A X^2 statistic computed from an estimated population table is too large, because the effective sample size is much smaller than the population size. Even after rescaling, its distribution is not exactly chi-squared. However, a chi-squared or F distribution for the rescaled statistic give reasonable approximations. The default is the F distribution, the "second-order Rao-Scott adjustment".
Using the dclus1 design object constructed in an earlier example we examine whether the proportion of schools meeting their "school-wide growth target" is different by school type. We use the default second-order adjustment and the first-order adjustment (the chi-squared approximation).
> svytable(~sch.wide + stype, dclus1)
stype
sch.wide E H M
No 406.1640 101.5410 270.7760
Yes 4467.8035 372.3170 575.3989
> svychisq(~sch.wide + stype, dclus1)
Pearson's X^2: Rao & Scott adjustment
data: svychisq(~sch.wide + stype, dclus1)
F = 5.1934, ndf = 1.495, ddf = 20.925, p-value = 0.02175
> svychisq(~sch.wide + stype, dclus1, statistic = "Chisq")
Pearson's X^2: Rao & Scott adjustment
data: svychisq(~sch.wide + stype, dclus1, statistic = "Chisq")
X-squared = 11.9409, df = 2, p-value = 0.005553
The other type of test is a Wald test based on the differences between the observed cell counts and those expected under independence (Koch et al, International Statistical Review 43: 59-78). I believe the statistic="Wald" test is the one used by SUDAAN. Using statistic="adjWald" reduces the statistic when the number of PSUs is small compared to the number of degrees of freedom. Rao & Thomas (JASA 82:630-636) recommend the adjusted version.
> svychisq(~sch.wide+stype, dclus1, statistic="adjWald")
Design-based Wald test of association
data: svychisq(~sch.wide + stype, dclus1, statistic = "adjWald")
F = 2.2296, ndf = 2, ddf = 13, p-value = 0.1471
> svychisq(~sch.wide+stype, dclus1, statistic="Wald")
Design-based Wald test of association
data: svychisq(~sch.wide + stype, dclus1, statistic = "Wald")
F = 2.4011, ndf = 2, ddf = 14, p-value = 0.1269
Both types of test are also available for designs with replicate
weights. As the number of clusters is not available with replicate
weights the degrees of freedom are based on the rank of the matrix of
replicate weights.