Sign test - overview
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Sign test | Pearson correlation | Two sample $t$ test - equal variances not assumed |
You cannot compare more than 3 methods |
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Independent variable | Variable 1 | Independent/grouping variable | |
2 paired groups | One quantitative of interval or ratio level | One categorical with 2 independent groups | |
Dependent variable | Variable 2 | Dependent variable | |
One of ordinal level | One quantitative of interval or ratio level | One quantitative of interval or ratio level | |
Null hypothesis | Null hypothesis | Null hypothesis | |
| H0: $\rho = \rho_0$
Here $\rho$ is the Pearson correlation in the population, and $\rho_0$ is the Pearson correlation in the population according to the null hypothesis (usually 0). The Pearson correlation is a measure for the strength and direction of the linear relationship between two variables of at least interval measurement level. | H0: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2. | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
| H1 two sided: $\rho \neq \rho_0$ H1 right sided: $\rho > \rho_0$ H1 left sided: $\rho < \rho_0$ | H1 two sided: $\mu_1 \neq \mu_2$ H1 right sided: $\mu_1 > \mu_2$ H1 left sided: $\mu_1 < \mu_2$ | |
Assumptions | Assumptions of test for correlation | Assumptions | |
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Test statistic | Test statistic | Test statistic | |
$W = $ number of difference scores that is larger than 0 | Test statistic for testing H0: $\rho = 0$:
| $t = \dfrac{(\bar{y}_1 - \bar{y}_2) - 0}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}} = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s^2_1$ is the sample variance in group 1, $s^2_2$ is the sample variance in group 2, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. The denominator $\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1 - \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1 - \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2 - \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$. | |
Sampling distribution of $W$ if H0 were true | Sampling distribution of $t$ and of $z$ 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. | Sampling distribution of $t$:
| Approximately the $t$ distribution with $k$ degrees of freedom, with $k$ equal to $k = \dfrac{\Bigg(\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}\Bigg)^2}{\dfrac{1}{n_1 - 1} \Bigg(\dfrac{s^2_1}{n_1}\Bigg)^2 + \dfrac{1}{n_2 - 1} \Bigg(\dfrac{s^2_2}{n_2}\Bigg)^2}$ or $k$ = the smaller of $n_1$ - 1 and $n_2$ - 1 First definition of $k$ is used by computer programs, second definition is often used for hand calculations. | |
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:
| $t$ Test two sided:
| Two sided:
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n.a. | Approximate $C$% confidence interval for $\rho$ | Approximate $C\%$ confidence interval for $\mu_1 - \mu_2$ | |
- | First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
| $(\bar{y}_1 - \bar{y}_2) \pm t^* \times \sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}$
where the critical value $t^*$ is the value under the $t_{k}$ distribution with the area $C / 100$ between $-t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20). The confidence interval for $\mu_1 - \mu_2$ can also be used as significance test. | |
n.a. | Properties of the Pearson correlation coefficient | n.a. | |
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n.a. | n.a. | Visual representation | |
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Equivalent to | Equivalent to | n.a. | |
Two sided sign test is equivalent to
| OLS regression with one independent variable:
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Example context | Example context | Example context | |
Do people tend to score higher on mental health after a mindfulness course? | Is there a linear relationship between physical health and mental health? | Is the average mental health score different between men and women? | |
SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
| Analyze > Correlate > Bivariate...
| Analyze > Compare Means > Independent-Samples T Test...
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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
| Regression > Correlation Matrix
| T-Tests > Independent Samples T-Test
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Practice questions | Practice questions | Practice questions | |