Example of Stability Study with a random batch factor

Interpret the results

The p-value that compares the models with and without the Month by Batch interaction is 0.059. Because the p-value is less than the significance level of 0.25, the analysis uses the model with the Month by Batch interaction. The shelf life, which is approximately 53 months, is an estimate of the how long the engineer can be 95% confident that 95% of the drug is above the lower specification limit. The estimate applies to any batch that the engineer randomly selects from the process.

The marginal residuals may not follow a normal distribution with constant variance. The points on the normal probability plot do not follow the line well. One reason for the nonnormal behavior of the marginal residuals is that, when the final model includes the batch by time interaction, the variance of the marginal residuals depends on the time variable and may not be constant. You can use the conditional residuals to check the normality of the error term in the model.

Factor Information

Factor	Type	Number of Levels	Levels
Batch	Random	8	1, 2, 3, 4, 5, 6, 7, 8

Model Selection with α = 0.25

Model	-2 LogLikelihood	Difference	P-Value
Month Batch Month*Batch	128.599
Month Batch	133.424	4.82476	0.059

Variance Components

Source	Var	% of Total	SE Var	Z-Value	P-Value
Batch	0.527409	72.91%	0.303853	1.735739	0.041
Month*Batch	0.000174	0.02%	0.000142	1.224102	0.110
Error	0.195739	27.06%	0.036752	5.325932	0.000
Total	0.723322

Model Summary

S	R-sq	R-sq(adj)
0.442424	96.91%	96.87%

Coefficients

Term	Coef	SE Coef	DF	T-Value	P-Value
Constant	100.060247	0.268706	7.22	372.378347	0.000
Month	-0.138766	0.005794	7.22	-23.950196	0.000

Random Effect Predictions

Term	BLUP	StDev	DF	T-Value	P-Value
Batch
1	1.359433	0.313988	12.45	4.329567	0.001
2	0.395375	0.313988	12.45	1.259203	0.231
3	0.109151	0.313988	12.45	0.347629	0.734
4	-0.409322	0.313988	12.45	-1.303623	0.216
5	-0.135643	0.313988	12.45	-0.432001	0.673
6	-1.064736	0.313988	12.45	-3.391006	0.005
7	0.049420	0.313988	12.45	0.157394	0.877
8	-0.303678	0.313988	12.45	-0.967164	0.352
Month*Batch
1	0.006281	0.008581	10.49	0.731925	0.480
2	0.019905	0.008581	10.49	2.319537	0.042
3	-0.013831	0.008581	10.49	-1.611742	0.137
4	0.003468	0.008581	10.49	0.404173	0.694
5	0.001240	0.008581	10.49	0.144455	0.888
6	0.000276	0.008581	10.49	0.032144	0.975
7	-0.010961	0.008581	10.49	-1.277272	0.229
8	-0.006378	0.008581	10.49	-0.743220	0.474

Marginal Fits and Diagnostics for Unusual Observations

Obs	Drug%	Fit	DF	Resid	Std Resid
10	101.564000	99.643950	7.04368	1.920050	2.375254	R
31	100.618000	98.811354	7.05273	1.806646	2.213787	R
55	98.481000	96.729866	8.87383	1.751134	2.033482	R

Shelf Life Estimation

Lower spec limit = 90
Shelf life = time period in which you can be 95% confident that at least 95% of response is
     above lower spec limit
Shelf life for all batches = 53.1818

Interpret the results