含假阴性/假阳性的比例置信区间计算问题咨询

Great question—this is a common scenario in diagnostic testing where observed counts don’t directly map to true positives or negatives. Here’s a clear, actionable approach to compute the confidence interval (CI) for the true positive probability π, accounting for known false positive (α) and false negative (β) rates:

Key Background & Simplification #

First, let’s clarify the relationship between your observed data and the true underlying probabilities:

• For any individual in your sample, the probability of testing positive is:
  p = π*(1 - β) + (1 - π)*α
  This combines the chance of a true positive being correctly identified (π*(1-β)) and a true negative being incorrectly labeled positive ((1-π)*α).
• Critically, since each individual’s test result is independent, the total number of observed positives k follows a Binomial(n, p) distribution. This lets us leverage standard binomial CI methods before transforming to π.

Step-by-Step Calculation #

Assume you have pre-estimated values for α (false positive rate) and β (false negative rate) from a gold-standard validation of your test. If you don’t have these, you’ll need to first validate your test to get these rates—they’re essential for this correction.

1. Compute a CI for the marginal test-positive probability p #

Use your preferred binomial CI method on k and n:

• Agresti-Coull: A robust, less conservative alternative to Clopper-Pearson, especially for small samples. Adjust k to k' = k + 2 and n to n' = n + 4, then compute the Wald CI:
  p' = k'/n'
SE = sqrt(p'*(1-p')/n')
CI_p = [p' - z*SE, p' + z*SE] (use z*=1.96 for 95% confidence)
• Clopper-Pearson: The exact (but conservative) binomial CI. You can calculate this using statistical software or online calculators.

2. Transform the p CI to get the π CI #

Since π = (p - α)/(1 - α - β) (let’s call C = 1 - α - β for simplicity), apply this linear transformation to the bounds of your p CI:

• π_low = max(0, (p_low - α)/C)
• π_high = min(1, (p_high - α)/C)
  We clamp the values to [0,1] because probabilities can’t fall outside this range.

Example Walkthrough #

Let’s say:

• Sample size n = 100, observed positives k = 20
• False positive rate α = 0.05, false negative rate β = 0.10
• C = 1 - 0.05 - 0.10 = 0.85

Step 1: Agresti-Coull CI for p #

• k' = 20 + 2 = 22, n' = 100 + 4 = 104
• p' = 22/104 ≈ 0.2115
• SE = sqrt(0.2115*0.7885/104) ≈ 0.040
• CI_p = [0.2115 - 1.96*0.040, 0.2115 + 1.96*0.040] ≈ [0.133, 0.290]

Step 2: Transform to π CI #

• π_low = (0.133 - 0.05)/0.85 ≈ 0.098
• π_high = (0.290 - 0.05)/0.85 ≈ 0.282
• Final 95% CI for true positive probability: [0.10, 0.28]

Handling Uncertainty in α and β #

If your estimates of α and β are not precise (e.g., from small validation studies), you’ll need to account for their uncertainty:

• Parametric Bootstrap:
  1. Resample your validation data to generate new estimates α* and β*
  2. Resample your sample data to generate a new observed count k*
  3. Compute π* = (k*/n - α*)/(1 - α* - β*)
  4. Repeat thousands of times, then take the 2.5th and 97.5th percentiles of the π* values as your CI.
• Delta Method: Calculate the combined variance of π by accounting for variances in p, α, and β, then construct a Wald CI. This is more mathematical but can be done with basic variance rules.

Important Notes #

• If C = 1 - α - β = 0, your test has no diagnostic value (the probability of testing positive is the same regardless of true status), so you can’t estimate π.
• Always clamp your final CI to [0,1]—negative values or values greater than 1 don’t make sense for probabilities.

内容的提问来源于stack exchange，提问作者user3208442