ANCOVA: overview
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ANCOVA  Wilcoxon signedrank test 


Independent variables  Independent variable  
One or more categorical with independent groups, and one or more quantitative control variables of interval or ratio level (covariates)  2 paired groups  
Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One quantitative of interval or ratio level  
THIS TABLE IS YET TO BE COMPLETED  Null hypothesis  
  The median of the difference scores is zero in the population
Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher.  
n.a.  Alternative hypothesis  
 
 
n.a.  Assumptions  
 
 
n.a.  Test statistic  
  Two different types of test statistics can be used; both will result in the same test outcome. We will denote the first option the $W_1$ statistic (also known as the $T$ statistic), and the second option the $W_2$ statistic.
In order to compute each of the test statistics, follow the steps below:
 
n.a.  Sampling distribution of $W_1$ and of $W_2$ if H0 were true  
  Sampling distribution of $W_1$:
If $N_r$ is large, $W_1$ is approximately normally distributed with mean $\mu_{W_1}$ and standard deviation $\sigma_{W_1}$ if the null hypothesis were true. Here $$\mu_{W_1} = \frac{N_r(N_r + 1)}{4}$$ $$\sigma_{W_1} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{24}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_1  \mu_{W_1}}{\sigma_{W_1}}$$ follows approximately a standard normal distribution if the null hypothesis were true. Sampling distribution of $W_2$: If $N_r$ is large, $W_2$ is approximately normally distributed with mean $0$ and standard deviation $\sigma_{W_2}$ if the null hypothesis were true. Here $$\sigma_{W_2} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{6}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_2}{\sigma_{W_2}}$$ follows approximately a standard normal distribution if the null hypothesis were true. If $N_r$ is small, the exact distribution of $W_1$ or $W_2$ should be used. Note: the formula for the standard deviations $\sigma_{W_1}$ and $\sigma_{W_2}$ is more complicated if ties are present in the data.  
n.a.  Significant?  
  For large samples, the table for standard normal probabilities can be used: Two sided:
 
n.a.  Example context  
  Is the median of the differences between the mental health scores before and after an intervention different from 0?  
n.a.  SPSS  
  Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 
n.a.  Jamovi  
  TTests > Paired Samples TTest
 
Practice questions  Practice questions  