What is the shelf life?

Christopher R. Bilder
Department of Statistics
Oklahoma State University
Stillwater, OK 74078-0595

Statistics Teaching and Resource Library, February 7, 2001

© 2001 by Christopher R. Bilder, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor.

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The Food and Drug Administration requires pharmaceutical companies to establish a shelf life for all new drug products through a stability analysis.  This is done to ensure the quality of the drug taken by an individual is within established levels.  The purpose of this out-of-class project or in-class example is to determine the shelf life of a new drug.  This is done through using simple linear regression models and correctly interpreting confidence and prediction intervals.  An Excel spreadsheet and SAS program are given to help perform the analysis. 

Key words: prediction interval, confidence interval, stability 

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Introduction

Pharmaceutical companies estimate the shelf life (and then expiration date) of a drug to determine the amount of time the drug is at acceptable potency, color, etc., levels.  The acceptable levels are set by the pharmaceutical company or the Food and Drug Administration.  The process in which the shelf life is determined is called a stability analysis.  The shelf life of a drug is loosely defined here as the length of time a drug can stay on the shelf without degrading to unacceptable levels.  For more on conducting a stability analysis, see Chow and Liu (1995).

Objectives

The objectives of this out-of-class project or in-class example are to give students a situation where simple linear regression can be used.  Although the actual determination of shelf life usually involves more complicated models (such as ANCOVA), this simplified exercise illustrates the concepts involved in determining the shelf life of a drug.  The activity also helps reinforce hypothesis testing, confidence interval, and prediction interval concepts in simple linear regression. 

Included here is a prototype activity that may be handed out directly to students or modified to suit instructor needs.  Note that the data included is not real, but the problem set-up is similar in content to an actual problem encountered by the author.  An answer key is included at the end of the prototype activity.

Assessment

The beginning of the activity describes what stability analysis is and the drug for which a shelf life is desired.  The data given is the potency of randomly selected tablets of the drug at particular time points.  Questions 1)-7) ask standard regression analysis questions, such as: finding the estimated regression model using time to predict potency, interpreting R², and finding prediction intervals.  Questions 8) and 9) give directions on how to find the shelf life of a drug.  Students are required to construct a scatter plot with the estimated regression line drawn upon it as shown in Figure 1.  In addition, confidence and prediction intervals bands are drawn on the plot.  The shelf life is the smallest time in which the 95% confidence interval bands intersect the 95% or 105% potency lines.  In Figure 1, the 95% potency line intersects the lower 95% confidence interval band at approximately 32.2 months.

Figure 1.  Scatter plot with an estimated regression line, confidence interval bands, and prediction interval bands.

Questions 10)-13) are thought provoking questions about stability analysis.  Question 10) begins with asking students to describe the difference between what confidence and prediction intervals are estimating for this activity.  Question 10) continues with asking students to specify which type of interval (confidence or prediction) should be used for determining shelf life.  Currently, confidence intervals are used; however, there are instances where prediction intervals may be of more interest.  See the answer key for a discussion.  Question 11) asks students to describe what happens to the shelf life when the confidence level decreases.  To answer this question, students need to understand both what happens to a confidence interval (increases or decreases in width) and what happens to the intersection point of the confidence interval lower bound and the 95% potency line.  Students tend to see the correct answer better by constructing examples for this question.  Question 12) asks students to identify the level of confidence (high or low) an individual probably would prefer in determining the shelf life.  Question 13) asks the same question, but from the drug manufacturers perspective.  Both questions 12) and 13) put deciding the confidence level into a realistic situation for students. 

Teaching Notes

Most statistical packages contain options to construct a plot similar to Figure 1, and a sample SAS program is included here.  For users of non-statistical software packages, this type of plot may be difficult to construct.  Included here is an easy-to-use Excel spreadsheet which constructed Figure 1.  Directions on how to use this spreadsheet are included within it.  

Questions 10)-13) can be difficult for some students to answer.  When I use this activity as an out-of-class project, I often will do some of these in class or assign some as extra credit. 

There are outliers at the times 6 and 60 months.  Additional questions may be added to the project regarding residual analysis. 

References

Chow, S. and Liu, J. (1995).  Statistical Design and Analysis in Pharmaceutical Science: Validation, Process Controls, and Stability.  New York: Marcel Dekker, Inc. 

 

Editor's note: Before 11-6-01, the "student's version" of an activity was called the "prototype".