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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 13K / d / Solution

The reduced model fit3 retains malignancy but omits both site indicators. The analysis of deviance for nested generalized linear models therefore tests

H_{0} : β_{B} = β_{C} = 0

against the additive model with a site effect. The likelihood-ratio statistic is the reduction in deviance,

7.4923 - 0.8505 = 6.6418.

The full model adds two parameters, so under

H_{0}

this is approximately

χ_{2}^{2}

. Since

6.6418 > χ_{2, 0.95}^{2} = 5.9915

but is less than the

0.99

quantile

9.2103

, the p-value is between

0.01

and

0.05

; in fact, because

χ_{2}^{2}

has survival function

e^{- x /2}

,

p = e^{- 6.6418/2} \approx 0.0361.

We reject the no-site-effect null at the

5%

level and find evidence that survival differs by site after adjustment for malignancy.

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