Weighted Least Squares for Binomial Distribution

Definition

Let $Y_i\sim B(m_i,{\pi }_i),i=1,\dots ,n$ and random variables $p_i=\frac{y_i}{m_i}$ and consider a variance stabilizing transformation $\psi (\cdot )$. Then, by Taylor expansion and Delta Method $E[\psi (p_i)]\approx \psi ({\pi }_i)$ and $\mathrm{Var}(\psi (p_i))\approx {\psi }^{\prime }({\pi }_i{)}^2\frac{{\pi }_i(1-{\pi }_i)}{m_i}$ hold. Now, the Pearson $X^2$ is defined as $X^2=\sum _{i=1}^n\frac{[\psi (p_i)-\psi ({\pi }_i){]}^2}{{\psi }^{\prime }({\pi }_i{)}^2{\pi }_i(1-{\pi }_i)/m_i}$ By minimizing $X^2$ we can find the WLS estimator $\hat{\pi }$. It can be seen as a Weighted Least Squares with $y_i=\psi (p_i)$, ${\hat{y}}_i=\psi ({\pi }_i)$, and $w_i=[{\psi }^{\prime }({\pi }_i{)}^2{\pi }_i(1-{\pi }_i)/m_i{]}^{-1}$

Widely Used Variance Stabilizing Transformations

• Logit function: $\psi (x)=\ln \left(\frac{x}{1-x}\right)$
• The arcsin squared-root function: $\psi (x)=\arcsin (\sqrt{x})$
• Empirical logistic function: $\psi (x)=\ln \left(\frac{x+0.5}{1-x+0.5}\right)$