Sign test
Ranks (nonparametric) · One sample · reference distribution: Binomial(n, ½)
When to use it
Test the median of a sample without assuming anything about the shape of the distribution; it only uses whether each value is above or below.
Null hypothesis
The population median equals m₀.
Assumptions
- Independent observations
- At least ordinal variable
Test statistic
S = \#\{\, x_i > m_0 \,\} \sim \text{Bin}\left(n, \tfrac{1}{2}\right)Effect size
Cohen’s g: how far the proportion of positive signs is from ½.
g = \hat{p} - \tfrac{1}{2}How to report it
15 of 20 above the reference value, sign test p = .041
In R and Python
R
binom.test(sum(x > m0), sum(x != m0), p = 0.5)Python
stats.binomtest(int((x > m0).sum()), int((x != m0).sum()), p=0.5)In Python, stats is scipy.stats and np is numpy.
Variants and alternatives
- Wilcoxon signed-rank, more powerful if the distribution is symmetric
Where it sits in the catalog
Ranks (nonparametric). Nonparametric tests assume no distribution for the data. They replace each value by its rank, its position in order (1st, 2nd, 3rd…), which protects them from extreme values and makes them fit ordinal data.
One sample. A single group compared with a reference value set in advance, such as a target or a standard.
In the decision tree
- What do you want to do? Compare groups, or one group with a reference value
- What kind of response did you measure? Numeric: a measurement, such as weight, time or score
- How many groups or measurements? One group, against a reference value
- Are the data approximately normal, or is the sample large (n ≳ 30)? No
- What do you want to do? Compare groups, or one group with a reference value
- What kind of response did you measure? Numeric: a measurement, such as weight, time or score
- How many groups or measurements? Two groups
- Are the groups made of different individuals or the same individuals? The same individuals, such as before and after (paired)
- Are the differences between pairs approximately normal? No, and not symmetric either