arcsinsqrtp

arcsinsqrtp refers to the composition arсsin(sqrt(p)) applied to a proportion p in the interval [0, 1]. The transformation maps [0, 1] to [0, π/2], since sqrt(p) ∈ [0, 1] and arcsin yields angles in that range. It is commonly known as the arcsine square root transformation or the angular transformation and is used primarily in statistics to stabilize the variance of proportion data.

Purpose and use: The transformation helps stabilize variance across the range of p, enabling the use of parametric methods that assume more constant variance (homoscedasticity) when analyzing proportions. It is frequently applied in the analysis of binomial proportions, meta-analyses of proportions, and ANOVA-like comparisons where residual variance should be roughly independent of the mean proportion.

Mathematical properties: If p_hat is an estimate of p from a binomial sample with n trials, then by the delta method the transformed variable y = arcsin(sqrt(p_hat)) has approximately constant variance of about 1/(4n) for large n, independent of p. The transformation is inverted by back-transforming p ≈ sin^2(y).

Related terms and alternatives: The arcsin square root transformation is one of several variance-stabilizing transformations for proportions. Others include the Freeman–Tukey double arcsine transformation, logit, and probit transformations. The choice depends on the context and the need for interpretability on the original scale.

Limitations: Interpretation on the transformed scale can be nonintuitive, and the variance stabilization is approximate for small sample sizes or extreme p values. Back-transformation can also complicate interpretation of results in practical terms.