Focus on displaying the factors with the most influence on the response(s) – the factors with the greatest significant effects in the ANOVA. To help you decide which factors and views to plot, consider selecting Perturbation from the View menu

The Perturbation graph uses the model terms to display the effect of changing each factor from the reference point (default centre), whilst holding the other factors constant

In this way the plot can be used to decide which factors have the greatest effect on the response as well as locating ideal settings (or a direction) for each factor to improve the response

In this example clearly B-Liquid Volume has the greatest (curve-linear) affect on Hardness; with this response reaching a maximum of approx 12kp at when Liquid Volume = 0.36 coded units (40%w/w). The other 3 factors have very little effect in relation to factor B.

A plot of the One Factor Liquid Volume using the slider bars on the Factors Tool box for the other factors seems to be an obvious choice of plot to describe the effects of the factors on Hardness.

When the ANOVA and Perturbation plot suggest two or more factors are important, consider the Interaction, Contour and 3D Surface plots under the View menu. For example, the strongest effects on the response Compressibility are those of C-Inlet Air Temperature and D-Spray Rate together with their interaction. B-Liquid Volume also has a small positive effect according to the Perturbation plot.

Choose the Contour plot under the View menu to easily determine that to improve or minimise Compressibility (its goal), both Inlet Air Temperature and Spray Rate need to be increased, while Liquid Volume needs to be reduced using its slide bar on the Factors Tool box.

View: 3D Surface displays the predicted compressibility response as a 3rd  dimension versus the two factors Inlet Air Temperature and Spray Rate. The 3D surface is rotatable in any direction using the Rotation tool
To change the factors you are plotting on any axis of any plot, simply right click on the appropriate factor in the Factors Tool box and select the axis of choice

You can change graph properties by right clicking on a graph and select Graph Preferences.  Alter axis scales, the number of contours and their values, font settings etc. Alternatively, add contours or flags (prediction values) to contour plots by right clicking over the graph and selecting the appropriate option; click and drag a contour; or double click on a contour to enter a desired response

Interpret your Results (Optimization: Multiple Responses)

Once you have modelled all your responses and interpret their model graphs in turn, Design Expert provides optimization tools to help locate settings and ranges to simultaneously meet the Criteria and Goals you set for the multiple responses and also for the factors

For Numerical Optimization, set the Criteria: Goal, Lower &/or Upper Limits for the factors and responses. Use Weights & Importance to respectively give more or less emphasis to an individual goal relative to others. Design Expert searches for and lists Solutions (settings for the factors) to match your criteria: from the most to the least desirable – desirability ranges from zero (at least one goal was unachievable) to one (all goals were easily met).

Solutions displayed as a report

Solutions visually presented as ramps

Click Graphs to visually locate the most desirable solution, indicated by a flag, or region and by how much desirability falls off as you deviate from this ideal. In this case, the Numerical optimization Solutions and Graphs suggest the most forcing conditions, or highest settings of B-Liquid Volume, C-Inlet Air Temperature and D-Spray Rate produce the most desirable results in terms of meeting the goals for all the responses. A-Batch Size can vary across its entire range without altering or affecting the desirable region (i.e., the responses are robust to batch size in this region)

For Graphical Optimization, set the Criteria: Lower &/or Upper Limits for the responses to establish a yellow feasible region which meets all the criteria. A limitation of this plot is that it leads to the perception that all results within the yellow region are good, while all results outside are bad (i.e., all or nothing). On either of the two graphs above, click and hold the left mouse button to drag a box over a desirable, or potentially robust, region of conformance in order to zoom-in on that region

Point Prediction presents the predicted values of each your responses together with the reliability surrounding a predicted average result (SE Mean & 95% Confidence Interval), or a predicted individual result (SE Pred & 95% Prediction Interval). You can display the predictions for the desirable Numerical Solutions or for specified factor settings using the Factors Tool – move the slider bars (Gauges) or enter values on the Sheet

If no numerical or graphical solutions are found, you may need to loosen your criteria and try again. You may also wish to consider extrapolated predicted solutions outside of your current ranges. To view these on a graph, right click on the graph; select Graph Preferences; and expand the X1 and X2 Axes. You should verify these extrapolated predictions

In addition to the flag posted when a numerical solution is selected, you can add a flag at any location on an optimization graph by right clicking and selecting Graph Preferences

Residuals = observed – model predicted data. They represent the noise left over after the systematic model effects are removed. Use the Diagnostics Tool to display these residuals in simple plots to check your model assumption

• If the residuals lie roughly on a straight line, then the noise is approx. Normally distributed
• If the residuals are equally spread out across the plot – the prediction range – & around zero then the noise is constant & centred around zero (no noise). The tram lines help identify outliers  or large residual values
• If there is a trend in the residuals vs. run order, this would indicate something not in the model was changing over time. E.g., a downward trend may indicate degrading starting material
• If the model fits well to the data, the predicted and actual should correspond closely and approximately lie on a straight line
• If the residuals are not normally and consistently spread (e.g. fan out/worsen as the predictions get bigger) use Box-Cox plot to help you choose a transform to try out (i.e. go back to Transform)
• If the effects of the factors have been adequately captured by the model, then there will be no remaining systematic relationship between residuals versus factor settings and the variation across the factor settings will be consistently spread

Under Analysis on the left-hand tree structure, click on each response you want to analyse and simply work your way along the analysis buttons from left-to-right. For the Transform, start with None and proceed to the next button. You may wish to reconsider a transformation after examining the Diagnostics.

Fit Summary: uses 3 tables of statistics to recommend the complexity or order of Model to fit to your data…

• Sequential Model Sum of Squares chooses the highest order of polynomial model required to significantly explain the variability in your data
• Lack of Fit Tests compares the model’s residual error to the pure error from the replicated design points to recommend the model with insignificant lack of fit
• Model Summary Statistics suggests the simplest model which both explains and predicts the greatest fraction of variation in the data; using the Adjusted R-Squared and Predicted R-Squared statistics respectively

Warning: Design Expert considers all the terms which make up the order of each polynomial model, however insignificant, to recommend the complexity of the model that best fits your data. The Quadratic model listed in the example above will include all main effects, all two factor interactions and all squared terms when only a subset of these would be necessary. For less experienced users we recommend you follow the steps outlined in the next section under Model, regardless of the model recommended under Fit Summary. Also, don’t worry about the WARNING message as it is simply letting you know if the design isn’t capable of estimating all possible information for the cubic model independently. It is unlikely that your data will require such a complex model unless the ranges of the factors are too wide. You may need to reconsider the ranges in order to generate a suitable model for prediction.

Model: Since the model recommended by the Fit Summary often includes insignificant effects or misses important effects we suggest you perform the following regardless of the Process Order proposed by Design Expert…

• Choose a Quadratic model for the Process Order
• Use Backward Selection to fit the full quadratic model – all main effects, two factor interaction and quadratic terms have a green M for model
• Click the ANOVA button to automatically and sequentially omit terms with the highest to lowest p-values greater than the Alpha Out threshold 0.1
• If you are asked: Would you like the hierarchy corrected automatically? choose Yes. This ensures that you preserve the hierarchy of your model (i.e., main effects of the factors involved in a significant interaction or quadratic effect will also be included in the model for a suitable testing of its effects)
  

Analysing your Results (ANOVA)

Adj R-Squared and Pred R-Squared measure the fraction of variation that can be explained and predicted by your model. Together with the other summary statistics, they provide a numerical assessment of model adequacy

Warning: close agreement between your replicate observations can artificially lead to evidence of significant lack of fit.

• The insignificant terms (p-value > 0.1) omitted from the model by the backward selection approach are listed first
• Check the ANOVA (Analysis of Variance) table to assess the importance of the model and the individual terms in the model
• Mean Square column refers to the variance or signal associated with each term (e.g., variation in Hardness due to Liquid Volume is 4.00, while residual or noise variation is just a 100th of its size at 0.04)
• F Value, next column, is the signal-to-noise variance ratio (e.g., the signal or effect due to Liquid Volume is 100 times that due to the noise)
• P-value, final column, is the probability of observing a signal-to-noise ratio as large as in the previous column purely by chance (i.e., the risk of you making the wrong the decision about the importance of the Liquid Volume effect is incredibly small <0.0001)
• The important terms in the model or effects on Hardness appear to be B-Liquid Volume, B2 (curvature due to Liquid Volume) and a smaller CD interaction between C-Inlet Air Temperature and D-Spray Rate
• There are also tests for evidence of Lack of Fit the ratio of variance due to effects not previously selected to include in your model with the Pure Error. If you fail to include large effects then this will inflate the Lack of Fit. Use effect and diagnostic plots (e.g.,  residuals (ei) vs. Factor) to identify potential effects to include to improve your model)

Annotated ANOVA – by default the above ANOVA results come with comments designed to help you interpret the output. If the annotations do not appear, select Annotated ANOVA on the View menu. Help to interpret any value on the output can be gained by highlighting the value and either pressing the F1 key or right clicking and selecting Help 

There are three response surface method (RSM) designs you will commonly choose from:

• Central Composite Design (CCD)
• Box-Behnken Design (BBD)
• D-Optimal Design 

1. Central Composite Design (CCD)

This is the most popular design for modelling curvature, or an optimum, as it is sequentially created by augmenting a two (±1) level design with centre points and axial points; which are a distance of -alpha & +alpha from the centre of the design in coded units

Click the CCD Options button to change the default design points. You can change how far the axial points are extended.

 
Consider the following options:

• Alpha = Rotatable desirable properties of 5 settings for each factor and equal reliability for predictions the same distance from the centre of your design. If these are too extreme or impractical try…
• Alpha = Face Centred (CCF) extends the axial points to the faces of your factorial (±1) design. There are only 3 settings for each factor, but the axial points are not extended beyond current ranges
• Alpha = Other permits you to control how far the axial points are extended (e.g., alpha < 1 brings the points inside the design creating a 5-level central composite inscribed (CCI) design

Rotatable CCD provides excellent prediction capability across the experimental space; CCF has better capability nearer the centre and CCI only nearest to the centre

CCD is insensitive to missing data or failed experiments

Click Factor Ranges in Terms of Alphas to set the axial points at your practical extremes DX will adjusts the ±1 factorial settings accordingly

Separate factorial & axial points into blocks to manage or to remove systematic bias between experiments

Click Type. If ≥ 5 factors, then DX uses a green fractional factorial core design if you select Full, or a red fractional factorial core if you select Small 

2. Box-Behnken Design (BBD)

Box-Behnken Design is design created solely to model curvature, or an optimum. There are only 3 settings for each factor. These are positioned at the midpoints of the edges of the design; emphasising an ideal as well as specific setting for each of your factors. BBD provides strong prediction capability near the centre of your experimental space (where the optimum is presumed to be), but is weaker at the corners of this space.

BBD requires fewer experiments to set up compared to a CCD, but is not recommended if you expect missing data or failed experiments

You can change the number of Centre points, but stick with the default unless resource is an issue, and you can divide the experiments into Blocks for reasons of management or bias

If a CCD or BBD (standard design) is not fit for purpose, practical or flexible then consider using a D-Optimal design.

3. D-Optimal Design

The key benefits of D-optimal designs:

• Resource Efficient requiring even fewer experiments compared with both BBD & CCD, and the capacity to further decrease the number of runs using your existing process knowledge
• Augments an existing set of data to improve the predictive power of your model for optimization, design repair, or inclusion of previously performed experiments (c.f. augmenting designs for optimisation)
• Constraints on the experimental space can be imposed by clicking Edit Constraints to exclude undesirable/unnecessary conditions, or to choose from a specific Candidate List of desirable experiments

The standard menu for generating D-Optimal designs allows you to create a model to suit your current state of knowledge (e.g., Quadratic to model curvature) and your pocket!

• Edit model: Specify the model you want to fit with only the effects of interest based on your experience: reduce a model & therefore the resources needed to estimate it, by removing terms you think are unlikely to be important. Double click on a term to remove it from the Model (M)
• You are then informed of the type of runs needed to fit (Model points) and test your model (lack of fit & Replicates) as well as the Total runs required
• The D-optimal algorithm selects a subset of design points from the list of Candidate points to maximize the information about your specified model

Enter one constraint per row in the spaces provided. In this example, the constraint ensures that settings of spray rate (D) cannot fall below a certain g/min to compensate for increasing batch size. This excludes a particular region of experimental space where you cannot get a response.

Click Evaluation and the Graphs button to graph the standard error plot with the constraints and the excluded area visible

Entering Factors and Responses

To enter factors & responses for each of the above three Response Surface designs

• Simply select the number of Numeric and Categoric factors and enter the factor names, units and extreme low and high levels. For a CCD, the distance of the axial points from the centre of the design (i.e., alpha) is automatically determined
• Click Continue to provide Block names (e.g., operators), if you chose to perform the runs in more than one block, or to enter the number, names and units of the responses
  

Running the Experiments and Entering the Data

For each experiment run in Run order, enter the results for each response into the Design (Actual) sheet. An alternative Run Sheet view, for carrying out & recording your results offline, is available under the View menu

Right click on a column heading to display options relating to that column:

• Std – sort your design into a standard order
• Run – sort into a randomised run order
• Block – display point type e.g., Factorial, Axial etc.
• Factor – edit information e.g., ranges, decimal place
• Response – e.g., simulate results
  

To handle optimization designs within the Design Expert DX7 software tool, we have further tipsheets to help you:

   Optimization Designs: Building the Design
   Optimization Designs: Analysing your Results
   Optimization Designs: Diagnosing your Results
   Optimization Designs: Interpreting your Results

These tipsheets are linked to this case study

• Move
• Close

Granulation Process

This top-spray granulation process has four key process and ingredient parameters:

• Magnesium Stearate
• Granulation Paste Type
• Filler (Dibasic Sodium Phosphate)
• Granulation Blend Time

and three key properties of tablets of a drug product:

• Hardness
• Degradation Rate
• Dissolution

A screening study has established improved ranges for the four key factors known to affect the process. Two non-critical or qualitative factors have been set at preferred (optimal) settings. An optimisation design is needed to locate a sweet spot for the process and predict ranges for the factors to deliver a design space that both conforms to the specified targets (Goals) and is a workable solution on plant

The increase in C-Inlet Air Temp brought about a statistically significant decrease in tablet hardness. A low setting of this parameter together with a low setting for D-Spray Rate raises Compressibility to its highest predicted value 17.37 displayed on the left. This suggests tightening the range for factors C and D to toward their higher settings should move the results away from the upper specification limit.

Use Point Prediction to predict whether the extreme settings of the factors identified by the analysis will generate future results unlikely to meet your targets:

For compressibility, low settings of factors C & D take us closest to the edge of failure. However even at these worse case settings, the 95% upper prediction limit 17.79 < 18% upper specification limit for this response.

The predicted results fall within acceptable limits (specification); suggesting that the process is robust over the preferred operational ranges

To help determine whether there are any individual out of specification results for each response (i.e., results beyond that considered practically acceptable) generated by the robustness study, click on Graph Columns under the Design node on the left-hand tree structure

Select Run from the X axis list and cycle through all the responses in the Y axis list. You can Colour By each response to highlight results from low-to-high values.

Check the range & pattern of response values against the response goal, target or specification. Also check the background variation – differences between the repeat centre point results – is not exceesive. If the results fall within acceptable ranges (specification) this would suggest the process or method is able to comply with acceptable quality over the preferred operational ranges and for all combinations of the parameters investigated. To ensure that the process or method is able to produce results that conform to specifications during normal usage and allowing for variation you should also consider the Point Prediction results generated at the extreme settings of the method parameters

Use the run order to display the results, so they are in the same order the experiments were conducted in – this gives you information about possible systematic trends in the data parameters

Analyse your Results (Effects, ANOVA and Model Graphs)

Under the Analysis node on the left-hand tree structure, click on each response you want to analyse and simply work your way along the analysis buttons from left-to- right. Please refer to the Tipsheet on 2-Level Designs for more details on analysing these types of designs.

For robustness studies the key output to focus on are the Effects Plots. The Half-Normal plot and Pareto Chart provide a visual means of assessing robustness. On the former plot determine whether the effects lie to the right of the line and away from the green triangles, which represent differences between the replicated points or background noise, while the Pareto Chart provides statistical thresholds to test the significance of the effects selected. If there are no statistically significant effects for each of the responses – in other words the effects are no larger than the noise – then the process or method is robust.

For Hardness, since all the effects lie approximately on a straight line and the size of effects are not statistically significant, or unimportant relative to the noise which stretches across the plot, this response is robust to changes in the factor ranges.

Note also that none of the individual results are out of specification (i.e., < 11kp) according to the Graph Columns plot for Hardness above

For Compressibility, there are two strong statistically significant effects due to C-Inlet Air Temperature and D-Spray Rate.

According to the Graph Columns plot the individual results are all within specification (i.e., < 18%), so the process may well be practically robust. It is worth looking at the Model Graphs for the two statistically significant effects to determine whether tighter control or tolerances on these parameters are required 

Choose New Design from the File menu

Select the lowest run design for the Number of Factors you want to study & enter the number of replicated Centre points (e.g., 2 – 4) you wish to run to establish the level of noise

Since we expect there to be no significant effects or interactions, then larger, higher resolution designs will typically not be required. All factors investigated at the screening stage should be included to demonstrate robustness

Continue to view the Aliases (for a 2^6-3 Red design, main effects are all aliased with 2-Factor Interactions); and Continue again to enter the details of the factors and then the responses (refer to the 2-Level Designs Tip Sheets if you need more detail on how to set up these types of design).

Enter the Predicted Acceptable Ranges

Enter the Output

Set the Robustness Targets

Run the Experiments and Enter the Data

For each experiment run in Run order, enter the results for each response into the Design (Actual) sheet. An alternative Run Sheet view, for carrying out & recording your results offline, is available under the View menu

If the robustness study fails to give you confidence that the process can be operated successfully within your predicted design space, then you can use the results of the robustness study to help you decide which factors are critical and require narrower ranges or tighter control. If there is excessive variation between the centre point results you may need to follow up with a variation management study or measurement systems analysis.

To handle robustness designs within the Design Expert DX7 software tool, we have further tipsheets to help you:

   Robustness Designs: Building the Design
   Robustness Designs: Analysing your Results
   Robustness Designs: Interpreting your Results

These tipsheets are linked to this case study

• Move
• Close

Granulation Process

This top-spray granulation process has four key process and ingredient parameters:

• Magnesium Stearate
• Granulation Paste Type
• Filler (Dibasic Sodium Phosphate)
• Granulation Blend Time

and three key properties of tablets of a drug product:

• Hardness
• Degradation Rate
• Dissolution

Set up and analyse the results of a resource efficient robustness study to evaluate the predicted acceptable ranges of the six equipment settings of a top spray granulator and assess its robustness to these small but deliberate changes in the six parameters