How to compute the probability associated with absurdly large Z-scores?

Question

Software packages for network motif detection can return enormously high Z-scores (the highest I've seen is 600,000+, but Z-scores of more than 100 are quite common). I plan to show that these Z-scores are bogus.

Huge Z-scores correspond to extremely low associated probabilities. The values of the associated probabilities are given on e.g. the normal distribution wikipedia page (and probably every stats textbook) for Z-scores of up to 6. So...

Question: How does one compute the error function $1-\mathrm{erf}(n/\sqrt{2})$ for n up to 1,000,000, say?

I'm particularly after an already implemented package for this (if possible). The best I've found so far is WolframAlpha, which manages to compute it for n=150 (here).

Answer 1

The question concerns the complementary error function

$$\textrm{erfc}(x) = \frac{2}{\sqrt{\pi}}\int_{x}^{\infty}\exp(-t^2) d t$$

for "large" values of $x$ ($=n/\sqrt{2}$ in the original question)--that is, between 100 and 700,000 or so. (In practice, any value greater than about 6 should be considered "large," as we will see.) Note that because this will be used to compute p-values, there is little value in obtaining more than three significant (decimal) digits.

To begin, consider the approximation suggested by @Iterator,

$$f(x) = 1 - \sqrt{1 - \exp \left(-x^2 \left(\frac{4 + ax^2}{\pi + ax^2}\right)\right)},$$

where

$$a = \frac{8(\pi-3)}{3(4-\pi)} \approx 0.439862.$$

Although this is an excellent approximation to the error function itself, it's a terrible approximation to $\textrm{erfc}$. However, there is a way to systematically fix that up.

For the p-values associated with such large values of $x$, we're interested in the relative error $f(x)/\textrm{erfc}(x) - 1$: we hope its absolute value would be less than 0.001 for three significant digits of precision. Unfortunately this expression is difficult to study for large $x$ due to underflows in double-precision computation. Here is one attempt, which plots the relative error versus $x$ for $0 \le x \le 5.8$:

The calculation becomes unstable once $x$ exceeds 5.3 or so and cannot deliver one significant digit past 5.8. This is no surprise: $\exp(-5.8^2) \approx 10^{-14.6}$ is pushing the limits of double-precision arithmetic. Because there is no evidence that the relative error is going to be acceptably small for larger $x$, we need to do better.

Performing the calculation in extended arithmetic (with Mathematica) improves our picture of what's going on:

The error increases rapidly with $x$ and shows no signs of leveling off. Past $x=10$ or so, this approximation doesn't even deliver one reliable digit of information!

However, the plot is starting to look linear. We might guess that the relative error is directly proportional to $x$. (This makes sense on theoretical grounds: $\textrm{erfc}$ is manifestly an odd function and $f$ is manifestly even, so their ratio ought to be an odd function. Thus we would expect the relative error, if it increases, to behave like an odd power of $x$.) This leads us to study the relative error divided by $x$. Equivalently, I choose to examine $x \cdot \textrm{erfc}(x)/f(x)$, because the hope is this should have a constant limiting value. Here is its graph:

Our guess appears to be borne out: this ratio does seem to be approaching a limit around 8 or so. When asked, Mathematica will supply it:

a1 = Limit[x (Erfc[x]/f[x]), x -> \[Infinity]]

The value is $a_1 = \frac{2 }{\sqrt{\pi }}e^{\frac{3 (-4+\pi )^2}{8 (-3+\pi )}} \approx 7.94325$. This enables us to improve the estimate: we take

$$f_1(x) = f(x) \frac{a_1}{x}$$

as the first refinement of the approximation. When $x$ is really large--greater than a few thousand--this approximation is just fine. Because it's still not going to be good enough for an interesting range of arguments between $5.3$ and $2000$ or so, let's iterate the procedure. This time, the inverse relative error--specifically, the expression $1 - \textrm{erfc}(x)/f_1(x)$--should behave like $1/x^2$ for large $x$ (by virtue of the previous parity considerations). Accordingly, we multiply by $x^2$ and find the next limit:

a2 = Limit[x^2 (a1 - x (Erfc[x]/f[x])), x -> \[Infinity]] 

The value is

$$a_2 = \frac{1}{32 \sqrt{\pi }} e^{\frac{3 (-4+\pi )^2}{8 (-3+\pi )}} \left(32-\frac{9 (-4+\pi )^3 \pi }{(-3+\pi )^2}\right) \approx 114.687.$$

This process can proceed as long as we like. I took it out one more step, finding

a3 = Limit[x^2 (a2 - x^2 (a1 - x (Erfc[x]/f[x]))), x -> \[Infinity]] 

with value approximately 1623.67. (The full expression involves a degree-eight rational function of $\pi$ and is too long to be useful here.)

Unwinding these operations yields our final approximation

$$f_3(x) = f(x)\left(a_1 - a_2/x^2 + a_3/x^4\right)/x.$$

The error is proportional to $x^{-6}$. Of import is the constant of proportionality, so we plot $x^6(1 - \textrm{erfc}(x) / f_3(x))$:

It rapidly approaches a limiting value around 2660.59. Using the approximation $f_3$, we obtain estimates of $\textrm{erfc}(x)$ whose relative accuracy is better than $2661/x^6$ for all $x \gt 0$. Once $x$ exceeds 20 or so, we have our three significant digits (or far more, as $x$ gets larger). As a check, here is a table comparing the correct values to the approximation for $x$ between $10$ and $20$:

 x  Erfc    Approximation      
10  2.088*10^-45    2.094*10^-45
11  1.441*10^-54    1.443*10^-54
12  1.356*10^-64    1.357*10^-64
13  1.740*10^-75    1.741*10^-75
14  3.037*10^-87    3.038*10^-87
15  7.213*10^-100   7.215*10^-100
16  2.328*10^-113   2.329*10^-113
17  1.021*10^-127   1.021*10^-127
18  6.082*10^-143   6.083*10^-143
19  4.918*10^-159   4.918*10^-159
20  5.396*10^-176   5.396*10^-176

In fact, this approximation delivers at least two significant figures of precision for $x=8$ on, which is just about where pedestrian calculations (such as Excel's NormSDist function) peter out.

Finally, one might worry about our ability to compute the initial approximation $f$. However, that's not hard: when $x$ is large enough to cause underflows in the exponential, the square root is well approximated by half the exponential,

$$f(x) \approx \frac{1}{2} \exp \left(-x^2 \left(\frac{4 + ax^2}{\pi + ax^2}\right)\right).$$

Computing the logarithm of this (in base 10) is simple, and readily gives the desired result. For example, let $x=1000$. The common logarithm of this approximation is

$$\log_{10}(f(x)) \approx \left(-1000^2 \left(\frac{4 + a \cdot 1000^2}{\pi + a \cdot 1000^2}\right) - \log(2)\right) / \log(10) \sim -434295.63047.$$

Exponentiating yields

$$f(1000) \approx 2.34169 \cdot 10^{-434296}.$$

Applying the correction (in $f_3$) produces

$$\textrm{erfc}(1000) \approx 1.86003\ 70486\ 32328 \cdot 10^{-434298}.$$

Note that the correction reduces the original approximation by over 99% (and indeed, $a_1/x \approx 1\text{%}$.) (This approximation differs from the correct value only in the last digit. Another well-known approximation, $\exp(-x^2)/(x\sqrt{\pi})$, equals $1.860038 \cdot 10^{-434298}$, erring in the sixth significant digit. I'm sure we could improve that one, too, if we wanted, using the same techniques.)

Answer 2

A simple upper bound

For very large values of the argument in the calculation of upper tail probability of a normal, excellent bounds exist that are probably as good as one will get using any other methods with double-precision floating point. For $z > 0$, let $$ \renewcommand{\Pr}{\mathbb{P}}\newcommand{\rd}{\mathrm{d}} S(z) := \Pr(Z > z) = \int_z^\infty \varphi(z) \rd z \>, $$ where $\varphi(z) = (2\pi)^{-1/2} e^{-z^2/2}$ is the standard normal pdf. I've used the notation $S(z)$ in deference to the standard notation in survival analysis. In engineering contexts, they call this function the $Q$-function and denote it by $Q(z)$.

Then, a very simple, elementary upper bound is $$\newcommand{\Su}{\hat{S}_u} \newcommand{\Sl}{\hat{S}_\ell} S(z) \leq \frac{\varphi(z)}{z} =: \Su(z) \> , $$ where the notation on the right-hand side indicates this is an upper-bound estimate. This answer gives a proof of the bound.

There are several nice complementary lower bounds as well. One of the handiest and easiest to derive is the bound $$ S(z) \geq \frac{z}{z^2+1} \varphi(z) =: \Sl(z) \> . $$ There are at least three separate methods for deriving this bound. A rough sketch of one such method can be found in this answer to a related question.

A picture

Below is a plot of the two bounds (in grey) along with the actual function $S(z)$.

How good is it?

From the plot, it seems that the bounds become quite tight even for moderately large $z$. We might ask ourselves how tight they are and what sort of quantitative statement in that regard can be made.

One useful measure of tightness is the absolute relative error $$ \newcommand{\err}{\mathcal{E}} \err(z) = \left|\frac{\Su(z) - S(z)}{S(z)}\right| \>. $$ This gives you the proportional error of the estimate.

Now, note that, since all of the involved functions are nonnegative, by using the bounding properties of $\Su(z)$ and $\Sl(z)$, we get $$ \err(z) = \frac{\Su(z) - S(z)}{S(z)} \leq \frac{\Su(z) - \Sl(z)}{\Sl(z)} = z^{-2} \> , $$ and so this provides a proof that for $z \geq 10$ the upper-bound is correct to within 1%, for $z \geq 28$ it is correct to within 0.1% and for $z \geq 100$ it is correct to within 0.01%.

In fact, the simple form of the bounds provides a good check on other "approximations". If, in the numerical calculation of more complicated approximations, we get a value outside these bounds, we can simply "correct" it to take the value of, e.g., the upper bound provided here.

There are many refinements of these bounds. The Laplace bounds mentioned here provide a nice sequence of upper and lower bounds on $S(z)$ of the form $R(z) \varphi(z)$ where $R(z)$ is a rational function.

Finally, here is another somewhat-related question and answer.

Answer 3

You can approximate it with much simpler functions - see this Wikipedia section for more information. The basic approximation is that $\textrm{erf}(x) \approx \textrm{sgn}(x)\sqrt{1 - \exp(-x^2 \frac{4/\pi + ax^2}{1+ax^2}})$

The article has an incorrect link for that section. The PDF referenced can be found in Sergei Winitzki's files - or at this link.