Introduction

This document describes the equations used to derive the asymptotic T to Z and F to Z statistic transformation as used in film. The problem is to find Z as a function of T or F (and degrees of freedom) with a relative accuracy of $10^{-3}$ or better.

The standard definitions for the Z statistic is:

$$p = \frac{1}{2} - \frac{1}{2} \ensuremath{\mathrm{erf}}\left( \frac{Z}{\sqrt{2}} \right)$$ (1)

where $Z$ is the Z statistic score, $p$ is the (complementary) cumulative probability value (the integral of the pdf from the value to $\infty$), and the standard error function $\ensuremath{\mathrm{erf}}$ is defined as:

$$\ensuremath{\mathrm{erf}}(x) = \frac{2}{\sqrt{\pi}} \int_0^x \exp(-w^2) \, dw$$ (2)

The standard definitions for the T and F distributions are given from the incomplete beta function:

$$p = \frac{1}{2} \beta_{inc} \left( \frac{D}{D + T^2} , \frac{D}{2} , \frac{1}{2} \right)$$ (3)

$$p = \frac{1}{2} \beta_{inc} \left( \frac{D_2}{D_2 + D_1F} , \frac{D_2}{2} , \frac{D_1}{2} \right)$$ (4)

where $T$ is the T statistic score with degrees of freedom $D$, and $F$ is the F statistic score with degrees of freedom $D_1$ and $D_2$. The incomplete Beta function, $\beta_{inc}$, is defined as:

$$\beta_{inc}(x,v,w) = \frac{1}{\beta(v,w)} \int_0^x y^{v-1} (1-y)^{w-1} \, dy$$ (5)

where $\beta$ is the complete beta function, which can be expressed in terms of the Gamma function $\Gamma$:

$$\beta(v,w) = \beta_{inc}(1,v,w)$$ (6)

$$= \frac{\Gamma(v)\Gamma(w)}{\Gamma(v+w)}$$ (7)

Using equations 1, 3 and 4, a Z score for every corresponding T or F score can be calculated by using $p$ as an intermediate value. However, in practice, even for moderate values of T or F, the value of $p$ becomes extremely small and underflows standard numerical precision. Therefore an asymptotic form of these equations is sought for very small values of $p$. This is achieved by finding expansions for $\log(p)$, which will be well behaved.