairs and reproducibility drills
Taking your code out for a walk

The most popular and most controversial was the first in the list, reasons to distrust microarray results.

The emphasis shifts from science to software development as you go down the list, though science and software are intertwined throughout the posts.

As part of this process, I’m winding down the blog that I started last July as part of the ReproducibleResearch.org site. I plan to keep the links to my old posts valid, but I do not know whether the new site will have a new blog. I wrote about reproducible research on this blog before starting the ReproducibleResearch.org site, and I will go back to writing about reproducible research here. (See reproducibility in the tag cloud.)

I wanted to point out an article by Steve Eddins posted this morning: Reproducible research in signal processing. His article comments on the article by Patrick Vandewalle, Jelena Kovačević, and Martin Vetterli announced recently on ReproducibleResearch.org.

Readers interested in reproducible research may also want to take a look at the Science in the open blog.

Related posts:

Irreproducible analysis
Using Photoshop on experimental results
Highlights from Reproducible Ideas

Related posts:

Cartoon guide to cancer research
Naked mole rats and cancer
How to treat a stingray wound

Related posts on cancer research

1. What did you work on?
2. What did you get done?
3. What did you do that you didn’t anticipate having to do?
4. What did you plan to do that you didn’t get done?
5. What do you plan to do?
6. What do you need from others?

When I started managing a group of programmers, I’d focus on #1 and #2. But in some ways #3 is the most important question. That question can alert you to a major time sink that’s not include in your project estimates. That question can let you know of problems beyond an individual developer’s ability to resolve. That question that can tell you it’s time to buy something you were planning on building yourself.

A first guess would be that metabolism (equivalently, heat produced) goes up in proportion to volume. If cells are all roughly the same size, then number of cells increases proportionately with volume. But heat is dissipated through the surface. Surface area increases in proportion to the square of length but volume increases in proportion to the cube of length. That means the ratio of surface area to volume decreases as overall size increases. The surface area to volume ratio for an elephant is much smaller than it is for a mouse. If an elephant’s metabolism per unit volume were the same as that of a mouse, the elephant’s skin would burn up.

So metabolism cannot be proportional to volume. What about surface area? Here we get into variety and controversy. Many people assume metabolism is proportional to surface area based on the argument above. This idea was first proposed by Max Rubner in 1883. Others emphasize data that supports the theory that suggests metabolism is proportional to surface area.

In the 1930’s, Max Kleiber proposed that metabolism increases according to body mass raised to the power ³/₄. (I’ve been a little sloppy here using body mass and volume interchangeably. Body mass is more accurate, though to first approximation animals have uniform density.) If metabolism were proportional to volume, the exponent would be 1. If it were proportional to surface area, the exponent would be ²/₃. But Kleiber’s law says it’s somewhere in between, namely ³/₄. The image below comes from a paper by Kleiber from 1947.

The graph shows that on a log-log plot, the metabolism rate versus body mass for a large variety of animals has slope approximately ³/₄.

So why the exponent ³/₄? There is a theoretical explanation called the metabolic scaling theory proposed by Geoffrey West, Brian Enquist, and James Brown. Metabolic scaling theory says that circulatory systems and other networks are fractal-like because this is the most efficient way to serve an animal’s physiological needs. To quote Enquist:

Although living things occupy a three-dimensional space, their internal physiology and anatomy operate as if they were four-dimensional. … Fractal geometry has literally given life an added dimension.

The fractal theory would explain the power law exponent exponent ³/₄ simply: it’s the ratio of the volume dimension to the fractal dimension. However, as I suggested earlier, this theory is controversial. Some biologists dispute Kleiber’s law. Others accept Kleiber’s law as an empirical observation but dispute the theoretical explanation of West, Enquist, and Brown.

To read more about metabolism and power laws, see chapter 17 of Complexity: A Guided Tour.

Related posts:

Networks and power laws
Rate of regularizing English verbs

See his post AntiPaste, because Pasting Code Is Harmful.

It’s a joke, but many a truth is told in jest.

Ward Cheney and Will Light give a candid definition of “classical” in the scientific sense in the introduction to their book on approximation theory:

… the “classical” portion of approximation theory — understood to be the parts of the subject that were already in place when the authors were students.

There you have it: whatever was known when you were in school is classical. Yes, this definition is entirely relative. And it describes common usage pretty well.

But it turns out the real rodents might be useful in cancer research. According to a recent 60-Second Science podcast, naked mole rats live in low-oxygen environments. The core of large tumors is also a low-oxygen environment, and so maybe studying naked mole rats can tell us something about cancer. So far researchers have found three genes in common between naked mole rats and cancer cells.

Suppose we have two springs in series with stiffness k₁ and k₂:

Then the combined stiffness k of the two springs satisfies 1/k = 1/k₁ + 1/k₂. Think about what this says in the extremes. If one of the springs were infinitely stiff, say k₂ = ∞. Then k = k₁. It would be as if the second spring were not there. Being infinitely stiff, we could think of it as an extension of the block it is attached to. Now think of one of the springs having no stiffness at all, say k₁ = 0. Then k = 0. One mushy spring makes the combination mushy.

Next think of two springs in parallel:

Now the combined stiffness of the two springs is k = k₁ + k₂. Again think of the two extremes. If one spring is infinitely stiff, say k₁ = ∞, then k = ∞ and the combination is infinitely stiff. And if one spring has no stiffness, say k₂ = 0, then k = k₁. We could imagine the spring with no stiffness isn’t there.

The stiffness of springs in series adds harmonically. The stiffness of the combination is half the harmonic mean of the two individual stiffnesses.

Electrical resistors combine in a way that is the opposite of mechanical springs. Resistors in parallel combine like springs in series, and vice versa.

If two resistors have resistance r₁ and r₂, the combined resistance r of the two resistors in parallel satisfies 1/r = 1/r₁ + 1/r₂. If one of the resistors has infinite resistance, say r₂ = ∞, then r = r₁. It would be as if the second resistor were not there. All electrons would flow through the first resistor.

If the two resistors were in series, then r = r₁ + r₂. If one resistor has infinite resistance, so does the combination. Electrons cannot flow through the combination if they cannot flow through one of the resistors. And if one resistor has zero resistance, say r₂ = 0, then r = r₁. Since the second resistor offers no resistance to the flow of electrons, it may as well not be there.

These physical problems illustrate why zeros as handled specially in the definition of means.

Image credit: Wikipedia

PS> function √($num) { [Math]::Sqrt($num) }
PS> √ 81
9

As Keith points out, these symbols are not practical since they’re difficult to enter, but they’re fun to play around with.

Here’s another example using the symbol for pounds sterling

for the function to convert British pounds to US dollars.

PS> function £($num) { 1.44*$num }
PS> £ 300.00
432

(As I write this, a British pound is worth $1.44 USD. If you wanted to get fancy, you could call a web service in your function to get the current exchange rate.)

I read once that someone (Larry Wall?) had semi-seriously suggested using the Japanese Yen currency symbol

for the “zip” function in Perl 6 since the symbol looks like a zipper.

Mathematica lets you use Greek letters as variable and function names, and it provides convenient ways to enter these characters, either graphically or via their TeX representations. I think this is a great idea. It could make mathematical source code much more readable. But I don’t use it because I’ve never got into the habit of doing so.

There are some dangers to allowing Unicode characters in programming languages. Because Unicode characters are semantic rather than visual, two characters may have the same graphical representation. Here are a couple examples. The Roman letter A (U+0041) and the capital Greek letter Α (U+0391) look the same but correspond to different characters. Also, the the Greek letter Ω (U+03A9) and the symbol Ω (U+2126) for Ohms (unit of electrical resistance) have the same visual representation but are different characters. (Or at least they may have the same visual representation. A font designer may choose, for example, to distinguish Omega and Ohm, but that’s not a concern to the Unicode Consortium.)

When you see software with a silly interface, odds are the interface details were not specified and the programmer chose the path of least effort. Why should I press # after entering my five-digit zip code on a phone? It’s logically possible to determine what my zip code is as soon as I enter the fifth digit, but it makes life easier for some phone system programmer if the pound sign is a universal end-of-input signal. Or why should I select an account at the ATM if I only have one account? Again, I’m sure this made life easier for some programmer.

Some programmers are lazy. But some are unsung heroes. They understand gritty details of their company that no one else knows about. They have to: they have to specify these details to dumb machines. Most people don’t want to solve problems down to the final detail and programmers — for better and for worse — are the ones who fill in the gaps.

Some of the details that programmers fill in regard what to do when things go wrong. A client says a program is supposed to collect the user’s Social Security number (SSN). Fine. But what if the user doesn’t have an SSN? What if they enter an invalid SSN? (Is there a way to know whether an SSN is valid?)  After a few questions like that, the client will throw up his hands and leave it up to the developer. But the developer cannot throw up his hands. He has to decide something, even if he passively decides to let the software crash in case of unexpected input.

Update: Changed “analyst” to “client” above. See discussion below. “Client” more accurately reflects what I meant.

Related posts:
Paper doesn’t abort
Where does the programming effort go?

How well does a sphere describe the Earth’s shape? The Earth’s polar diameter is about 43 kilometers shorter than its equatorial diameter, a difference of about 0.3%.This is due to the equatorial bulge caused by the Earth’s rotation.

What’s a more accurate description of the Earth’s shape? An oblate spheroid.

What is an oblate spheroid? It’s the shape you get by spinning an ellipse around it’s minor axis. That says if you were to take a cross-section of the Earth containing the polar axis, the shape you get would be an ellipse. The polar axis would be the minor axis and the equatorial axis would be the major axis. But if you were to take a cross-section through the equator, or any plane parallel to the equator, you’d get a circle.

What is a prolate spheroid? A prolate spheroid is what you get by spinning an ellipse around its major axis.

What is an ellipsoid? An ellipsoid satisfies the following equation.

A sphere is an ellipsoid with a = b = c. An oblate spheroid is an ellipsoid with a = b > c. A prolate spheroid is an ellipsoid with a = b < c. A scalene ellipsoid is an ellipsoid for which a, b, and c are all distinct.

How good is the oblate spheroid model? The error in approximating the Earth’s shape as an oblate spheroid is less than 100 meters, two orders of magnitude better than the spherical model.

How are other planets shaped? The other planets in our solar system are also oblate spheroids with Saturn being the most oblate: the polar diameter of Saturn is about 10% shorter than its equatorial diameter.

Related post: Finding distances using longitude and latitude