Abstract
The literature about Prediction Interval (PI) and Tolerance Interval (TI) in linear mixed models is usually developed for specific designs, which is a main limitation to their use. This paper proposes to reformulate the two‐sided PI to be generalizable under a wide variety of designs (one random factor, nested and crossed designs for multiple random factors, and balanced or unbalanced designs). This new methodology is based on the Hessian matrix, namely, the inverse of (observed) Fisher Information matrix, and is built with a cell mean model. The degrees of freedom for the total variance are calculated with the generalized Satterthwaite method and compared to the Kenward‐Roger's degrees of freedom for fixed effects. Construction of two‐sided TIs are also detailed with one random factor, and two nested and two crossed random variables. An extensive simulation study is carried out to compare the widths and coverage probabilities of Confidence Intervals (CI), PIs, and TIs to their nominal levels. It shows excellent coverage whatever the design and the sample size are. Finally, these CIs, PIs, and TIs are applied to two real data sets: one from orthopedic surgery study (intralesional resection risk) and the other from assay validation study during vaccine development.
Highlights
Throughout medical and pharmaceutical research, linear mixed models are frequently used
Eight different statistical intervals are calculated for each simulated data set: the 95% Confidence Intervals (CI), the 95% Prediction Interval (PI) and the 95%Tolerance Interval (TI) with confidence levels from 50% to 90%
The following sections will summarize the results on graphs with the coverage probabilities, while the comparison of the widths of these intervals is given online as supplementary materials (CI, PI, and the largest TI with 90% confidence level)
Summary
Throughout medical and pharmaceutical research, linear mixed models are frequently used. When investigating the effect of new drugs, statistical reports and medical papers usually report the estimated treatment effects with their corresponding Confidence Intervals (CIs). CIs are typically used to assess the uncertainty on an unknown population parameter, eg, the estimated average response for a given population. For release of a pharmaceutical product, on the other hand, suitable analytical release methods for impurities and the active ingredient need to be developed and qualified. The manufacturer needs to be confident that the measured release value is within a given percentage of the true (antigen or impurity) concentration. During an analytical method qualification, the bias (the systematic error, related to the trueness) and the random variability (related to the precision) are assessed. The accuracy of an analytical method describes the combined effects of trueness and precision
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