ANCOVA  overview
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ANCOVA  KruskalWallis 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 $I$ independent groups ($I \geqslant 2$)  
Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One 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 all $I$ 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 all $I$ 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, $\ldots$, group $I$ sample is an independent SRS from population $I$. That is, within and between groups, observations are independent of one another  
n.a.  Test statistic  
  $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i}  3(N + 1)$  
n.a.  Sampling distribution of $H$ if H0 were true  
  For large samples, approximately the chisquared distribution with $I  1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used.  
n.a.  Significant?  
  For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
 
n.a.  Example context  
  Do people from different religions tend to score differently on social economic status?  
n.a.  SPSS  
  Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
 
n.a.  Jamovi  
  ANOVA > One Way ANOVA  KruskalWallis
 
Practice questions  Practice questions  