Sign test  overview
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Sign test  Pearson correlation  $z$ test for the difference between two proportions  Cochran's Q test 


Independent variable  Variable 1  Independent/grouping variable  Independent/grouping variable  
2 paired groups  One quantitative of interval or ratio level  One categorical with 2 independent groups  One within subject factor ($\geq 2$ related groups)  
Dependent variable  Variable 2  Dependent variable  Dependent variable  
One of ordinal level  One quantitative of interval or ratio level  One categorical with 2 independent groups  One categorical with 2 independent groups  
Null hypothesis  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}: $\pi_1 = \pi_2$
$\pi_1$ is the population proportion of 'successes' for group 1; $\pi_2$ is the population proportion of 'successes' for group 2  H_{0}: $\pi_1 = \pi_2 = \ldots = \pi_I$
$\pi_1$ is the population proportion of 'successes' for group 1; $\pi_2$ is the population proportion of 'successes' for group 2; $\pi_I$ is the population proportion of 'successes' for group $I$  
Alternative hypothesis  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: $\pi_1 \neq \pi_2$ H_{1} right sided: $\pi_1 > \pi_2$ H_{1} left sided: $\pi_1 < \pi_2$  H_{1}: not all population proportions are equal  
Assumptions  Assumptions of test for correlation  Assumptions  Assumptions  



 
Test statistic  Test statistic  Test statistic  Test statistic  
$W = $ number of difference scores that is larger than 0  Test statistic for testing H0: $\rho = 0$:
 $z = \dfrac{p_1  p_2}{\sqrt{p(1  p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
$p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, $n_2$ is the sample size of group 2 Note: we could just as well compute $p_2  p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1$  If a failure is scored as 0 and a success is scored as 1:
$Q = k(k  1) \dfrac{\sum_{groups} \Big (\mbox{group total}  \frac{\mbox{grand total}}{k} \Big)^2}{\sum_{blocks} \mbox{block total} \times (k  \mbox{block total})}$ Here $k$ is the number of related groups (usually the number of repeated measurements), a group total is the sum of the scores in a group, a block total is the sum of the scores in a block (usually a subject), and the grand total is the sum of all the scores. Before computing $Q$, first exclude blocks with equal scores in all $k$ groups.  
Sampling distribution of $W$ if H_{0} were true  Sampling distribution of $t$ and of $z$ if H_{0} were true  Sampling distribution of $z$ if H_{0} were true  Sampling distribution of $Q$ 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 $t$:
 Approximately the standard normal distribution  If the number of blocks (usually the number of subjects) is large, approximately the chisquared distribution with $k  1$ degrees of freedom  
Significant?  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:
 If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
 
n.a.  Approximate $C$% confidence interval for $\rho$  Approximate $C\%$ confidence interval for $\pi_1  \pi_2$  n.a.  
  First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
 Regular (large sample):
   
n.a.  Properties of the Pearson correlation coefficient  n.a.  n.a.  
 
     
Equivalent to  Equivalent to  Equivalent to  Equivalent to  
Two sided sign test is equivalent to
 OLS regression with one independent variable:
 When testing two sided: chisquared test for the relationship between two categorical variables, where both categorical variables have 2 levels  Friedman test, with a categorical dependent variable consisting of two independent groups  
Example context  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 proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.  Subjects perform three different tasks, which they can either perform correctly or incorrectly. Is there a difference in task performance between the three different tasks?  
SPSS  SPSS  SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 Analyze > Correlate > Bivariate...
 SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chisquared test instead. The $p$ value resulting from this chisquared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Analyze > Descriptive Statistics > Crosstabs...
 Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
 
Jamovi  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
 Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chisquared test instead. The $p$ value resulting from this chisquared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Frequencies > Independent Samples  $\chi^2$ test of association
 Jamovi does not have a specific option for the Cochran's Q test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the $p$ value that would have resulted from the Cochran's Q test. Go to:
ANOVA > Repeated Measures ANOVA  Friedman
 
Practice questions  Practice questions  Practice questions  Practice questions  