Sign Test

1. Applications

Applications of the Sign Test:

• Single population quantile or median test.
• Single population proportion test.
• Paired population median test.

Sign Test is applicable for small sample sizes and when the data is not normally distributed.

2. Test Procedure

Given $n$ data points $x_1,x_2,\dots ,x_n$, a null hypothesis $H_0:\text{median}=m_0$, and an alternative hypothesis $H_1:\text{median}\ne m_0$, the Sign Test is used to test the null hypothesis under a given significance level $\alpha$.

1. Go through the data points and remove ties, i.e. data points equal to $m_0$. Redefine the sample size as $n^{\prime }$.

2. Compute the number of data points greater than $m_0$, denoted as $N_{+}$, and the number of data points less than $m_0$, denoted as $N_{-}$.

3. The test statistic is given by $S=N_{+}$ by convention. Under the null hypothesis, $S\stackrel{H_0}{\sim }\text{Binomial}(n^{\prime },0.5)$.

4. Compute the p-value:

   $$p=2\cdot \mathbb{P}(S\ge s_{obs}|H_0)=2\sum _{k=s_{obs}}^{n^{\prime }}(\genfrac{}{}{0ex}{}{n^{\prime }}{k})0.5^{n^{\prime }},$$

   where $s_{obs}$ is the observed value of $S$ under the sample data.

   • If $H_a:\text{median}>m_0$, then $p=\mathbb{P}(S\ge s_{obs})$.
   • If $H_a:\text{median}<m_0$, then $p=\mathbb{P}(S\le s_{obs})$.

3. Large Sample Approximation

Large sample approximation of the Sign Test:

• Empirically, for a Sign Test on the median, if $n\ge 10$, then the test statistic $S$ can be approximated by a normal distribution.
• In this case, the Sign Test is equivalent to the $\hat{p}$ Z-test.
• For binomial distribution, we may also consider the adjusted continuity correction for the normal approximation.

4. Examples

Example (Quantile Sign Test). A survey was conducted on the heights of $n=10$ students. Conduct a Sign Test to test the null hypothesis that the 3/4 quartile of the heights is 180 cm.

Solution

Consider the null hypothesis $H_0:Q_3=180$ cm, and alternative hypothesis $H_1:Q_3\ne 180$ cm.

First check for ties. Then $N_{+}=8$ and $N_{-}=2$. Here let $S=N_{+}=8$.

Under the null hypothesis, $S\stackrel{H_0}{\sim }\text{Binomial}(n,1/4)$.

Compute the p-value:

$$p=2\mathbb{P}(S\ge 8|H_0)=2\sum _{k=8}^{10}\left(\genfrac{}{}{0ex}{}{10}{k}\right)0.25^{k}0.75^{10-k}=0.0008316<\mathrm{0.05.}$$

Reject the null hypothesis.

Example (Paired Sign Test). There are two types of tires, A and B. We want to test if A is significantly more durable than B. The data of $n=10$ cars are given.

• If we can assume that the data is from a normal distribution, we can use the paired t-test to test if the mean differs significantly.
• Otherwise, we can use the Sign Test, without assuming the distribution of the data, and test if the median differs significantly.

Solution

Consider the null hypothesis $H_0:\text{median}(A)=\text{median}(B)$, and alternative hypothesis $H_1:\text{median}(A)>\text{median}(B)$.

First check for ties, i.e. if $A_i=B_j$, and remove them.

Then compute the statistic:

$$S=\sum _{i=1}^n\mathbb{I}(A_i>B_i)=8.$$

Compute the p-value:

$$p=\mathbb{P}(S\ge 8|H_0)=\sum _{k=8}^{10}\left(\genfrac{}{}{0ex}{}{10}{k}\right)0.5^{10}=0.0547>\mathrm{0.05.}$$

Fail to reject the null hypothesis.

Related Notes

• Wilcoxon Signed Rank Test
• Nonparametric Tests Overview