Which of the following statements is true regarding Poisson regression?

Poisson regression is commonly used for modeling count data and is built upon a few key statistical principles. The correct statement regarding Poisson regression is that, if the model is adequate, the deviance follows a chi-square distribution.

The deviance is a measure of the goodness of fit of a model; it compares the likelihood of the fitted model to that of a saturated model (which perfectly fits the data). When the model is a good fit, the deviance can be approximated by a chi-square distribution under the null hypothesis that the model fits the data well. This property is crucial in assessing the adequacy of the model through hypothesis testing.

In contrast, the other statements present inaccuracies regarding the Poisson regression framework. The square root link function is not standard in Poisson regression; a log link is typically used because it ensures that the predicted counts are non-negative. Additionally, a small Pearson chi-square statistic actually indicates that the model may be under-dispersed rather than over-dispersed. Overdispersion occurs when the variance exceeds the mean, leading to a larger Pearson chi-square value instead.

Each of these aspects is fundamental to understanding how Poisson regression operates and the implications of the model fit, making the statement regarding the deviance