Statistics Meeting Book (June 20, 2018)

with corresponding confidence interval dPOD + t SE(dPOD)

(8) where ‘t’ is the corresponding student-t quantile for n -1 degrees of freedom, e.g., t = 2.201 for n = 10 and 95% confidence. Note that the interval in eq.(6) is always contained wholly within that of eq.(8), which is more conservative. Both the intervals in eqs.(6) and (8) can exceed the maximum limits of (-1, +1), and should be truncated to + 1 as the need arises. A more significant problem is that these intervals are ‘large-sample’ approximations and become degenerate if b and c are too small. Generally acceptable results will only be obtained if b + c is at least 3 or more in magnitude. In order to deal with the degeneracy problem, while still retaining the simplicity of formulation, we will apply ‘continuity-correction’, as mentioned above. For the Wald formula of eq.(5) and ε = ½, the SE(dPOD) estimate becomes SE(dPOD) = √{ [ (b + c + 1) / (n + 1) – ((b – c) / (n + 1)) 2 ] / n } (9) The corresponding adjustment for eq.(7) is SE(dPOD) = √{ [ (b + c + 1) / (n + 1) – ((b – c) / (n + 1)) 2 ] / (n -1) } (10) For the Wald formula of eq.(5) and ε = 1, the SE(dPOD) estimate becomes SE(dPOD) = √{ [ (b + c + 2) / (n + 2) – ((b – c) / (n + 2)) 2 ] / n } (11) The corresponding adjustment for eq.(7) is SE(dPOD) = √{ [ (b + c + 2) / (n + 2) – ((b – c) / (n + 2)) 2 ] / (n -1) } (12)

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