The Effects of an Arcsin Square Root Transform on a Binomial Distributed Quantity.

Imaging Science and Biomedical Engineering Division

ABSTRACT This document provides proofs of the following: • The binomial distribution can be approximated with a Gaussian distribution at large values of N . • The arcsin square-root transform is the variance stabilising transform for the binomial distribution. • The Gaussian approximation for the binomial distribution is more accurate at smaller values of N after the variance stabilising transform has been applied. The conclusion contains some comments concerning the relationship between the variance stabilising transform and the improved accuracy of the Gaussian approximation, which holds for both the binomial and Poisson distributions.

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    • "unmarried people, unemployed people, renting households), and D ia are relevant denominators (total adult populations, working populations or total households). Then one possible variance stabilising transformation is z ia =Z 0.5 ia with var(z ia ) ≈ a /D ia (Hogan and Tchernis, 2004); other possibilities are the arc-sin square root transform (Bromiley and Thacker, 2002) and the folded exponential (Piepho, 2003). While one might retain binomial sampling directly the use of normal approximations may often provide greater modelling flexibility; in more general terms, this is one justification underlying the use of data augmentation for binary data (Albert and Chib, 1993). "
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