Sign Test

Non-parametric statistical tests tend to be more general, and easier to explain and apply, due to the lack of assumptions about the distribution of the population or population parameters. One such statistical method is known as the sign test.

The sign test can be used to test the hypothesis that there is "no difference in medians" between the continuous distributions of two random variables $X$ and $Y$, in the situation when we can draw paired samples from $X$ and $Y$. As outlined above, the sign test is a non-parametric test which makes very few assumptions about the nature of the distributions under examination. Because of this fact, it has very general applicability but may lack the statistical power of other tests.

The One-Sample Sign Test

This test concerns the median $\tilde { \mu }$ of a continuous population. The idea is that the probability of getting a value below the median or a value above the median is $\frac{1}{2}$. We test the null hypothesis:

${ H }_{ 0 }:\tilde { \mu } ={ \tilde { \mu } }_{ 0 }$

against an appropriate alternative hypothesis:

${ H }_{ 1 }:\tilde { \mu } \neq ,>,<{ \tilde { \mu } }_{ 0 }$

We count the number of values in the sample that are above ${ \tilde { \mu } }_{ 0 }$ and represent them with the $+$ sign and the ones falling below ${ \tilde { \mu } }_{ 0 }$ with the $-$.

For example, suppose that in a sample of students from a class the ages of the students are:

$\{ 23.5, 24.2, 19.2, 21, 34.5, 23.5, 27.7, 22, 38, 21.8, 25, 23 \}$

Test the claim that the median is less than $24$ years of age with a significance level of $\alpha = 0.05$. The hypothesis is then written as:

${ H }_{ 0 }:{ \tilde { \mu } }_{ 0 }=24$

${ H }_{ 1 }:{ \tilde { \mu } }_{ 0 }<24$

The test statistic $x$ is then the number of plus signs. In this case we get:

$\{ -,+,-,-,+,-,+,-,+,-,+,- \}$

Therefore, $x=5$.

The variable $X$ follows a binomial distribution with $n=12$ (number of values) and $p=\frac{1}{2}$. Therefore:

$\begin{aligned}P\left\{ X\le 5 \right\} &=0.0002+0.0029+0.0161+0.0537+0.1208+0.1934\\ &=0.3872\end{aligned}$

Since the $p$-value of $0.3872$ is larger than the significance level $\alpha = 0.05$, the null-hypothesis cannot be rejected. Therefore, we conclude that the median age of the population is not less than $24$ years of age. Actually in this particular class, the median age was $24$, so we arrive at the correct conclusion.

The Sign Test

The sign test involves denoting values above the median of a continuous population with a plus sign and the ones falling below the median with a minus sign in order to test the hypothesis that there is no difference in medians.