The Sign Test

Source

What is the purpose of the sign test?

To test if the population median equals a given value (e.g. the sample median).

What are the assumptions made?

  • Independence: Each sample is independent of the others.
  • Identical Distributions
  • Continuity: There are no ties 
    • Needed for the hypothesis test
    • Means there is exactly one point x such that F(x) =1/2 and that point is the median ($theta$) 

How do you use it?

  1. Setup the null hypothesis:
    \[ \begin{aligned}
    H_0 &: m = m_0
    \end{aligned} \]
$m$ is the true median, $m_0$ is the proposed median.
  1. Setup the alternative hypothesis as one of:
    \[\begin{aligned}
    H_a &: m \neq m_0\\
    H_a &: m > m_0\\
    H_a &: m < m_0
    \end{aligned}\]
  2. Collect a random sample from the population.
  3. Assign a 1 or 0 to each value in the data:
    • $sign = 0 \; if value < m_0$
    • $sign = 1 \; if value > m_0$
    • remove value from sample if $value = m_0$
  4. Sum all the signs to get $k$:
    \[\begin{aligned}
    x &= \sum_{i=0}^{n-1} sign(value_i)
    \end{aligned} \]
  5. Find $k$ on the binomial distribution:
    • $n$ is the current sample size
    • $p = 0.5$ (If $H_0$ is true, half are above, half are below)
  6. Find the p-value:
    • if $H_a$ uses <, add probabilities for $x \leq k$
    • if $H_a$ uses >, add probabilities for $x \geq k$
    • if $H_a$ uses $neq$, $p = 2 \times sum(x \geq k)$
  7. Draw a conclusion:
If $p-value < alpha$ (0.05 for 95%), reject $H_0$. Otherwise don't reject.