Confidence, prediction, and tolerance in linear mixed models.

Area of Science:

• Statistics
• Biostatistics
• Statistical Modeling

Background:

• Existing literature on Prediction Intervals (PI) and Tolerance Intervals (TI) in linear mixed models is often limited to specific experimental designs.
• This limitation restricts the broad applicability of these statistical tools in diverse research settings.
• A need exists for generalized methods applicable to a wide array of linear mixed model designs.

Purpose of the Study:

• To reformulate two-sided Prediction Intervals (PI) for generalizability across various linear mixed model designs.
• To detail the construction of two-sided Tolerance Intervals (TI) for models with multiple random factors.
• To provide a unified and robust methodology for constructing intervals in complex statistical models.

Main Methods:

• A novel methodology based on the Hessian matrix (inverse of the observed Fisher Information matrix) is proposed for PI.
• The cell mean model is utilized as the foundation for the new PI construction.
• The generalized Satterthwaite method is employed for calculating degrees of freedom for total variance, with comparisons to Kenward-Roger methods.

Main Results:

• An extensive simulation study demonstrated that the proposed methods achieve excellent coverage probabilities for Confidence Intervals (CI), PIs, and TIs.
• The performance was consistent across different designs (one random factor, nested, crossed) and sample sizes.
• The generalized Satterthwaite method for degrees of freedom showed comparable performance to Kenward-Roger's method.

Conclusions:

• The developed methodology offers a generalizable approach to constructing PIs and TIs in linear mixed models.
• The methods are validated through simulations and applied to real-world datasets from orthopedic surgery and vaccine development.
• This work enhances the utility of statistical intervals in complex and varied research applications.