In a previous post, we discussed how p-values are computed in the case of Student t-test in LispStat, R, Stata and Python. John Monahan has more to say about the Student’s t distribution in his book Numerical Methods in Statistics,^{1} and how it can be approximated.
Specifically, let’s denote $Q(t\mid k) = \Pr(X > t)$, assuming $X$ follows a Student’s t distribution with $k$ df. As we noted earlier, the fact that $k$ ($\eta$ in our previous post) is an integer or not plays a role when one looks for an analytical solution. If $k$ is an integer, Abramowitz and Stegun gave the following formula:^{2}
$$ Q(t\mid k) = \begin{cases} \frac{1}{2} - \frac{\theta}{\pi}, \quad k=1 \cr \frac{1}{2} - \frac{1}{\pi}\left[\theta + \sin\theta\left(\cos\theta + \frac{2}{3}\cos^3\theta + \dots + \frac{2\times 4\times\dots\times (k-3)}{1\times 3\times\dots\times (k-2)}\cos^{k-2}\theta\right)\right], \quad k\ \text{odd} \cr \frac{1}{2} - \sin\theta\left[1 + \frac{1}{2}\cos^2\theta + \frac{1\times 3}{2\times 4}\cos^4\theta + \dots + \frac{1\times 3\times\dots\times (k-3)}{2\times 4\times\dots\times (k-2)}\cos^{k-2}\theta\right], \quad k\ \text{even} \end{cases} $$
where $\theta = \text{arctan}(t/\sqrt{k})$.^{3} With large value of $k$, the incomplete beta function may be used as follows:
$$ Q(t\mid k) = \frac{1}{2}I_x(k/2,1/2) \quad \text{for}\; x = k/(k+t^2). $$
Note that the use of a Beta distribution was also at hand in our previous post. Furthermore, for large value of $t$, some of the trigonometric expressions above may be avoided by using the fact that $\sin(\theta) = 1/\sqrt{1+w}$ and $\cos(\theta) = \sqrt{\frac{w}{1+w}}$, with $w=k/t^2$. When $k$ is even, we have:
$$ Q(t\mid k) = \frac{\sqrt{1+w} - B}{2\sqrt{1+w}}, \quad \text{where}\ B = \sum_{j=1}^{n/2-1} c_j\left(\frac{w}{1+w}\right)^j $$
with $c_j = (2j-1)c_{j-1}/(2j)$, and $c_0 = 1$. To avoid cancellation, we can use the following power series: $\sqrt{1+w} = 1 + \frac{1}{2}w - \frac{1}{8}w^2 + \frac{1}{16}w^3 - \frac{5}{128}w^4 + \frac{7}{256}w^5 - \dots$, which allows to compute $\sqrt{1+w}-B$ by matching powers of $w$.
Similarly, when $k$ is odd,
$$ Q(t\mid k) = \frac{1}{\pi}\left[\frac{\pi}{2} - \text{arctan}\left(\frac{t}{\sqrt{k}}\right) - \frac{\sqrt{w}}{1+w}\sum_{j=1}^{(n-3)/2} d_jw^j\right], $$
where $d_j = 2jd_{j-1}/(2j+1)$ and $d_0 = 1$. Again, the following series can be used: $\sqrt{1+w} = \frac{\pi}{2} - \text{arctan}\left(\frac{t}{\sqrt{k}}\right) = \frac{1}{\sqrt{w}}\left[1-\frac{w}{3} + \frac{w^2}{5} - \frac{w^3}{7} + \frac{w^4}{9} - \dots \right]$.
The tail probabilities for Student’s $t$ distribution can also be represented using the Gamma distribution as:
$$ Q(t\mid k) = \frac{\Gamma\big((k+1)/2\big)}{\Gamma(k/2)\sqrt{\pi}}u^{-k/2}\sum_{j=0}^\infty\frac{c_j}{(k+2)u^j}, $$
where $c_0 = 1$, $c_j = \frac{2j-1}{2j}c_{j-1}$, and $u = 1 + \frac{t^2}{k}$.
Currently, there’s no way to compute tail probability from the Student’s $t$ distribution in Racket math library, although it has everything we need for the Gamma or Beta distribution. Let’s do this by hand using Racket’s builtin facilities to compute the regularized incomplete Beta function:
(require math/special-functions)
(define (pt t df [two-tailed? #f])
(let ([beta (* 0.5 (beta-inc (* 0.5 df) 0.5 (/ df (+ df (sqr t))) #f #t))])
(if two-tailed? (* 2 beta) beta)))
(pt 2.41 36)
; => 0.010594171035128275
R returns $Q(2.41 \mid 36) = 0.01059417$ for the one-tailed probability.
John F. Monahan. Numerical Methods in Statistics. Cambridge University Press (2011). ↩︎
Abramowitz and Stegun. Handbook of Mathematical Functions ↩︎
As noted by Monahan, for large values of $t/\sqrt{k}$, which may happen for very large test statistic under low df distribution, this expression may yield to potential cancellation. ↩︎