Sign test - overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table
Sign test | Binomial test for a single proportion | Spearman's rho |
You cannot compare more than 3 methods |
---|---|---|---|
Independent variable | Independent variable | Variable 1 | |
2 paired groups | None | One of ordinal level | |
Dependent variable | Dependent variable | Variable 2 | |
One of ordinal level | One categorical with 2 independent groups | One of ordinal level | |
Null hypothesis | Null hypothesis | Null hypothesis | |
| H0: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis. | H0: $\rho_s = 0$
Here $\rho_s$ is the Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level. In words, the null hypothesis would be: H0: there is no monotonic relationship between the two variables in the population. | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
| H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | H1 two sided: $\rho_s \neq 0$ H1 right sided: $\rho_s > 0$ H1 left sided: $\rho_s < 0$ | |
Assumptions | Assumptions | Assumptions | |
|
|
| |
Test statistic | Test statistic | Test statistic | |
$W = $ number of difference scores that is larger than 0 | $X$ = number of successes in the sample | $t = \dfrac{r_s \times \sqrt{N - 2}}{\sqrt{1 - r_s^2}} $ Here $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores. | |
Sampling distribution of $W$ if H0 were true | Sampling distribution of $X$ if H0 were true | Sampling distribution of $t$ 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. | Binomial($n$, $P$) distribution.
Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis). | Approximately the $t$ distribution with $N - 2$ degrees of freedom | |
Significant? | Significant? | Significant? | |
If $n$ is small, the table for the binomial distribution should be used: Two sided:
If $n$ is large, the table for standard normal probabilities can be used: Two sided:
| Two sided:
| Two sided:
| |
Equivalent to | n.a. | n.a. | |
Two sided sign test is equivalent to
| - | - | |
Example context | Example context | Example context | |
Do people tend to score higher on mental health after a mindfulness course? | Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? | Is there a monotonic relationship between physical health and mental health? | |
SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
| Analyze > Correlate > Bivariate...
| |
Jamovi | Jamovi | 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
| Frequencies > 2 Outcomes - Binomial test
| Regression > Correlation Matrix
| |
Practice questions | Practice questions | Practice questions | |