  We build on recent works on Stein's method for functions of multivariate
normal random variables to derive bounds for the rate of convergence of some
asymptotically chi-square distributed statistics. We obtain some general bounds
and establish some simple sufficient conditions for convergence rates of order
$n^{-1}$ for smooth test functions. These general bounds are applied to
Friedman's statistic for comparing $r$ treatments across $n$ trials and the
family of power divergence statistics for goodness-of-fit across $n$ trials and
$r$ classifications, with index parameter $\lambda\in\mathbb{R}$ (Pearson's
statistic corresponds to $\lambda=1$). We obtain a $O(n^{-1})$ bound for the
rate of convergence of Friedman's statistic for any number of treatments
$r\geq2$. We also obtain a $O(n^{-1})$ bound on the rate of convergence of the
power divergence statistics for any $r\geq2$ when $\lambda$ is a positive
integer or any real number greater than 5. We conjecture that the $O(n^{-1})$
rate holds for any $\lambda\in\mathbb{R}$.
