Chapter 27 A Generalized Linear Model for Binomial Response Data

For all $i = 1, \ldots, n$, $y_i\sim \text{Binomial}(m_i, \pi_i)$, where $m_i$ is known number of trials for observation $i$, $\pi_i = \frac{\exp(x_i'\beta)}{1+\exp(x_i'\beta)}$, and $y_1, \ldots, y_n$ are independent. The Binomial log likelihood is $$\ell(\beta\mid y) = \sum_{i = 1}^n [y_ix_i'\beta - m_i\log(1 + \exp(-x_i'\beta))] + \text{constant}$$ We can compare the fit of a logitstic regression model known as saturated model. The MLE of $\pi_i$ under the logistic regression model is $\hat\pi_i = \frac{\exp(x_i'\hat\beta)}{1 + \exp(x_i'\hat\beta)}$, and the MLE of $\pi_i$ under saturated model is $y_i/m_i$. Then the likelihood ratio statistic for testing the logistic regression model as the reduced model VS. the saturated model as the full model is $$2 \sum_{i=1}^{n}\left[y_{i} \log \left(\frac{y_{i} / m_{i}}{\hat{\pi}_{i}}\right)+\left(m_{i}-y_{i}\right) \log \left(\frac{1-y_{i} / m_{i}}{1-\hat{\pi}_{i}}\right)\right]$$ which is called the Deviance Statistics, the Residual Deviance or just the Deviance.

A Lack-of-fit Test: when $n$ is large, and/or $m_1, \ldots m_n$, are each suitablely large, the Deviance Statistic is approximately $\chi_{n-p}^2$ if the logistic regression model is correct.

Deviance Residual: $$d_{i} \equiv \operatorname{sign}\left(y_{i} / m_{i}-\hat{\pi}_{i}\right) \sqrt{2\left[y_{i} \log \left(\frac{y_{i}}{m_{i} \hat{\pi}_{i}}\right)+\left(m_{i}-y_{i}\right) \log \left(\frac{m_{i}-y_{i}}{m_{i}-m_{i} \hat{\pi}_{i}}\right)\right]}$$ The residual deviance statistic is the sum of the squared deviance residuals $(\sum_{i = 1}^n d_i^2)$.

Pearson’s Chi-Square Statistic: Another lack of fit statistic that is approximately $\chi_{n-p}^2$ under the null is Pearson’s Chi-Square Statistic: $$\begin{aligned} X^{2} = \sum_{i=1}^{n}\left(\frac{y_{i}-\widehat{E}\left(y_{i}\right)}{\sqrt{\widehat{\operatorname{Var}}\left(y_{i}\right)}}\right)^{2} = \sum_{i=1}^{n}\left(\frac{y_{i}-m_{i} \hat{\pi}_{i}}{\sqrt{m_{i} \hat{\pi}_{i}\left(1-\hat{\pi}_{i}\right)}}\right)^{2} . \end{aligned}$$ The term $r_i = \frac{y_{i}-m_{i} \hat{\pi}_{i}}{\sqrt{m_{i} \hat{\pi}_{i}\left(1-\hat{\pi}_{i}\right)}}$ is known as Pearson residual.

Residual Diagnostics: For large $m_i$ values, both $d_i$ and $r_i$ should be approximately distributed as standard normal random variables if the logistic regression model is correct.

o=g1m(cbind(tumor, total-tumor)~dose, 
      family=binomial(link=logit), data=d)
summary(o)

Overdispersion: in the GLM framework, its often the case that $Var(y_i)$ is the function of $E(y_i)$. That is the case for logistic regression where $Var(y_i) = m_i\pi(1-\pi_i) = m_i\pi_i - (m_i\pi_i)^2/m_i = E(y_i) - [E(y_i)]^2/m_i$. If the variability of our response is greater than we should expect based on our estimates of the mean, we say that there is overdispersion.

Quasi-likelihood Inference: in the binomial case, we make all the same assumptions as before except that we assume $Var(y_i) = \phi m_i\pi_i(1-\pi_i)$ for some unknown dispersion parameter $\phi > 1$. The dispersion parameter can be estimated by $\hat\phi = \sum_{i=1}^n d_i^2/(n-p)$ or $\hat\phi = \sum_{i=1}^n r_i^2/(n-p)$.

• The estimated variance of $\hat\beta$ is multiplied by $\hat\phi$.
• For Wald type inferences, the standard normal null distribution is replaced by $t$ with $n - p$ degrees of freedom.
• Any test statistic $T$ that was assumed $\chi_q^2$ under $H_0$ is replaced with $T/(q\hat\phi)$ and compared to an $F$ distribution with $q$ and $n-p$ degrees of freedom.

oq=g1m(cbind(tumor, total-tumor)~dosef, 
       family=quasibinomial(link=logit),data=d)
summary(oq)