Multinomial tests for the fit of iid observationsX1…,Xnto a specified distributionFare based on the countsNiof observations falling inkcellsE1, …,Ekthat partition the range of theXj. The earliest such test is based on the Pearson (1900) chi-squared statistic:X2= Σki=1(Ni–npi)2/npi, wherepi=PF(XjinEi) are the cell probabilities under the null hypothesis. A common competing test is the likelihood ratio test based onLR= 2 Σki=1Nilog(Ni/npi). Cressie and Read (1984) introduced a class of multinomial goodness-of-fit statistics,Rλ, based on measures of the divergence between discrete distributions. This class includes bothX2(when λ = 1) andLR(when λ = 0). All of theRλhave the same chi-squared limiting null distribution. The power of the commonly used members of the class is usually approximated from a noncentral chi-squared distribution that is also the same for all λ. We propose new approximations to the power that vary with the statistic chosen. Both the computation and results on asymptotic error rates suggest that the new approximations are greatly superior to the traditional power approximation for statisticsRλother than the PearsonX2. The derivation of the limiting null distribution for the Cressie—Read statistics, following that forLR, is based on a Taylor series expansion ofRλ, in whichX2is the dominant term. The same expansion produces the traditional noncentral chi-squared power approximation by considering sequences of alternative distributions for theXjthat approach the hypothesisFat a suitable rate. Our power approximations are obtained from a Taylor series expansion that is valid for arbitrary sequences of alternatives. When linear and quadratic terms are retained, an accurate but computationally difficult approximation,Aλ, in terms of linear combinations of noncentral chi-squares is obtained. A second approximation,Bλ, in terms of a single noncentral chi-squared distribution results from averaging the coefficients inAλ, This simple approximation performs well. In the important case of the statisticLR, Aλ=Bλand this new noncentral chi-squared approximation is very accurate. Retaining only linear terms in the expansion produces an approximationLλbased on a normal distribution; this is generally much inferior toAλand Bλ.