With constant hazard $\theta$, observation $i$ contributesWriting $D={\sum }_i{v}_i$ and $T={\sum }_i{x}_i$, the log-likelihood is $D\log \theta -\theta T$, whose derivative is $D/\theta -T$. Hencethe number of events divided by total person-time.

Matching controls strong prognostic differences between patients and can greatly improve precision by making treatment comparisons within homogeneous pairs. It also protects against confounding when treatment assignment is not perfectly randomized globally.

Its disadvantage is loss of information: pairs with censoring before a comparison or tied outcomes contribute little or nothing, and the analysis cannot estimate the effects of pair-level matching variables. Poor matches and the logistics of paired enrollment can also reduce efficiency relative to a well-adjusted unpaired analysis.

The Cox proportional-hazards model iswhere $h_0$ is an unspecified baseline hazard and $z_i$ indicates treatment. With no tied events, the Cox partial likelihood isEquality of the two hazard functions is $H_0:\beta =0$, testable with a likelihood-ratio, score, or Wald test from this likelihood.

The first two mechanisms are standard independent, noninformative censoring. The third relabels the event time itself and therefore is not independent censoring for the original latent event time; it is more naturally interpreted as assigning a cause in a competing risks model. The detailed calculations follow in the three subparts.

Without censoring, every $v_i=1$ and $\mathbb{E}{T}_i=1/\lambda$. ThereforeThis is events divided by total person-time at risk.

If $U\sim \mathrm{Uniform}(0,1)$, the inverse transform sampling formulahas survival function $\mathbb{P}(T>t)=e^{-\lambda t}$ and hence the required exponential distribution.

With equal allocation and true alternative rates ${\lambda }_T,{\lambda }_C$, the asymptotic variance isThe normal approximation requires $|\log R{R}_A|/\sqrt{V_A}\ge z_{1-\alpha /2}+z_{1-\beta }$. Hencerounded upward. Writing ${\lambda }_T=R{R}_A{\lambda }_C$ expresses it using a specified control rate and the target rate ratio.

Since $X_j\sim \mathrm{Poisson}(n_j{\lambda }_j)$, the rate MLE has asymptotic variance ${\lambda }_j/n_j$. The delta method for the logarithm givesIndependence of the arms therefore yieldsUnder $H_0:{\lambda }_T={\lambda }_C=\lambda$, the asymptotic variance of the log rate ratio is

Targeting the Neyman ratio asymptotically minimizes the standard error for a fixed total sample size, so it cannot be worse than a fixed 1:1 design when the target probabilities are known. Targeting the ethical ratio can increase the standard error because it optimizes failures rather than information, although it may still outperform 1:1 for some probabilities.

In response-adaptive randomization, the probabilities are estimated during the trial. Delayed outcomes, time trends, unstable early estimates, and random final arm sizes complicate logistics and inference; naive Wald standard errors may also ignore adaptation.