Abstract
The use of transformations to stabilize the variance of binomial or Poisson data is familiar(Anscombe [1], Bartlett [2, 3], Curtiss [4], Eisenhart [5]). The comparison of transformed binomial or Poisson data with percentage points of the normal distribution to make approximate significance tests or to set approximate confidence intervals is less familiar. Mosteller and Tukey [6] have recently made a graphical application of a transformation related to the square-root transformation for such purposes, where the use of "binomial probability paper" avoids all computation. We report here on an empirical study of a number of approximations, some intended for significance and confidence work and others for variance stabilization. For significance testing and the setting of confidence limits, we should like to use the normal deviate KK exceeded with the same probability as the number of successes xx from nn in a binomial distribution with expectation npnp, which is defined by 12π∫K−∞e−12t2dt=Prob{x≤k|midbinomial,n,p}.12π∫K−∞e−12t2dt=Prob{x≤k|midbinomial,n,p}. The most useful approximations to KK that we can propose here are NN (very simple), N+N+ (accurate near the usual percentage points), and N∗∗N∗∗ (quite accurate generally), where N=2(√(k+1)q−√(n−k)p).N=2(√(k+1)q−√(n−k)p). (This is the approximation used with binomial probability paper.) N+=N+N+2p−112√E,E=lesser ofnpandnq,N∗=N+(N−2)(N+2)12(1√np+1−1√nq+1),N∗∗=N∗+N∗+2p−112√E⋅E=lesser ofnpandnq.N+=N+N+2p−112√E,E=lesser ofnpandnq,N∗=N+(N−2)(N+2)12(1√np+1−1√nq+1),N∗∗=N∗+N∗+2p−112√E⋅E=lesser ofnpandnq. For variance stabilization, the averaged angular transformation sin−1√xn+1+sin−1√x+1n+1sin−1√xn+1+sin−1√x+1n+1 has variance within ±6±6 of 1n+12(angles in radians),821n+12(angles in degrees),1n+12(angles in radians),821n+12(angles in degrees), for almost all cases where np≥1np≥1. In the Poisson case, this simplifies to using √x+√x+1√x+√x+1 as having variance 1.
Citation
Murray F. Freeman. John W. Tukey. "Transformations Related to the Angular and the Square Root." Ann. Math. Statist. 21 (4) 607 - 611, December, 1950. https://doi.org/10.1214/aoms/1177729756
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