可选停止下始终有效推断的闭式样本量校正
A closed-form sample size correction for always-valid inference with optional stopping
AI总结：
针对A/B测试中连续监测的顺序检验，提出闭式校正因子k*，通过调整固定样本量使经验功效接近目标值，节省8-20%样本预算。
摘要：
允许连续监测的顺序检验在A/B实验中很常见。这些检验的功效计算需要模拟，这在实验平台上跨多个指标难以扩展。相反，一种常见的样本量确定启发式方法会膨胀固定样本量，直到计划终点处的边际拒绝概率达到$1-\beta$。这个最后点规则是保守的，因为始终有效（AV）功效是在运行期间任何时间边界跨越的概率，而不仅仅是终点。我们给出了一个闭式校正因子$k^*(\alpha, \beta, t_0)$，用初等函数和二元正态CDF表示，其中$t_0 = m/n_z$是预热比例。闭式近似仅通过边界在计划终点处的值和斜率依赖于边界，并且可以针对任何光滑凹边界进行评估。我们研究了三种情况：Waudby-Smith等人（2023）和Maharaj等人（2023）的置信序列，以及Johari等人（2022）的混合序贯概率比检验。将总样本量设为$k^* \cdot n_z$，其中$n_z$是分配比$r$下的固定样本量，在高斯模拟中使经验功效与目标值相差约3个百分点以内。校正因子仅通过$t_0 = m/n_z(r)$依赖于分配比$r$。我们研究了预热参数的敏感性，并表明在校正因子在操作范围内节省了8-20%的最后点样本预算。
英文摘要：
Sequential tests that allow continuous monitoring are common in A/B experimentation. Power calculations for these tests require simulations that are hard to scale across many metrics on an experimentation platform. Instead, a common sizing heuristic inflates the fixed-sample size until the marginal rejection probability at the planned endpoint reaches $1-β$. This last-point rule is conservative because always-valid (AV) power is the probability of a boundary crossing at any time during the run, not at the endpoint alone. We give a closed-form correction factor $k^(α, β, t_0)$ expressed in elementary functions and the bivariate normal CDF, where $t_0 = m/n_z$ is the burn-in fraction. The closed-form approximation depends on the boundary only through its value and slope at the planned endpoint and can be evaluated for any smooth concave boundary. We work out three cases: the confidence sequences of Waudby-Smith et al. (2023) and Maharaj et al. (2023), and the mixture sequential probability ratio test of Johari et al. (2022). Setting the total sample size to $k^ \cdot n_z$, where $n_z$ is the fixed-sample size for allocation ratio $r$, hits empirical power within approximately 3 percentage points of target in Gaussian simulations. The correction factor depends on the allocation ratio $r$ only through $t_0 = m/n_z(r)$. We study sensitivity to the burn-in parameter and show that the correction saves 8--20% of the last-point sample budget across the operating range.
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