Non-Parametric Tests

Non-parametric tests do not assume a particular distribution for the data. They are used when normality assumptions are questionable, when data are ordinal, or when sample sizes are very small. This lesson covers the sign test and the Wilcoxon signed-rank test.

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When to Use Non-Parametric Tests

Situation	Parametric test	Non-parametric alternative
One-sample location test (normal data)	One-sample t-test	Sign test or Wilcoxon signed-rank test
Paired data (normal differences)	Paired t-test	Sign test or Wilcoxon signed-rank test
Two independent samples (normal)	Two-sample t-test	Mann-Whitney U test

Non-parametric tests are generally less powerful than parametric tests when the parametric assumptions hold, but they are more robust when assumptions are violated.

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The Sign Test

The sign test is the simplest non-parametric test. It tests the median of a distribution.

One-Sample Sign Test

$H_0$H0​: The population median is $m_0$m0​. $H_1$H1​: The population median is not $m_0$m0​ (or $> m_0$>m0​ or $< m_0$<m0​).

Procedure:

1. For each observation, compute $x_i - m_0$xi​−m0​.
2. Count the number of positive differences ($S^+$S+) and negative differences ($S^-$S−).
3. Discard any zero differences; let $n$n be the remaining sample size.
4. Under $H_0$H0​, $S^+ \sim B(n, 0.5)$S+∼B(n,0.5).
5. Find the p-value using the binomial distribution.

Worked Example

A manufacturer claims the median weight of packets is 500 g. A sample of 12 packets gives:

498, 502, 497, 501, 503, 499, 504, 496, 505, 500, 498, 502

Differences from 500: -2, +2, -3, +1, +3, -1, +4, -4, +5, 0, -2, +2

Remove the zero: $n = 11$n=11. Count: $S^+ = 6$S+=6, $S^- = 5$S−=5.

For a two-tailed test, the test statistic is $\min(S^+, S^-) = 5$min(S+,S−)=5.

Using $B(11, 0.5)$B(11,0.5): $P(S \leq 5) + P(S \geq 6)$P(S≤5)+P(S≥6). Since the distribution is symmetric, $P(S \leq 5) = P(S \geq 6) = 0.5$P(S≤5)=P(S≥6)=0.5. The two-tailed p-value = 1. We do not reject $H_0$H0​.

This result makes sense — the counts are nearly equal, providing no evidence against the claimed median.

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Paired Sign Test

For paired data $(x_i, y_i)$(xi​,yi​), compute $d_i = x_i - y_i$di​=xi​−yi​ and apply the sign test to the differences with $m_0 = 0$m0​=0.

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The Wilcoxon Signed-Rank Test

The Wilcoxon signed-rank test is more powerful than the sign test because it uses both the signs and the magnitudes of the differences.

Assumptions

1. The differences are independent.
2. The distribution of differences is symmetric about the median.
3. Data are at least on an ordinal scale.

Procedure