Figure 1: Least-square line method for the time period of 0–9 months. (ALL FIGURES COURTESY OF AUTHORS)
Regression-control-chart-method. The regression-control-chart method is used to compare the results within the batch and detect present OOT results. For the
purpose of this method, the tenth batch was examined. Several least-square regression lines were fit to the suitable data
(5). The first regression line was constructed from the three results for assay at the first three time points (0, 3, and
6 months). With extrapolation of that regression line, the expected values for Y and the Y residuals were calculated (see
Figure 1) (6). The procedure was then repeated by gradually adding all the other consecutive time points.
Table I: Regression-control chart for the tenth batch.
The next step was to calculate the mean and standard deviation (σ) of the Y residuals of the regression line. As a result,
a sum of five means and standard deviations corresponding to the time periods (0–6, 0–9, 0–12, 0–18, and 0–24 months) were
constructed. To identify the present OOT result, the z-score test was used to calculate the z-value for each Y residual at
each time point. The z-value is based on the means and standard deviations of the defined time periods (see Table I). The z-value was limited to -3 < z <+3, where 99.73% of the future results are expected to enter the interval within these
Table II: The regression-control chart method limits from the tolerance interval (TI) and z-values for the corresponding TI
Testing the precision of the TI in comparison to the z-score test, the TI for the same five time periods was calculated according
to the suitable equation with defined certainty (α=0.027) and confidence (γ=0.95) (7). To compare the two methods for each
TI value, a corresponding z-value was calculated (see Table II).
Figure 2: Representation of the historical data with the use of the by-time-point method.
By-time-point method. The by-time-point method is used to determine whether a result is within expectations on the basis of experiences from other
batches measured at the same stability time point. To minimize the α and β error (6), batches that comprise the historical
data (Batches 1 to 9) were individually tested for present OOT results. In addition, calculations for the mean and standard
deviation of the values for the tested attribute were made from the historical data for each time point individually (see
Table III: Mean value and standard deviation of the historical data and z-value for Batch 10.
To identify the present OOT result, the z-score test was used to calculate the z-value for each value of Y (the assay result)
of the tenth batch. Analysis was made at each time point using the mean and the standard deviation from the historical data
corresponding to the tested time point (see Table III). The value of z was again limited to -3 < z < +3.