Among the 12 pairs with control time first, the eight in which that first time is an event contribute $(1+\beta)^{-1}$ each. Among the eight pairs with treated time first, the four in which that first time is an event contribute $\beta/(1+\beta)$ each. Later events in one-eye risk sets contribute one. Hence
$$
L(\beta)\propto
\left(\frac1{1+\beta}\right)^8
\left(\frac\beta{1+\beta}\right)^4
=\frac{\beta^4}{(1+\beta)^{12}}.
$$
Differentiating the log likelihood gives $4/\beta-12/(1+\beta)=0$, and therefore
$$
\boxed{\widehat\beta=\frac12.}
$$

Without randomization, treatment side can be associated with prognosis. Always treating the left eye confounds treatment with systematic left-right differences; choosing the worse eye creates severe <confounding by indication>, baseline imbalance, and possible <regression toward the mean>. The within-patient comparison then no longer identifies a treatment effect without stronger adjustment assumptions.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 207 / 5 / b / Solution

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Confounding by indication occurs when treatment choice depends on disease severity or prognosis. The treated group can then have worse outcomes because it was at higher baseline risk, even if treatment is beneficial.


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 Incoming links: Confounding by indication