Fitting a triangular distribution

Sometimes you only need a rough fit to some data and a triangular distribution will do. As the name implies, this is a distribution whose density function graph is a triangle. The triangle is determined by its base, running between points a and b, and a point c somewhere in between where the altitude intersects the base. (c is called the foot of the altitude.) The height of the triangle is whatever it needs to be for the area to equal 1 since we want the triangle to be a probability density.

One way to fit a triangular distribution to data would be to set a to the minimum value and b to the maximum value. You could pick a and b are the smallest and largest possible values, if these values are known. Otherwise you could use the smallest and largest values in the data, or make the interval a little larger if you want the density to be positive at the extreme data values.

How do you pick c? One approach would be to pick it so the resulting distribution has the same mean as the data. The triangular distribution has mean

(a + b + c)/3

so you could simply solve for c to match the sample mean.

Another approach would be to pick c so that the resulting distribution has the same median as the data. This approach is more interesting because it cannot always be done.

Suppose your sample median is m. You can always find a point c so that half the area of the triangle lies to the left of a vertical line drawn through m. However, this might require the foot c to be to the left or the right of the base [a, b]. In that case the resulting triangle is obtuse and so sides of the triangle do not form the graph of a function.

For the triangle to give us the graph of a density function, c must be in the interval [a, b]. Such a density has a median in the range

[b – (ba)/√2, a + (ba)/√2].

If the sample median m is in this range, then we can solve for c so that the distribution has median m. The solution is

c = b – 2(bm)2 / (ba)

if m < (a + b)/2 and

c = a + 2(am)2 / (ba)

otherwise.

Extremely small probabilities

One objection to modeling adult heights with a normal distribution is that the former is obviously positive but the latter can be negative. However, by this model negative heights are astronomically unlikely. I’ll explain below how one can take “astronomically” literally in this context.

A common model says that men’s and women’s heights are normally distributed with means of 70 and 64 inches respectively, both with a standard deviation of 3 inches. A woman with negative height would be 21.33 standard deviations below the mean, and a man with negative height would be 23.33 standard deviations below the mean. These events have probability 3 × 10-101 and 10-120 respectively. Or to write them out in full

0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003

and

0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001.

As I mentioned on Twitter yesterday, if you’re worried about probabilities that require scientific notation to write down, you’ve probably exceeded the resolution of your model. I imagine most probability models are good to two or three decimal places at most. When model probabilities are extremely small, factors outside the model become more important than ones inside.

According to Wolfram Alpha, there are around 1080 atoms in the universe. So picking one particular atom at random from all atoms in the universe would be on the order of a billion trillion times more likely than running into a woman with negative height. Of course negative heights are not just unlikely, they’re impossible. As you travel from the mean out into the tails, the first problem you encounter with the normal approximation is not that the probability of negative heights is over-estimated, but that the probability of extremely short and extremely tall people is under-estimated. There exist people whose heights would be impossibly unlikely according to this normal approximation. See examples here.

Probabilities such as those above have no practical value, but it’s interesting to see how you’d compute them anyway. You could find the probability of a man having negative height by typing pnorm(-23.33) into R or scipy.stats.norm.cdf(-23.33) into Python. Without relying on such software, you could use the bounds

\frac{x}{\sqrt{2\pi}(x^2 + 1)} \exp(-x^2/2) < \Phi^c(x) < \frac{1}{\sqrt{2\pi}\,x} \exp(-x^2/2)

with x equal to -21.33 and -23.33. For a proof of these bounds and tighter bounds see these notes.

Finding the best dose

In a dose-finding clinical trial, you have a small number of doses to test, and you hope find the one with the best response. Here “best” may mean most effective, least toxic, closest to a target toxicity, some combination of criteria, etc.

Since your goal is to find the best dose, it seems natural to compare dose-finding methods by how often they find the best dose.  This is what is most often done in the clinical trial literature. But this seemingly natural criterion is actually artificial.

Suppose a trial is testing doses of 100, 200, 300, and 400 milligrams of some new drug. Suppose further that on some scale of goodness, these doses rank 0.1, 0.2, 0.5, and 0.51. (Of course these goodness scores are unknown; the point of the trial is to estimate them. But you might make up some values for simulation, pretending with half your brain that these are the true values and pretending with the other half that you don’t know what they are.)

Now suppose you’re evaluating two clinical trial designs, running simulations to see how each performs. The first design picks the 400 mg dose, the best dose, 20% of the time and picks the 300 mg dose, the second best dose, 50% of the time. The second design picks each dose with equal probability. The latter design picks the best dose more often, but it picks a good dose less often.

In this scenario, the two largest doses are essentially equally good; it hardly matters how often a method distinguishes between them. The first method picks one of the two good doses 70% of the time while the second method picks one of the two good doses only 50% of the time.

This example was exaggerated to make a point: obviously it doesn’t matter how often a method can pick the better of two very similar doses, not when it very often picks a bad dose. But there are less obvious situations that are quantitatively different but qualitatively the same.

The goal is actually to find a good dose. Finding the absolute best dose is impossible. The most you could hope for is that a method finds with high probability the best of the four arbitrarily chosen doses under consideration. Maybe the best dose is 350 mg, 843 mg, or some other dose not under consideration.

A simple way to make evaluating dose-finding methods less arbitrary would be to estimate the benefit to patients. Finding the best dose is only a matter of curiosity in itself unless you consider how that information is used. Knowing the best dose is important because you want to treat future patients as effectively as you can. (And patients in the trial itself as well, if it is an adaptive trial.)

Suppose the measure of goodness in the scenario above is probability of successful treatment and that 1,000 patients will be treated at the dose level picked by the trial. Under the first design, there’s a 20% chance that 51% of the future patients will be treated successfully, and a 50% chance that 50% will be. The expected number of successful treatments from the two best doses is 352. Under the second design, the corresponding number is 252.5.

(To simplify the example above, I didn’t say how often the first design picks each of the two lowest doses. But the first design will result in at least 382 expected successes and the second design 327.5.)

You never know how many future patients will be treated according to the outcome of a clinical trial, but there must be some implicit estimate. If this estimate is zero, the trial is not worth conducting. In the example given here, the estimate of 1,000 future patients is irrelevant: the future patient horizon cancels out in a comparison of the two methods. The patient horizon matters when you want to include the benefit to patients in the trial itself. The patient horizon serves as a way to weigh the interests of current versus future patients, an ethically difficult comparison usually left implicit.

Random walks and the arcsine law

Suppose you stand at 0 and flip a fair coin. If the coin comes up heads, you take a step to the right. Otherwise you take a step to the left. How much of the time will you spend to the right of where you started?

As the number of steps N goes to infinity, the probability that the proportion of your time in positive territory is less than x approaches 2 arcsin(√x)/π. The arcsine term gives this rule its name, the arcsine law.

Here’s a little Python script to illustrate the arcsine law.

import random
from numpy import arcsin, pi, sqrt

def step():
    u = random.random()
    return 1 if u < 0.5 else -1

M = 1000 # outer loop    
N = 1000 # inner loop

x = 0.3 # Use any 0 < x < 1 you'd like.

outer_count = 0
for _ in range(M):
    n = 0
    position= 0 
    inner_count = 0
    for __ in range(N):
        position += step()
        if position > 0:
            inner_count += 1
    if inner_count/N < x:
        outer_count += 1

print (outer_count/M)
print (2*arcsin(sqrt(x))/pi)

Miscellaneous math resources

Every Wednesday I’ve been pointing out various resources on my web site. So far they’ve all been web pages, but the following are all PDF files.

Probability and statistics:

Other math:

See also journal articles and technical reports.

Last week: Probability approximations

Next week: Code Project articles

Probability approximations

This week’s resource post lists notes on probability approximations.

Do we even need probability approximations anymore? They’re not as necessary for numerical computation as they once were, but they remain vital for understanding the behavior of probability distributions and for theoretical calculations.

Textbooks often leave out details such as quantifying the error when discussion approximations. The following pages are notes I wrote to fill in some of these details when I was teaching.

See also blog posts tagged Probability and statistics and the Twitter account ProbFact.

Last week: Numerical computing resources

Next week: Miscellaneous math notes

More data, less accuracy

Statistical methods should do better with more data. That’s essentially what the technical term “consistency” means. But with improper numerical techniques, the the numerical error can increase with more data, overshadowing the decreasing statistical error.

There are three ways Bayesian posterior probability calculations can degrade with more data:

  1. Polynomial approximation
  2. Missing the spike
  3. Underflow

Elementary numerical integration algorithms, such as Gaussian quadrature, are based on polynomial approximations. The method aims to exactly integrate a polynomial that approximates the integrand. But likelihood functions are not approximately polynomial, and they become less like polynomials when they contain more data. They become more like a normal density, asymptotically flat in the tails, something no polynomial can do. With better integration techniques, the integration accuracy will improve with more data rather than degrade.

With more data, the posterior distribution becomes more concentrated. This means that a naive approach to integration might entirely miss the part of the integrand where nearly all the mass is concentrated. You need to make sure your integration method is putting its effort where the action is. Fortunately, it’s easy to estimate where the mode should be.

The third problem is that software calculating the likelihood function can underflow with even a moderate amount of data. The usual solution is to work with the logarithm of the likelihood function, but with numerical integration the solution isn’t quite that simple. You need to integrate the likelihood function itself, not its logarithm. I describe how to deal with this situation in Avoiding underflow in Bayesian computations.

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If you’d like help with statistical computation, let’s talk.

Probability resources

Each Wednesday I post a list of notes on some topic. This week it’s probability.

See also posts tagged probability and statistics and the Twitter account ProbFact.

Last week: Python resources

Next week: Regular expression resources

After a coin comes up heads 10 times

Suppose you’ve seen a coin come up heads 10 times in a row. What do you believe is likely to happen next? Three common responses:

  1. Heads
  2. Tails
  3. Equal probability of heads or tails.

Each is reasonable in its own context. The last answer is correct assuming the flips are independent and heads and tails are equally likely.

But as I argued here, if you see nothing but heads, you have reason to question the assumption that the coin is fair. So there’s some justification for the first answer.

The reasoning behind the second answer is that tails are “due.” This isn’t true if you’re looking at independent flips of a fair coin, but it could reasonable in other settings, such as sampling without replacement.

Say there are a number of coins on a table, covered by a cloth. A fixed number are on the table heads up, and a fixed number tails up. You reach under the cloth and slide a coin out. Every head you pull out increases the chances that the next coin will be tails. If there were an equal number of heads and tails under the cloth to being with, then after pulling out 10 heads tails are indeed more likely next time.

Related post: Long runs

First two impressions of statistics

When I was a postdoc I asked a statistician a few questions and he gave me an overview of his subject. (My area was PDEs; I knew nothing about statistics.) I remember two things that he said.

  1. A big part of being a statistician is knowing what to do when your assumptions aren’t met, because they’re never exactly met.
  2. A lot of statisticians think time series analysis is voodoo, and he was inclined to agree with them.

Blue Bonnet Bayes

Blue Bonnet™ used to run commercials with the jingle “Everything’s better with Blue Bonnet on it.” Maybe they still do.

Perhaps in reaction to knee-jerk antipathy toward Bayesian methods, some statisticians have adopted knee-jerk enthusiasm for Bayesian methods. Everything’s better with Bayesian analysis on it. Bayes makes it better, like a little dab of margarine on a dry piece of bread.

There’s much that I prefer about the Bayesian approach to statistics. Sometimes it’s the only way to go. But Bayes-for-the-sake-of-Bayes can expend a great deal of effort, by human and computer, to arrive at a conclusion that could have been reached far more easily by other means.

Related: Bayes isn’t magic

Image via Gallery of Graphic Design

Common sense and statistics

College courses often begin by trying to weaken your confidence in common sense. For example, a psychology course might start by presenting optical illusions to show that there are limits to your ability to perceive the world accurately. I’ve seen at least one physics textbook that also starts with optical illusions to emphasize the need for measurement. Optical illusions, however, take considerable skill to create. The fact that they are so contrived illustrates that your perception of the world is actually pretty good in ordinary circumstances.

For several years I’ve thought about the interplay of statistics and common sense. Probability is more abstract than physical properties like length or color, and so common sense is more often misguided in the context of probability than in visual perception. In probability and statistics, the analogs of optical illusions are usually called paradoxes: St. Petersburg paradox, Simpson’s paradox, Lindley’s paradox, etc. These paradoxes show that common sense can be seriously wrong, without having to consider contrived examples. Instances of Simpson’s paradox, for example, pop up regularly in application.

Some physicists say that you should always have an order-of-magnitude idea of what a result will be before you calculate it. This implies a belief that such estimates are usually possible, and that they provide a sanity check for calculations. And that’s true in physics, at least in mechanics. In probability, however, it is quite common for even an expert’s intuition to be way off. Calculations are more likely to find errors in common sense than the other way around.

Nevertheless, common sense is vitally important in statistics. Attempts to minimize the need for common sense can lead to nonsense. You need common sense to formulate a statistical model and to interpret inferences from that model. Statistics is a layer of exact calculation sandwiched between necessarily subjective formulation and interpretation. Even though common sense can go badly wrong with probability, it can also do quite well in some contexts. Common sense is necessary to map probability theory to applications and to evaluate how well that map works.

Inverted sense of risk

Watching the news gives you an inverted sense of risk.

We fear bad things that we’ve seen on the news because they make a powerful emotional impression. But the things rare enough to be newsworthy are precisely the things we should not fear. Conversely, the risks we should be concerned about are the ones that happen too frequently to make the news.