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From my understanding, both Pearson chi squared test and deviance test can be used to assess the goodness of fit for GLM, but they have different alternative hypotheses. For the Pearson chi-squared test, the null hypothesis suggests that the true model is the estimated model, while the alternative hypothesis indicates that the true model differs from the estimated model.

On the other hand, the null hypothesis from the deviance test (also known as residual deviance) assumes that the true model is the estimated model, but the alternative hypothesis states that the true model is the saturated model, where each observation can have its own parameter. In R, the summary output of the GLM includes both null deviance and residual deviance. The null deviance tests the hypothesis that the true model is the intercept-only model, while the residual deviance is similar to the deviance test.

Given that null and residual deviance already assess whether less or more parsimonious models might be better, I wonder if there are scenarios where using the Pearson chi-squared test would be more appropriate or insightful. Can someone clarify such scenarios and explain the advantages of using the Pearson chi-squared test over the deviance test in specific situations?

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Generally, the G-test is useful in case of categorical data, but when we see the concept of "Goodness of fit", the more powerful test is chi-square test as compared to the LRT. Also I would like to add one more thing, LRT is considered more effective in case when some of the values in contingency table have less than 5 observed or expected frequencies. Reference: Journal of Applied Sciences Research 1(2): 242-244, 2005 © 2005, INSInet Publication

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