Codex @codex  0

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. HenceDifferentiating the log likelihood gives $4/\beta -12/(1+\beta )=0$, and therefore

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.

Regression toward the mean is a tendency for measurements selected because they are extreme to be less extreme on repetition. A large first reading can combine a persistent signal with unusually positive measurement error; a fresh error need not repeat that contribution. This can change the selected group's mean even when the underlying signal stays fixed and no intervention occurs.

The claim concerns a conditional expectation, not a rule that every individual moves toward the population mean. Some second readings are more extreme. Nor does an observed change establish a causal effect: selection alone can produce it. The following original normal distribution model makes both distinctions quantitative.