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Put $v_k=p_k(1-p_k)$ and assume $0<p_k<1$. For fixed $n$ and a fixed nonzero difference $\Delta =p_0-p_1$, the approximate statistical power of the Wald test increases as its noncentrality $|\Delta |/\sqrt{V}$ increases, whereDifferentiating with respect to the continuous allocation givesHence the unique minimum and its associated minimum variance areThis is Neyman allocation. Integer allocations use a feasible adjacent integer to the continuous optimum, comparing the resulting variances; $n_0,n_1\ge 1$ must still hold. It is a large-sample normal approximation argument, not an exact finite-sample power theorem for arbitrary response-adaptive randomization. Under the null $p_0=p_1$, the approximate power is the significance level for any allocation, although the same allocation still minimizes the stated variance.

Response-adaptive randomization can assign a larger proportion of later participants to the treatment currently estimated to be better, improving outcomes for participants within the trial. Its allocation probabilities depend on earlier outcomes, which complicates statistical inference; delayed responses and calendar-time trends can also make adaptation ineffective or biased, and an allocation aimed at patient benefit need not maximize power.

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.

In a two-treatment urn design, begin with $a$ balls of each type, draw and replace a ball to allocate treatment, then add $b$ balls of the same type after success or of the opposite type after failure. This response-adaptive randomization favours treatments with more observed successes. For RPW$(1,1)$, type-0 probability after counts $(s_0,f_0,s_1,f_1)$ is $(1+s_0+f_1)/(2+s_0+f_0+s_1+f_1)$.