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 = 0.1269Both 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.

Thomas Lumley Last modified: Fri Sep 2 13:54:46 PDT 2005