Obtaining z as input to Rosenthal formula on previous page for a nonparametric analysis using three or more groups
If you really want to quote an effect size for the pooled analysis ie comparing all the groups at once then you could estimate z as below (again following Field, 2005):
- Obtain the one or two-sided p-value for the chi-square for the data from using the Kruskal-Wallis and/or chi-square test (it will have more than one df)
- Find the z value that has the same one or two-sided p-value as that obtained just now for the chi-square. You can do this using a function in any stats package or in Excel.
For example in SPSS: suppose the two-sided p-value of my chi-square from doing either a Kruskal-Wallis or chi-square test is 0.02. If I run the below
COMPUTE PIN=0.02. COMPUTE P=1-(PIN/2). COMPUTE ZOUT=IDF.NORMAL(P,0,1). EXE.
give s z of 2.326 so if I say have a total sample size of 40 the the effect size = 2.326/sqrt(4)= 0.367. Alternatively you could run this from the gui using compute.
If you are using a one-tailed test and say had a one-sided p=0.01 then we could instead tweak the above and use the below
COMPUTE PIN=0.01. COMPUTE P=1-(PIN). COMPUTE ZOUT=IDF.NORMAL(P,0,1). EXE.
which gives a z=1.645.