the shelf life?
Department of Statistics
Stillwater, OK 74078-0595
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.
and Drug Administration requires pharmaceutical companies to
establish a shelf life for all new drug products through a
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
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
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
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.
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.
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
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.
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
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.
outliers at the times 6 and 60 months. Additional questions may
be added to the project regarding residual analysis.
Chow, S. and Liu, J. (1995). Statistical Design and
Analysis in Pharmaceutical Science: Validation, Process Controls,
New York: Marcel Dekker, Inc.
note: Before 11-6-01, the "student's version" of an
activity was called the "prototype".