Stability analysis is the study of how drug product potency degrades over time. The primary statistical quantity of interest is the expiration date or shelf life. Typically, a drug product is manufactured in batches. When estimating the shelf life of a medication, it is necessary to evaluate how the batches differ with respect to the potency degradation of the drug product over time. Specifically, three different model types are considered.

### Model 1: Common Regression across All Batches

This model is appropriate when the intercepts and slopes do not differ across batches and the data therefore can be pooled in order to estimate a common intercept and common slope.

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Model 2: Separate Intercepts and Common Slope

This model is appropriate when the intercepts differ across batches, but the slopes are common across batches.

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Model 3: Separate Intercepts and Separate Slopes.

This model is appropriate when the intercepts and slopes differ across batches. Additionally, the Mean Square Error (MSE) is not pooled across different batches, that is, homogeneity of variance across batches is not assumed.

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Parameter Estimation

Parameters are estimated by ordinary least squares. This can be accomplished by solving the normal equations:

### t-Statistic

The t-statistic associated with the null hypothesis that the ith parameter is equal to 0 is given by:

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Confidence Interval for a Single Parameter

A two-sided 95% confidence interval for the ith parameter is given by:

### Predicted Values

The predicted value or estimated mean response for a given covariate vector Xi is given by:

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Confidence Interval

The two-sided 95% confidence interval for the mean response for a given covariate vector Xi

is given by:

For a one-sided interval, then the 95th quantile of t distribution is used.

### Prediction Interval

The two-sided 95% prediction interval for a new observation Xi is given by:

For a one-sided interval, then the 95th quantile of the t distribution is used.

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**Model Selection**

Statistica selects the model by first evaluating if the slopes are equal across different batches.

If the slopes' p-value is significant (by default, less than 0.25), the separate intercepts and separate slopes model is used to estimate shelf life.

If the slopes' p-value is not significant, Statistica determines if the intercepts are equal across different batches. If the intercepts' p-value is significant (by default, less than 0.25), the separate intercepts, common slope model is used. If neither the slopes' p-value nor the intercepts' p-value is significant (by default, less than 0.25) the common intercept, common slope model is used.

The p-values are computed by using the type 1 sums of squares decomposition of the full linear model with effects placed into the model in the following order: time, batch, batch by time interaction. The slopes' p-value is associated with the batch by time interaction and

the intercepts' p-value is associated with the batch effect.

### Shelf Life Estimation

Shelf life estimation is performed by determining where the 95% lower and/or upper confidence or prediction interval intersects with the user-defined specification limit(s). If either Model 2 or Model 3 has been selected, shelf life is estimated separately for each batch. Shelf life is then defined as the shelf life associated with the worst-case batch, that is, the smallest shelf life estimate. By default, Statistica will base shelf life computations on a two-sided 95% confidence interval.

### Residual Analysis

A user can optionally perform a residual analysis to determine the selected model's adequacy.

A residual is defined as the difference between the observed and predicted value, that is:

Additionally, **studentized** residuals are computed. The formula for the studentized residuals is given by:

A **normal probability** plot of the (studentized) residuals is constructed as follows. First, the (studentized) residuals are rank ordered. From these ranks, Statistica computes z values (i.e., standardized values of the normal distribution) based on the assumption that the data come from a normal distribution:

These z values are plotted on the y-axis in the plot. If the observed (studentized) residuals (plotted on the x-axis) are normally distributed, all values should fall onto a straight line. If the residuals are not normally distributed, they will deviate from the line. Outliers can also become evident in this plot. If there is a general lack of fit and the data seem to form a clear pattern (e.g., an S shape) around the line, the variable may have to be transformed in some way (e.g., a log transformation to "pull?in" the tail of the distribution, etc.)

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