A Bootstrap Prediction Confidence Band for QMRA Beta-Poisson Dose-Response Models

Let P I ( d ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGjbaabeaakiaacIcacaWGKbGaaiykaaaa@3A10@ denote the probability of infection at a given mean dose d . In the quantitative microbial risk assessment (QMRA) framework, the beta-Poisson dose-response model P I ( d ) = 1 ？ 1 F 1 ( α , α + β , ？ d ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGjbaabeaakiaacIcacaWGKbGaaiykaiabg2da9iaaigda cqGHsislcaqGGaWaaSbaaSqaaiaaigdaaeqaaOGaamOramaaBaaale aacaaIXaaabeaakiaacIcacqaHXoqycaGGSaGaeqySdeMaey4kaSIa eqOSdiMaaiilaiabgkHiTiaadsgacaGGPaaaaa@4A5E@ where 1 F 1 ( . , . , . ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSbaaSqaai aaigdaaeqaaOGaamOramaaBaaaleaacaaIXaaabeaakiaacIcacaGG UaGaaiilaiaac6cacaGGSaGaaiOlaiaacMcaaaa@3D71@ denotes the Kummer confluent hypergeometric function and α,β are the model parameters, remains the most popular plausible dose response model in practice. One commonly accepted way of constructing the confidence band about the dose-response curve in QMRA literature is to follow a bootstrap procedure based on the maximum likelihood estimates of the model parameters α and β. Here, it is shown that this bootstrap confidence bands reported in the literature represent the confidence intervals for the mean value of the probability of adverse effect, (i.e., P I ( d ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGjbaabeaakiaacIcacaWGKbGaaiykaaaa@3A10@ which may represent the probability of infection at mean dose levels), not the confidence intervals for prediction. Therefore, the existing literature bootstrap (95%) confidence bands normally