Estimation, reference ranges and goodness of fit for the three-parameter log-normal distribution

Area of Science:

• Biostatistics
• Medical Statistics
• Quantitative Medicine

Background:

• Continuous variables in medicine often follow non-normal distributions.
• The three-parameter log-normal distribution (3PL) is a suitable model for such data.
• Accurate statistical modeling is crucial for clinical measurements and reference range calculations.

Purpose of the Study:

• To provide methods for estimating parameters and confidence intervals for the 3PL.
• To introduce a simple, non-iterative estimation method for the shift parameter.
• To adapt the Shapiro-Wilk test for assessing departures from the 3PL and compare its power.

Main Methods:

• Development of estimation techniques for 3PL parameters and their functions.
• Description of a non-iterative method for estimating the shift parameter.
• Modification of the Shapiro-Wilk test for 3PL goodness-of-fit testing and power comparison.

Main Results:

• The study presents practical methods for estimating 3PL parameters and confidence intervals.
• A straightforward non-iterative estimation for the shift parameter is demonstrated.
• A modified Shapiro-Wilk test is proposed for detecting deviations from the 3PL, with comparative power analysis.

Conclusions:

• The three-parameter log-normal distribution is a valuable tool for analyzing continuous medical data.
• The proposed methods facilitate robust parameter estimation and hypothesis testing for the 3PL.
• The modified Shapiro-Wilk test offers a useful approach for assessing model fit in clinical measurements.