Cramér's $V$ on Rao-Scott adjusted Pearson $\chi^2$ I have survey data with design weights for stratified sampling. My ultimate goal is to estimate Cramér's $V$ for contingency tables, a Pearson $\chi^2$ based measure. To account for the weights, I am thinking about using Rao-Scott adjustment to estimate $\chi^2$ statistics. However, then it is not clear how to get to Cramér's $V$.
Is it accurate to estimate $V$ from a first-order Rao-Scott adjusted $\chi^2$, just like I would estimate it from an unadjusted $\chi^2$? Or should $V$ be adjusted as well?
Thank you!
Some notes:
Link (Original Article)
http://www.amstat.org/sections/srms/proceedings/y2007/Files/JSM2007-000874.pdf (Some summary)
 A: I don't think Cramer's V makes much sense with complex survey data, as your $N$ in the denominator is not well defined. There is a number of replacements for it that can be offered: the design degrees of freedom (# of strata minus # of PSUs); the effective sample size (= sample size / design effect); the denominator d.f. of the Rao-Scott 2nd order correction; or there may be something yet else. If Alastair Scott does not discuss Cramer's V, then it likely just does not make sense.
With complex survey data, hypothesis testing can only be conducted using the Rao-Scott corrected test statistic and its (weird fractional) degrees of freedom, anyway. So coming up with interesting transformations of the statistic may be an interesting scholastic exercise, but it will hardly give you any additional insights.
A: I've just been asked how to do this for the survey package. It looks as though Cramer's $V$ and Udny's $\phi$ do have reasonable definitions as (super)population quantities.
In particular, in the $2\times 2$ case
$$\phi=\frac{n_{00}n_{11}-n_{01}n_{10}}{\sqrt{n_{0\cdot}n_{1\cdot}n_{\cdot 0}n_{\cdot 1}}}$$
and this quantity is just the Pearson correlation and for 2x2 tables $V=\phi$
A plug-in estimator of this for the population would be
$$\hat\phi=\frac{\hat N_{00}\hat N_{11}-\hat N_{01}\hat N_{10}}{\sqrt{\hat N_{0\cdot}\hat N_{1\cdot}\hat N_{\cdot 0}\hat N_{\cdot 1}}}.$$
This is also what you get if you compute it from the Pearson $X^2$ statistic on the population table, which fits with the original motivation of Rao-Scott statistics as approximate corrections to statistics inappropriately computed from the estimated population table.
svycramerV<-function(formula,design,...){
    tbl<-svytable(formula,design,...)
    chisq<-chisq.test(tbl, correct=FALSE)$statistic
    N<-sum(tbl)
    V<-chisq/N/min(dim(tbl)-1)
    names(V)<-"V"
    sqrt(V)
}

So I think there's a reasonable case for defining $\hat V$ or $\hat \phi$ from the estimated population contingency table, as long as there isn't a standard definition that contradicts it.
