Division of Cancer Epidemiology and Genetics, Department of Health and Human Services, National Cancer Institute, National Institutes of Health, Bethesda, MD, USA.
BMC Med Res Methodol. 2023 Jun 29;23(1):153. doi: 10.1186/s12874-023-01973-x.
The rule of thumb that there is little gain in statistical power by obtaining more than 4 controls per case, is based on type-1 error α = 0.05. However, association studies that evaluate thousands or millions of associations use smaller α and may have access to plentiful controls. We investigate power gains, and reductions in p-values, when increasing well beyond 4 controls per case, for small α.
We calculate the power, the median expected p-value, and the minimum detectable odds-ratio (OR), as a function of the number of controls/case, as α decreases.
As α decreases, at each ratio of controls per case, the increase in power is larger than for α = 0.05. For α between 10 and 10 (typical for thousands or millions of associations), increasing from 4 controls per case to 10-50 controls per case increases power. For example, a study with power = 0.2 (α = 5 × 10) with 1 control/case has power = 0.65 with 4 controls/case, but with 10 controls/case has power = 0.78, and with 50 controls/case has power = 0.84. For situations where obtaining more than 4 controls per case provides small increases in power beyond 0.9 (at small α), the expected p-value can decrease by orders-of-magnitude below α. Increasing from 1 to 4 controls/case reduces the minimum detectable OR toward the null by 20.9%, and from 4 to 50 controls/case reduces by an additional 9.7%, a result which applies regardless of α and hence also applies to "regular" α = 0.05 epidemiology.
At small α, versus 4 controls/case, recruiting 10 or more controls/cases can increase power, reduce the expected p-value by 1-2 orders of magnitude, and meaningfully reduce the minimum detectable OR. These benefits of increasing the controls/case ratio increase as the number of cases increases, although the amount of benefit depends on exposure frequencies and true OR. Provided that controls are comparable to cases, our findings suggest greater sharing of comparable controls in large-scale association studies.
有一个经验法则，即每例病例获得超过 4 个对照时，统计效能的提高很小，这是基于Ⅰ型错误α=0.05。然而，评估成千上万或上百万个关联的关联研究使用较小的α，并且可能有大量的对照。我们研究了当α较小，超过每例病例 4 个对照时，增加对照数对功效的提高，以及对 p 值的降低。
我们计算了作为α降低的函数，在每个对照/病例的比例下，功效、中位数预期 p 值和最小可检测比值（OR）。
随着α的降低，在每个对照/病例的比例下，增加的功效大于α=0.05 时的功效。对于 10 到 10（典型的成千上万或上百万个关联）之间的α，从每例病例 4 个对照增加到每例病例 10-50 个对照会增加功效。例如，一个研究的功效为 0.2（α=5×10），每个病例有 1 个对照，当每个病例有 4 个对照时，功效为 0.65，但当每个病例有 10 个对照时，功效为 0.78，当每个病例有 50 个对照时，功效为 0.84。对于在小 α 时，获得每个病例超过 4 个对照会使功效增加超过 0.9 的情况，预期的 p 值可以在α以下降低 1-2 个数量级。从 1 个对照增加到 4 个对照使最小可检测 OR 向零接近减少了 20.9%，从 4 个对照增加到 50 个对照减少了另外的 9.7%，这一结果适用于任何α，因此也适用于“常规”α=0.05 流行病学。
在小 α 时，与 4 个对照/病例相比，招募 10 个或更多对照/病例可以提高功效，将预期的 p 值降低 1-2 个数量级，并显著降低最小可检测 OR。随着病例数的增加，增加对照/病例比例的这些好处会增加，尽管受益的程度取决于暴露频率和真实 OR。只要对照与病例可比，我们的发现表明在大规模关联研究中更广泛地共享可比对照。
