Design resolution and aliasing. Central composite design. Supports a full quadratic model; main effects, two-factor interactions and curvature are estimated independently.
How to read this. R² is the share of the variation the model explains; adjusted R² penalises extra terms and is the fairer figure to quote. RMSE is the typical prediction error in the response's own units. A model p-value below 0.05 means the model as a whole explains more than chance would.
6. Effect estimates
Titre (g/L)
Term
Coefficient
Std. error
t
p
Sig.
Intercept
4.173
0.044
95.25
—
pH
0.119
0.028
4.26
0.0011
**
Temperature
-0.351
0.028
-12.53
<0.0001
***
DO Setpoint
0.081
0.028
2.89
0.0135
*
Feed Rate
0.552
0.028
19.71
<0.0001
***
pH×Temperature
0.058
0.030
1.93
0.0769
pH×DO Setpoint
0.004
0.030
0.13
0.9017
pH×Feed Rate
0.200
0.030
6.73
<0.0001
***
Temperature×DO Setpoint
0.035
0.030
1.18
0.2617
Temperature×Feed Rate
-0.196
0.030
-6.60
<0.0001
***
DO Setpoint×Feed Rate
-0.005
0.030
-0.17
0.8692
pH²
-0.289
0.074
-3.90
0.0021
**
Temperature²
-0.264
0.074
-3.56
0.0039
**
DO Setpoint²
-0.024
0.074
-0.32
0.7510
Feed Rate²
-0.334
0.074
-4.51
0.0007
***
Significant at p < 0.05: Feed Rate, Temperature, pH×Feed Rate, Temperature×Feed Rate, Feed Rate², pH, pH², Temperature², DO Setpoint. Terms not listed could not be distinguished from experimental noise in this study; that is not proof they have no effect, only that this design did not resolve one.
Significance codes: *** p < 0.001, ** p < 0.01, * p < 0.05.
7. Analysis of variance
Titre
Source
DF
Sum of squares
Mean square
F
p
Model
14
13.266
0.948
67.05
<0.0001
Residual
12
0.170
0.014
—
—
Lack of fit
10
0.167
0.017
12.84
0.0743
Pure error
2
0.003
0.001
—
—
Total
26
13.435
—
—
—
Lack of fit is not significant (p = 0.0743), so the model form is adequate for the data: the scatter around the model is consistent with pure experimental error measured from the 1 replicated setting(s).
8. Figures
Figure 1. Effects Pareto — terms ranked by |t|, with the p = 0.05 significance threshold.Figure 2. Residuals versus fitted values — should show no pattern.Figure 3. Normal Q-Q plot of residuals — points should follow the line.Figure 4. Response surface over the two selected factors, other factors held at the predicted optimum.
Figures are reproduced from the analysis session that generated this report. Residual and Q-Q plots are the model's assumption checks: clear curvature, funnelling or heavy tails indicate the model should not be trusted for prediction.
9. Predicted optimum and recommended setpoints
Factor
Recommended setpoint
Unit
Coded
pH
7.093
—
0.467
Temperature
32.193
°C
-0.923
DO Setpoint
59.194
%
0.946
Feed Rate
6.000
mL/L/d
1.000
Response
Goal
Predicted at optimum
Unit
Titre
maximize
4.754
g/L
These are model predictions, not measurements. The optimum is interpolated from the fitted model and must be verified by confirmation runs at the stated setpoints before it is adopted as an operating condition or written into a control strategy.
10. Confirmation runs
Confirmation check — observed vs predicted (±2σ)
Response
Predicted
Measured mean
±2σ band
Verdict
Titre
4.754
4.757 (n=3)
4.516 to 4.992
✓ within
Each confirmation mean is compared with the prediction at the setpoint it was actually run at, and all fall within ±2σ. These runs are consistent with the model; this is supporting evidence for this setpoint, not a formal validation.
11. Assumptions and limitations
Conclusions apply only within the factor ranges listed in section 3; extrapolation beyond them is not supported by this study. The model assumes independent, approximately normal residuals with constant variance — see the diagnostic figures. Effects not found significant are not established as absent. Predicted optima require experimental confirmation.
12. Design space and quality context
This study supports a Quality-by-Design approach (ICH Q8(R2)) by varying the listed factors across defined ranges to characterise their effect on the responses, informing the design space and the identification of critical process parameters. Used for analytical work, the same approach supports analytical procedure development under ICH Q14.
Status of this document. This is a generated report, not a regulated record system. It is not itself 21 CFR Part 11 compliant: it carries no system audit trail, access control or electronic signature. To use it in a regulated decision, review and approve it under your own site SOP, retain it in your validated document system, and hold the underlying data as the raw record.
13. Review and approval
Generated with DOE Pro at bioprocesstools.com on 01/11/2026. Design: Central Composite Design, 27 runs, 4 factors. Predicted optima are model estimates and require experimental confirmation before adoption as operating conditions.
