Table 13.8 reports the results of a study of fish hatching under three environments. Eggs from seven clutches were randomly assigned to three treatments, and the response was whether an egg hatched by day 10. The three treatments were (1) carbon dioxide and oxygen removed, (2) carbon dioxide only removed, and (3) neither removed.

a. Let π_it denote the probability of hatching for an egg from clutch i in treatment t. Assuming independent binomial observations, fit the model

logit(π_it) = β₁ z₁ + β₂ z₂ + β₃ z₃,

where z_t = 1 for treatment t and 0 otherwise. What does your software report for β̂₁, and what should it be? (Note that treatment 1 has no successes.)

b. Analyze these data using an approach that allows overdispersion. Interpret. Indicate whether evidence of overdispersion occurs for treatments 2 and 3.

Table 13.8:

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Treatment 1 Treatment 2 Number Treatment 3 Number Hatched Number Total Total Total Clutch Hatched Hatched 6 13 1 2 13 10 13 3 4 10 10 25 16 16 30 16 32 23 5 5 28 21 10 20 20