The model is,

$$\operatorname{logit}(\pi(x)) = \alpha + \beta x $$

with independent binomial outcomes of $y_0$ successes in $n_0$ trials and $y_1$ successes in $n_1$ trials.

Now, I have to construct Wald and Score/LM tests for testing $\beta = 0$. However, I'm not very sure how to go about this. I know that for the Wald test,

$$z = \frac{\widehat{\beta}}{\text{SE}}, \text{ where } SE = \frac 1 {\sqrt{I\left(\widehat{\beta}\right)}} = \sqrt{\operatorname{Cov}\left(\widehat{\beta\,}\right)}$$

For the LM/Score test,

$$\frac{[u(\beta_0)]^2}{I(\beta_0)} = \left. \left(\frac{\partial^2 L(\beta)}{\partial \beta_0^2}\right) \right/ \left(-\operatorname E\left[\frac{\partial^2 L(\beta)}{\partial \beta_0^2} \right] \right)$$

This is probably dumb but I am not sure how to find $\operatorname{Cov}(\beta)$, or $L(\beta)$ and how to evaluate its partial derivative at 0. Any pointers would be helpful, thanks!


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