Sign test overview

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Sign test	Binomial test for a single proportion $z$ test for a single proportion $z$ test for the difference between two proportions Goodness of fit test Chi-squared test for the relationship between two categorical variables One sample $z$ test for the mean One sample $t$ test for the mean Paired sample $t$ test Two sample $z$ test Two sample $t$ test - equal variances not assumed Two sample $t$ test - equal variances assumed One way ANOVA Two way ANOVA Pearson correlation Regression (OLS)Logistic regression Mann-Whitney-Wilcoxon test Kruskal-Wallis test Sign test McNemar's test Cochran's Q test Marginal Homogeneity test / Stuart-Maxwell test Friedman test Wilcoxon signed-rank test One sample Wilcoxon signed-rank test Spearman's rho
Independent variable
2 paired groups
Dependent variable
One of ordinal level
Null hypothesis
H₀: P(first score of a pair exceeds second score of a pair) = P(second score of a pair exceeds first score of a pair) If the dependent variable is measured on a continuous scale, this can also be formulated as: H₀: the population median of the difference scores is equal to zero A difference score is the difference between the first score of a pair and the second score of a pair.
Alternative hypothesis
H₁ two sided: P(first score of a pair exceeds second score of a pair) $\neq$ P(second score of a pair exceeds first score of a pair) H₁ right sided: P(first score of a pair exceeds second score of a pair) > P(second score of a pair exceeds first score of a pair) H₁ left sided: P(first score of a pair exceeds second score of a pair) < P(second score of a pair exceeds first score of a pair) If the dependent variable is measured on a continuous scale, this can also be formulated as: H₁ two sided: the population median of the difference scores is different from zero H₁ right sided: the population median of the difference scores is larger than zero H₁ left sided: the population median of the difference scores is smaller than zero
Assumptions
Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Test statistic
$W = $ number of difference scores that is larger than 0
Sampling distribution of $W$ if H₀ were true
The exact distribution of $W$ under the null hypothesis is the Binomial($n$, $P$) distribution, with $n =$ number of positive differences $+$ number of negative differences, and $P = 0.5$. If $n$ is large, $W$ is approximately normally distributed under the null hypothesis, with mean $nP = n \times 0.5$ and standard deviation $\sqrt{nP(1-P)} = \sqrt{n \times 0.5(1 - 0.5)}$. Hence, if $n$ is large, the standardized test statistic $$z = \frac{W - n \times 0.5}{\sqrt{n \times 0.5(1 - 0.5)}}$$ follows approximately the standard normal distribution if the null hypothesis were true.
Significant?
If $n$ is small, the table for the binomial distribution should be used: Two sided: Check if $W$ observed in sample is in the rejection region or Find two sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $W$ observed in sample is in the rejection region or Find right sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $W$ observed in sample is in the rejection region or Find left sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$ If $n$ is large, the table for standard normal probabilities can be used: Two sided: Check if $z$ observed in sample is at least as extreme as critical value $z^$ or Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Right sided: Check if $z$ observed in sample is equal to or larger than critical value $z^$ or Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$ Left sided: Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
Equivalent to
Two sided sign test is equivalent to McNemar's test: $W = b$, and $z^2 = X^2$ Friedman test with two related groups
Example context
Do people tend to score higher on mental health after a mindfulness course?
SPSS
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples... Put the two paired variables in the boxes below Variable 1 and Variable 2 Under Test Type, select the Sign test
Jamovi
Jamovi does not have a specific option for the sign test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the two sided $p$ value that would have resulted from the sign test. Go to: ANOVA > Repeated Measures ANOVA - Friedman Put the two paired variables in the box below Measures
Practice questions

Sign test - overview