ANCOVA: overview
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ANCOVA  MannWhitneyWilcoxon 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)  One categorical with 2 independent groups  
Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One categorical of ordinal level  
THIS TABLE IS YET TO BE COMPLETED  Null hypothesis  
  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
 
n.a.  Alternative hypothesis  
  If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
 
n.a.  Assumptions  
  Group 1 sample is a simple random sample (SRS) from population 1, group 2 sample is an independent SRS from population 2. That is, within and between groups, observations are independent of one another  
n.a.  Test statistic  
  Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2.  
n.a.  Sampling distribution of $W$ and of $U$ if H0 were true  
  Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: the formula for the standard deviations $\sigma_W$ and $\sigma_U$ 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.  Equivalent to  
  If no ties in the data: two sided MannWhitneyWilcoxon test is equivalent to KruskalWallis test with an independent variable with 2 levels ($I = 2$)  
n.a.  Example context  
  Do men tend to score higher on social economic status than women?  
n.a.  SPSS  
  Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
 
n.a.  Jamovi  
  TTests > Independent Samples TTest
 
Practice questions  Practice questions  