9231 · 4.4
Non-parametric tests
Non-parametric tests are statistical methods for testing hypotheses that don't require your data to fit a specific pattern, like the bell curve of a normal distribution. They work by comparing the ranks of data points, not their exact values.
Need to know
What you need to know
- **Hypotheses:** $H_0: m = m_0$ vs $H_1: m \neq m_0$ (or $m > m_0$, or $m < m_0$), where $m$ is the population median.
- **Procedure:** For each data point $x_i$, record a '+' if $x_i > m_0$ and a '−' if $x_i < m_0$. Ignore any data points where $x_i = m_0$ and reduce the sample size $n$ accordingly.
- **Test Statistic:** Let $N_+$ be the number of '+' signs and $N_-$ be the number of '−' signs. The test statistic $S$ is the smaller of $N_+$ and $N_-$.
- **Distribution:** Under $H_0$, the number of pluses (or minuses) follows a binomial distribution, $X \sim B(n, 0.5)$. We can use this to find a p-value, $P(X \le S)$.
Explanation
Testing Without Assumptions
- State the null and alternative hypotheses regarding the population median(s).
- Calculate the test statistic by ranking the data or counting signs, discarding any zero differences.
- Find the critical value from the statistical tables for your chosen significance level.
- Compare your test statistic with the critical value to decide whether to reject the null hypothesis.