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. Running the below
COMPUTE PIN=0.02. COMPUTE P=1-(PIN/2). COMPUTE ZOUT=IDF.NORMAL(P,0,1). EXE.
gives a z of 2.326 so for a total sample size of 40 the the effect size = 2.326/sqrt(40)= 0.367. Alternatively you could run this from the gui choosing the 'compute' option.
If you are using a one-tailed test and, for example, 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.