# aliquote.org

The Student t-test is a well-known statistical test when it comes to comparing the mean of two groups, or of one group to a reference value. It is an interesting case in the statistical armory since it requires two estimates (the mean and the standard deviation) from the same sample as well as strong assumptions, but if we leave that aside it is also an interesting challenge from an algorithmic perspective.

Indeed, you may recall that the test statistic is computed as $(\bar x - \mu_0) / (s_x / \sqrt{n})$ in the case of a single sample of size $n$, for which $\bar x$ and $s_x$ denote the sample mean and standard deviation which are estimates for the corresponding population mean ($\mu$) and variance ($\sigma^2$), while $\mu_0$ is the parameter we are interested in under the null hypothesis $H_0:, \mu = \mu_0$. In this case, $s^2$ follows a chi-square distribution with $\eta=n-1$ degrees of freedom, while $\bar x$ follows a gaussian distribution with parameter $\mu$ and $\sigma^2/n$.1 The Student $t$ distribution has a probability density function that can be expressed either as a combination of Gamma distributions, or using the following Beta distribution:

$$f(t) = \frac{1}{\sqrt{\eta}B(\frac{1}{2},\frac{\eta}{2})}\left( 1+\frac{t^2}{\eta}\right)^{-\frac{\eta+1}{2}}.$$

Depending on whether $\eta$ is odd or even, specific asymptotic expansions can be used but only in the case where $\eta$ is an integer value, which is not always the case, e.g., when using Welch or Satterthwaite approximations.

If you look into the R source code, you will find a standard helper function, src/library/stats/R/t.test.R, which computes the test statistic, its p-value and associated confidence interval. Nothing really surprising here, you could write a function that does the same job, providing you take care of computing the variance correctly (cf. possible catastrophic cancellation discussed in a previous post). What’s more difficult is to get the p-value, and the confidence interval de facto, since it involved a continuous distribution that is quite complicated.

If you look more carefully, you will notice that the probability distribution function pt is implemented in C in src/nmath/pt.c:

> pt
function (q, df, ncp, lower.tail = TRUE, log.p = FALSE)
{
if (missing(ncp))
.Call(C_pt, q, df, lower.tail, log.p)
else .Call(C_pnt, q, df, ncp, lower.tail, log.p)
}
<bytecode: 0x7fc000643fe0>
<environment: namespace:stats>


The pnt.c file is used in the case of a non-central distribution.

Now, what’s inside the C code?

• First, a normal approximation (pnorm) is used when n is large (n > 4e5 for R version ≤ 2.6), and in more recent versions of R a modified Beta approximation in the case $x^2>10^{100}n$;
• in the other case, the p-value is estimated from a Beta distribution, specifically a $\text{Beta}(x^2/(n+x^2), 0.5, n/2)$ when $n>x^2$, or a $\text{Beta}(1/(1 + x^2/n), n/2, 0.5)$ if $n \le x^2$.

The last two conditions aim at limiting possible underflow and come from Abramowitz & Stegun’s textbook, Handbook of Mathematical Functions, specifically the following formula for the incomplete Beta function:

$$I_x(a, b) = \frac{x^a(1-x)^b}{aB(a, b)} \left[ 1 + \sum_{n=0}^\infty \frac{B(a+1,n+1)}{B(a+b,n+1)} x^{n+1} \right],$$

where $B(a, b)$ denotes a Beta distribution with shape parameters $a$ and $b$, which are called $\alpha$ and $\beta$ on Wikipedia. In this case, $a = n/2$ and $b= 1/2$.

Beside R, there’s algorithm AS 3 that’s lurking on StatLib, but you will find a more detailed discussion the CACM, specifically Algorithm AS 395. In fact, LispStat relies on the latter, and again, a distinction is made depending on whether the degrees of freedom ($\eta=n-2$) and the test statistic, $t$, are considered small ($\eta<20$ and $t < 4$) or large enough, in which case a Beta distribution $B(1/2, n/2)$ is used.1 In fact, there are other corner cases that are treated, like the cases where:

• $\eta < 2$ and $\eta \ne 1$ (the degrees of freedom are not necessarily integers) — despite the values, this is treated as what is considered large in the above statement and relies on a $B(1/2, n/2)$; otherwise, LispStat relies on asymptotic series, as described p. 619 of Algorithm 395;
• nested summation of cosine series are used in all cases where none of the following conditions are verified: $\eta > \lfloor n\rfloor$, or $n \ge 20$ and $t < n$, or $n > 20$ — this is what is considered “small” in the above description.

Apparently, Stata relies on a Gamma distribution and, per the documentation, the function ttail computes the reverse cumulative (survivor) Student’s t distribution (i.e., $\Pr(T>t)$) as $\int_t^\infty \frac{\Gamma\big((\eta+1)/2\big)}{\sqrt{\pi\eta}\phantom{:}\Gamma(\eta/2)}(1+x^2/\eta)^{-(\eta+1)/2}dx$, with $2^{-10}\le\eta\le 2^{17}$, $\eta\in\mathbb{R}^+$. Finally, Python uses an incomplete Beta in the scipy package for the stats/ttest_* functions. It is defined in the special submodule, as C code (cephes/stdtr.c).

1. The parameterization of the Gaussian or Beta distribution may vary depending on the software you use. ↩︎