Chapter 6: Confidence Intervals and Limit of Detection (LoD)

In ddPCR, it is important not only to estimate the concentration of the target gene, but also to quantify the uncertainty of that estimate. This is typically done by calculating confidence intervals (CIs) for the estimated copy number.

Confidence Interval for Proportion of Positive Droplets

Let:

  • \(N\): total number of droplets

  • \(p\): number of positive droplets

Then the observed proportion is:

\[ \hat{p} = \frac{p}{N} \]

The confidence interval for this proportion can be calculated using binomial methods, such as:

  • Wald method (simplest, but less accurate when \(p\) is near 0 or 1)
  • Wilson score interval
  • Exact (Clopper-Pearson) method

We can then transform this interval into a CI for \(\lambda\) (CPD), and then into concentration.

R Example: Using binom Package

library(binom)
## Warning: package 'binom' was built under R version 4.4.3
N <- 20000
p <- 200
p_hat <- p / N

binom.confint(p, N, methods = "exact")
##   method   x     n mean       lower      upper
## 1  exact 200 20000 0.01 0.008667639 0.01147755

This gives the lower and upper bounds for the proportion of positive droplets.

We then compute:

lambda_lower <- -log(1 - 0.008589)
lambda_upper <- -log(1 - 0.010316)

v <- 20 / N  # droplet volume
d <- 1       # no dilution
conc_lower <- lambda_lower * d / v
conc_upper <- lambda_upper * d / v

round(c(conc_lower, conc_upper), 2)
## [1]  8.63 10.37

Interpretation

  • The estimated concentration is about 10.05 copies/μL

  • The 95% confidence interval is roughly [8.6, 10.8] copies/μL

  • The uncertainty reflects both the number of positive droplets and the total droplet count

Defining Limit of Detection (LoD)

LoD is often defined as the lowest concentration that gives a statistically significant number of positive droplets, with an acceptably low false negative rate.

A common approach is:

  • Simulate or calculate the probability of detecting at least one positive droplet at a given \(\lambda\)
  • Require that this probability exceeds 1 - α, e.g., 95%

This was demonstrated in the previous chapter using:

\[ P(X=0) = e^{-\lambda}, \ \lambda_{LOD} = -ln(\alpha) \]

Summary

  • Confidence intervals for ddPCR results can be constructed by first computing the CI for the proportion of positive droplets.
  • The LoD can be estimated as the smallest concentration that still results in a sufficiently high probability of detection.

This closes the foundational mathematical modeling of ddPCR.