统计术语专业解释

Definition: The Minimum Detectable Effect (MDE), also called the Minimum Detectable Effect Size (MDES), is the smallest true effect (difference or change) that a study is designed to detect with a specified statistical power (1−β) at a given Type I error rate (α), given the study design and assumed variability. Formally, it is the smallest δ such that P(reject H0 | true effect = δ) ≥ 1−β.

Key properties

• It is a design-time quantity determined by α, power, sample size/allocation, variance, test sidedness, and analysis method.
• It is expressed in the natural units of the metric (absolute MDE); a relative MDE is the absolute MDE divided by a baseline level.
• It reflects detectability, not the observed effect or practical importance.

General formula

• For many common tests, an accurate approximation is: MDE ≈ (z1−α/2 + z1−β) × SE(Δ̂) where SE(Δ̂) is the standard error of the estimator of the effect (Δ̂), computed under assumed values. For one-sided tests use z1−α instead of z1−α/2.

Common cases

• Difference in means (two independent groups): SE(Δ̂) = sqrt(σ1^2/n1 + σ2^2/n2). If σ1 = σ2 = σ and n1 = n2 = n: MDE ≈ (z1−α/2 + z1−β) × σ × sqrt(2/n).
• Difference in proportions (two independent groups): A practical planning approximation uses a baseline rate p (or a pooled estimate): SE(Δ̂) ≈ sqrt(p(1−p) × (1/n1 + 1/n2)), so MDE ≈ (z1−α/2 + z1−β) × sqrt(p(1−p) × (1/n1 + 1/n2)). More exact calculations use variance under both H0 and H1 and may require iteration if p under treatment is p+δ.

Related considerations

• Allocation ratio r = nt/nc, total N: SE scales as sqrt(1/nt + 1/nc); equal allocation (r = 1) minimizes MDE for fixed N.
• Multiple comparisons or sequential looks require α-adjustments (e.g., Bonferroni, group-sequential), which increase the MDE.
• Clustered or correlated data inflate SE via a design effect: DE = 1 + (m−1)×ICC; multiply SE by sqrt(DE).
• Standardized MDES (e.g., Cohen’s d) equals the MDE divided by the assumed standard deviation.

Interpretation

• If a study’s MDE is δ, the design has the specified power to detect effects of size ≥ δ; smaller true effects are likely to go undetected.
• Reporting MDE alongside null results clarifies which effect sizes can be ruled out by the study.