Multivariate formulation continued

From last time, the initial analysis of the full factorial formulation analysis was looking pretty good.  Looking at the experimental set up, there's the opportunity to actually get quite a bit more information than what would normally come from a full factorial.

The protein concentration and pH were both completed at three levels.  As a result, the square terms of these may be included in the model analysis.  Comparison with the full factorial results will then provide some indication if our model is improving or not.  The ANOVA results suggest the model has improved the F ratio as well as reducing the mean square error to our model.
Results from ANOVA 

Source	DF	Sum of Squares	Mean Square	F Ratio
Model	21	626.26088	29.8219	72.3149
Error	58	23.91862	0.4124	Prob > F
C. Total	79	650.17950		<.0001*

The parameter estimates provide some further indication of how to clean up the model.  Based upon this, our model should include the concentration, pH, ion, excipient along with some pairwise interactions and the square of the pH.
Parameter Estimates

Term		Estimate	Std Error	t Ratio	Prob>|t|
Intercept		45.828532	0.294567	155.58	<.0001*
Concentration		-0.508365	0.088588	-5.74	<.0001*
pH		2.7446617	0.088588	30.98	<.0001*
Ion[L1]		1.7889512	0.101366	17.65	<.0001*
Ion[L2]		-1.497137	0.101366	-14.77	<.0001*
Excipient[L1]		-0.966234	0.101366	-9.53	<.0001*
Excipient[L2]		0.7916596	0.101366	7.81	<.0001*
(Concentration-2.0125)*(pH-1.9875)		-0.111525	0.109247	-1.02	0.3116
(Concentration-2.0125)*Ion[L1]		-0.233058	0.124446	-1.87	0.0661
(Concentration-2.0125)*Ion[L2]		0.0613867	0.124446	0.49	0.6237
(pH-1.9875)*Ion[L1]		0.142317	0.124446	1.14	0.2575
(pH-1.9875)*Ion[L2]		-0.379905	0.124446	-3.05	0.0034*
(Concentration-2.0125)*Excipient[L1]		0.0169423	0.124446	0.14	0.8922
(Concentration-2.0125)*Excipient[L2]		0.0169423	0.124446	0.14	0.8922
(pH-1.9875)*Excipient[L1]		-0.229905	0.124446	-1.85	0.0698
(pH-1.9875)*Excipient[L2]		0.1756503	0.124446	1.41	0.1635
Ion[L1]*Excipient[L1]		0.4063566	0.143037	2.84	0.0062*
Ion[L1]*Excipient[L2]		0.0211714	0.143037	0.15	0.8828
Ion[L2]*Excipient[L1]		-0.086236	0.143037	-0.60	0.5489
Ion[L2]*Excipient[L2]		-0.238088	0.143037	-1.66	0.1014
(pH-1.9875)*(pH-1.9875)		-0.868243	0.152065	-5.71	<.0001*
(Concentration-2.0125)*(Concentration-2.0125)		0.2873127	0.152065	1.89	0.0638

Using the refined model then, the ANOVA shows the model's ability to describe the variance has gone up significantly and the effects are are all statistically significant.
ANOVA Results from Fit with Refined Model

Source	DF	Sum of Squares	Mean Square	F Ratio
Model	9	615.11995	68.3467	136.4611
Error	70	35.05955	0.5009	Prob > F
C. Total	79	650.17950		<.0001*

Effect Tests

Source	Nparm	DF	Sum of Squares	F Ratio	Prob > F
Concentration	1	1	13.85908	27.6711	<.0001*
pH	1	1	396.90170	792.4550	<.0001*
Ion	2	2	149.62521	149.3711	<.0001*
Excipient	2	2	43.08481	43.0116	<.0001*
pH*Ion	2	2	3.79747	3.7910	0.0273*
pH*pH	1	1	13.62475	27.2032	<.0001*

The Prediction Profiler in JMP allows the model to be visualized and provide some indications of how the factor varies.  At the mid-point for pH and concentration, without the presence of the excipients the predicted Th value is 51.28 C.  For the system in question, the Th is trying to be pushed to a maximum (higher Th = greater protein stability).    Upon increasing the pH and adding sucrose, the stability can be pushed significantly higher (Th = 55.07 C).  Further increase to the factor can be obtained by reducing the protein concentration.
Initial Prediction Profile of Th with JMP

Maximizing Th

A significant factor affecting the feasibility of the formulation is the viscosity of the solution.  In the next blog, I'll show how the models from each of these provides an acceptable formulation.