Improving coverage probabilities for parametric tolerance intervals via bootstrap calibration

Abstract

Statistical tolerance intervals are commonly employed in biomedical and pharmaceutical research, such as in lifetime analysis, the assessment of biosimilarity of branded and generic versions of biopharmaceutical drugs, and in quality control of drug products to ensure that a specified proportion of the products are covered within established acceptance limits. Exact two-sided parametric tolerance intervals are only available for the normal distribution, while exact one-sided parametric tolerance limits are available for a limited number of distributions.Approximations to two-sided parametric tolerance intervals often use the Bonferroni correction on the one-sided tolerance interval calculation; however, this often incurs a higher coverage probability than the nominal level. Recently, the usage of a bootstrap calibration has been demonstrated as a way to improve coverage probabilities of tolerance intervals for very specific and complex distributional settings. We present a focused treatment on using a single-layer bootstrap calibration to improve the coverage probabilities of two-sided parametric tolerance intervals. Simulation results clearly demonstrate the improved coverage probabilities towards the nominal level over the uncalibrated setting.Applications to medical data for various parametric distributions also highlight the utility of constructing these calibrated tolerance intervals.

PICaSSO Histologic Remission Index (PHRI) in ulcerative colitis: development of a novel simplified histological score for monitoring mucosal healing and predicting clinical outcomes and its applicability in an artificial intelligence system

“…To assess the inter-rater agreement of the histological scorings, we used one-way intraclass correlation (ICC) coefficient by means of R package irr (http://cran.r-project.org/package=irr). In order to test the hypothesis of the ICC being >0.5 against the alternative, we needed a minimum of 40 histology images to reach the power of 0.8 with a type I error of 0.05 36. According to Landis and Koch benchmarks,37 ICC of <0.2, 0.2 to 0.4, >0.4 to ≤0.6, >0.6 to 0.8 and >0.8 was considered ‘poor’, ‘fair’, ‘moderate’, ‘good’, ‘substantial’ and ‘almost perfect’, respectively.…”

“…In order to test the hypothesis of the ICC being >0.5 against the alternative, we needed a minimum of 40 histology images to reach the power of 0.8 with a type I error of 0.05. 36 According to Landis and Koch benchmarks, 37 ICC of <0.2, 0.2 to 0.4, >0.4 to ≤0.6, >0.6 to 0.8 and >0.8 was considered ‘poor’, ‘fair’, ‘moderate’, ‘good’, ‘substantial’ and ‘almost perfect’, respectively. Results of all statistical tests were considered significant at p<0.05.…”

Fixed-accuracy confidence interval estimation of P(X > c) for a two-parameter gamma population

“…This yields a reference sample of size n = 5255. The UACR data were analyzed in Zou and Young (2020), where the authors had established a one-sided upper tolerance limit of 0.7358 mg/g and suggested to classify adolescents with UACR values falling below it as "strictly normal." Interested readers may refer to the paper, and more background information is omitted here.…”

“…For illustrative purposes, we also treated the UACR data of size n = 5255 as our population, to which the exponential distribution gave a good fit (Zou and Young, 2020); that is, the shape parameter α = 1, but the rate parameter β was assumed unknown. We set c = 0.7358 and proceeded to estimate p = P β (X > c), which can be interpreted as the proportion of healthy adolescents with mildly increased UACR.…”

Content‐adjusted tolerance intervals via bootstrap calibration

“…According to the review of the literature above, it can be found that the emphasis of Bootstrap methods has always been on calibrating CIs, prediction intervals, or some nonstandard TI procedures, with scant attention given to the calibration of TIs for non‐normal distributions (see, e.g., Fernholz & Gillespie, 2001; Zou & Young, 2020).…”