Central tolerance regions and reference regions for multivariate normal populations

Dong, Xiaoyu; Mathew, Thomas

doi:10.1016/j.jmva.2014.10.009

Cited by 11 publications

(15 citation statements)

References 20 publications

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“…This interval is called the equal-tailed or central (P,γ) tolerance interval. 15 A formula for values of c 6 is available 11 and can be computed using the function of the R package . 24,25 This interval can be viewed as a γ confidence simultaneous lower confidence bound on quantile μ−z(1+P)/2 σ and upper confidence bound on quantile μ+z(1+P)/2 σ.…”

Section: Equal-tailed Tolerance Intervalsmentioning

confidence: 99%

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Reference range: Which statistical intervals to use?

Liu

Bretz

Cortina‐Borja

2020

Stat Methods Med Res

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Reference ranges, which are data-based intervals aiming to contain a pre-specified large proportion of the population values, are powerful tools to analyse observations in clinical laboratories. Their main point is to classify any future observations from the population which fall outside them as atypical and thus may warrant further investigation. As a reference range is constructed from a random sample from the population, the event ‘a reference range contains [Formula: see text] of the population’ is also random. Hence, all we can hope for is that such event has a large occurrence probability. In this paper we argue that some intervals, including the P prediction interval, are not suitable as reference ranges since there is a substantial probability that these intervals contain less than [Formula: see text] of the population, especially when the sample size is large. In contrast, a [Formula: see text] tolerance interval is designed to contain [Formula: see text] of the population with a pre-specified large confidence γ so it is eminently adequate as a reference range. An example based on real data illustrates the paper’s key points.

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Section: Equal-tailed Tolerance Intervalsmentioning

confidence: 99%

“…Several authors have proposed to use tolerance intervals as reference ranges. 5 , 14 , 15 With almost 80 years of research on tolerance intervals or regions, various parametric and nonparametric procedures are readily available for use as reference ranges.…”

Section: Introductionmentioning

confidence: 99%

Reference range: Which statistical intervals to use?

Liu

Bretz

Cortina‐Borja

2020

Stat Methods Med Res

View full text Add to dashboard Cite

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“…Subject matter expertise appears essential before more definite recommendations can be made; see also the discussion in Section 2.2. of Dong and Mathew. 22…”

Section: Nonparametric Multivariate Hyperrectangular Tolerance Regionsmentioning

confidence: 99%

“…However, a (0.95,0.95) tolerance interval will be wider than a 95% prediction interval; this could result in too many false negatives . Analogous to the central tolerance interval derived for a univariate normal distribution, an ellipsoidal central tolerance region can be derived in the multivariate case, assuming multivariate normality; see the early work of John 21 and the more recent work of Dong and Mathew 22 for the derivation and further details. It should be noted that a 100 β % prediction region is also referred to as a β−expectation tolerance region; a term that we shall use in the present work.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric hyperrectangular tolerance and prediction regions for setting multivariate reference regions in laboratory medicine

Young

Mathew

2020

Stat Methods Med Res

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Reference regions are widely used in clinical chemistry and laboratory medicine to interpret the results of biochemical or physiological tests of patients. There are well-established methods in the literature for reference limits for univariate measurements; however, limited methods are available for the construction of multivariate reference regions, since traditional multivariate statistical regions (e.g. confidence, prediction, and tolerance regions) are not constructed based on a hyperrectangular geometry. The present work addresses this problem by developing multivariate hyperrectangular nonparametric tolerance regions for setting the reference regions. The approach utilizes statistical data depth to determine which points to trim and then the extremes of the trimmed dataset are used as the faces of the hyperrectangular region. Also presented is a strategy for determining the number of points to trim based on previously established asymptotic results. An extensive coverage study shows the favorable performance of the proposed procedure for moderate to large sample sizes. The procedure is applied to obtain reference regions for addressing two important clinical problems: (1) assessing kidney function in adolescents and (2) characterizing insulin-like growth factor concentrations in the serum of adults.

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“…Perhaps, the most common approaches have been to simply use specified percentiles from a reference sample or calculate prediction intervals. However, tolerance intervals have been advocated as a more sound approach given that they provide information about the entire population …”

Section: Applicationsmentioning

confidence: 99%

Improving coverage probabilities for parametric tolerance intervals via bootstrap calibration

Zou

Young

2020

Statistics in Medicine

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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.

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Central tolerance regions and reference regions for multivariate normal populations

Cited by 11 publications

References 20 publications

Reference range: Which statistical intervals to use?

Reference range: Which statistical intervals to use?

Nonparametric hyperrectangular tolerance and prediction regions for setting multivariate reference regions in laboratory medicine

Improving coverage probabilities for parametric tolerance intervals via bootstrap calibration

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