📂Statistical Test

Sign Test in Statistics

Hypothesis Testing ¹

$n$ random samples (../1715) are given as ordered pairs $\left\{ \left( X_{k} , Y_{k} \right) \right\}_{k=1}^{n}$. Let the probability that $X_{k}$ is greater than $Y_{k}$ under the distributions of the two populations $X_{1} , \cdots , X_{n}$ and $Y_{1} , \cdots , Y_{n}$ be $p$. The following hypothesis test about $p$ is called the sign test.

• $H_{0} : p = 0.5$, the distributions of the two populations are the same.
• $H_{1} : p \ne 0.5$, the distributions of the two populations are not the same.

Test statistic

For the indicator function $I$ define the following test statistic $T$. $$ T = \sum_{k=1}^{n} I \left( X_{k} > Y_{k} \right) $$ $T$ is the number of cases among the $n$ samples in which $X_{k}$ is greater than $Y_{k}$, and under the assumption that the null hypothesis is true it follows the binomial distribution $B \left( n , p \right)$.

Explanation

The sign test is, as the name suggests, a test that uses the count of ordered pairs with a positive sign as the test statistic; among nonparametric methods that are relatively unconstrained by the limitations of the data, it requires the fewest assumptions. Techniques such as the Mann–Whitney test focus on the location parameters of the two populations rather than comparing their distributions themselves; the sign test, literally, seems to permit any arbitrary distribution.

However, this interpretation has a significant caveat. While it is indeed clever to invoke a binomial distribution despite having no information about the underlying distributions, even in the case of two normal distributions $N \left( 0 , 1 \right)$ and $N \left( 0 , 2 \right)$ that differ only in population variance, the sign test cannot distinguish between them.