Time is exposure: doubling the public duration should double the expected count without changing the viewing rate. Modeling the rate is therefore preferable to treating time as an additive linear predictor. In a Poisson regression the exact implementation should be

lo g E (N_{i}) = lo g t_{i} + β_{0} + β_{1} I_{i},

using

lo g t_{i}

as an offset; directly supplying noninteger

N_{i} / t_{i}

without exposure weights is not generally likelihood-equivalent.

b

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Solution

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Overdispersion means that conditional variance exceeds the Poisson mean. Under a correctly specified Poisson model, the Pearson statistic is approximately chi-squared with

149

residual degrees of freedom. Here

X^{2} = 149 (1.3119) ≃ 195.5, \frac{X ^{2} - 149}{2 ( 149 )} ≃ 2.69 > 1.645,

(1)

so a one-sided five-percent test rejects. The displayed model-based confidence interval is too narrow. A quasi-Poisson fit can multiply standard errors by

1.3119

; negative binomial regression or a random-effects count model can model the extra variation directly.

c

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Solution

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There is one row per video. Giving every video its own categorical coefficient saturates the linear predictor. The investment column is a linear combination of the video indicator columns, so the design matrix loses full rank and the investment effect can be shifted into the video effects without changing any fitted value. The model is nonidentifiable.