Sign test  overview
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Sign test  Pearson correlation  Paired sample $t$ test 


Independent variable  Variable 1  Independent variable  
2 paired groups  One quantitative of interval or ratio level  2 paired 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  
 H_{0}: $\rho = \rho_0$
$\rho$ is the unknown Pearson correlation in the population, $\rho_0$ is the 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.  H_{0}: $\mu = \mu_0$
$\mu$ is the population mean of the difference scores; $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0. A difference score is the difference between the first score of a pair and the second score of a pair.  
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
 H_{1} two sided: $\rho \neq \rho_0$ H_{1} right sided: $\rho > \rho_0$ H_{1} left sided: $\rho < \rho_0$  H_{1} two sided: $\mu \neq \mu_0$ H_{1} right sided: $\mu > \mu_0$ H_{1} left sided: $\mu < \mu_0$  
Assumptions  Assumptions of test for correlation  Assumptions  


 
Test statistic  Test statistic  Test statistic  
$W = $ number of difference scores that is larger than 0  Test statistic for testing H0: $\rho = 0$:
 $t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
$\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to the null hypothesis, $s$ is the sample standard deviation of the difference scores, $N$ is the sample size (number of difference scores). The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$.  
Sampling distribution of $W$ if H_{0} were true  Sampling distribution of $t$ and of $z$ if H_{0} were true  Sampling distribution of $t$ if H_{0} 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 $\sqrt{np(1p)} = \sqrt{n \times 0.5(1  0.5)}$. 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 $t$:
 $t$ distribution with $N  1$ 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:
 $t$ Test two sided:
 Two sided:
 
n.a.  Approximate $C$% confidence interval for $\rho$  $C\%$ confidence interval for $\mu$  
  First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
 $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N1}$ 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 $\mu$ can also be used as significance test.  
n.a.  Properties of the Pearson correlation coefficient  Effect size  
 
 Cohen's $d$: Standardized difference between the sample mean of the difference scores and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{s}$$ Indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0$  
n.a.  n.a.  Visual representation  
    
Equivalent to  Equivalent to  Equivalent to  
Two sided sign test is equivalent to
 OLS regression with one independent variable:

 
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 difference between the mental health scores before and after an intervention different from $\mu_0$ = 0?  
SPSS  SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 Analyze > Correlate > Bivariate...
 Analyze > Compare Means > PairedSamples T Test...
 
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
 TTests > Paired Samples TTest
 
Practice questions  Practice questions  Practice questions  