Two-Level Factorial and Fractional Factorial Designs are useful for either screening the vital few factors affecting your process from the many tested, or to evaluate the robustness of your process to small variations of the parameters in a design space predicted to conform to your acceptance criteria. Typically these are resource-efficient studies comprising 10 – 20 experiments.

To handle these design types within the Design Expert DX7 software tool, we have further tipsheets to help you:

   Two-Level Designs: Building the Design
   Two-Level Designs: Analysing your Results
   Two-Level Designs: Diagnosing your Results
   Two-Level Designs: Interpreting your Results

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Build the Design (Factorial Tab: 2-Level Factorial)

 

Choose New Design from the file menu. Select the cell corresponding to the Number of Factors you wish to investigate and the number of Runs you want to perform.

Select the number of Replicates, Blocks and Centre Points required (see Design Tips below)

Click Continue to view the alias pattern and decide allocation of factors to letters; and again to enter the details of the factors and responses

Colour key: The fractionation (superscript), colour and resolution (subscript) indicate how much information you can expect from the design about the key effects

White (Completely Safe) – all effects are capable of being independently estimated

Green (Safe, Go) – main effects & 2-Factor Interactions can be estimated separately of other key effects

Yellow (Caution) – main effects are aliased with 3-Factor Interactions, but 2-Factor Interactions are aliased with one another

Red (Stop, Think) – main effects are aliased with 2-Factor Interactions 

Replicates of factorial combinations, and/or replicated centre points, provide an estimate of the background variability (pure error). Including centre points also enables you to test whether a potential optimum exists within your current ranges (e.g., max hardness/min degradation etc.)

Blocks manage the experiments when they must be performed by different operators, equipment or on different days and these variations are expected to introduce bias

Factor Types can either be Numeric i.e on a continuous scale between Low & High Settings, or Categoric i.e. with discrete settings (e.g., type of lubricant)

Fractionating: The more you fractionate to save resource, the more the effects you would like to investigate get partnered up, or aliased, and so the poorer the resolution of your design

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

 

Display: Right-click on a column heading to display options:

• Std - sort your design into standard order
• Run - sort into a randomised run order
• Type - display type of point e.g. Factorial
• Factor - edit information e.g. ranges
• Response - e.g. simulate results

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Analysing your Results (Effects)

 

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. For the Transform, start with None and proceed by clicking on the Effects button.
 

 

The Half-Normal plot & Pareto Chart provide a visual means of identifying important effects and assessing their statistical significance. Click on the effects you think are important – on the former plot these are the effects farthest to the right 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

 

 

Example: In this example, A: Magnesium Stearate, B: Granulation Paste Type & D: Granulation Blend Time, together with the AD interaction (dependent relationship between Magnesium Stearate & Granulation Blend Time) are the largest effects and statistically significant. Although the AB interaction on the half-normal plot is to the right of the line passing through the green triangles (noise) & the small effects which appear no different from the noise, it is not a statistically significant effect according to the Pareto Chart.

Aliasing: In the case of a fractionated design, you should always check the aliasing. Either click the Alias List button, or right click on an effect in a plot (e.g., AD is aliased with BC here). Since A and D are both significant large effects, it is more likely to be the AD rather than the BC interaction that contributes largely to this effect. Remember, main effects and 2-Factor Interactions are more common than 3-Factor Interactions. Choose the aliased effect which makes most sense to you to carry forward to the next step of the analysis. If in doubt you can always add or “augment” further runs to de-alias or untangle effects. (See the augmentation tipsheet)

Hierarchy: If Design Expert asks “Would you like the hierarchy corrected automatically?” respond “Yes”. This ensures that if you select an interaction, but not the main effects of the factors involved, DX will automatically include the main effects in order to preserve the hierarchy of your model and a suitable testing of its effects

Analysing your Results (ANOVA)

 

Mean Square column refers to the variance or signal associated with each term (e.g., variation in hardness due to Magnesium Stearate is 136.95, while residual or noise variation is just 1.12).

F Value, next column, is the signal-to-noise variance ratio (e.g., the signal or effect due to Magnesium Stearate is 122.26 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 an effect)

There are also tests for evidence of:

Curvature compares the average of the centre point results with the prediction at the centre of your design (c.f. Intercept Coefficient Estimate). The difference is the Centre Point Coefficient Estimate

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

Curvature: close agreement between your replicate observations can artificially lead to evidence of significant lack of fit or curvature. If you confirm there is real evidence of curvature, then interpret the Model Graphs with caution. Predictions made inside the low and high settings will be unreliable

ANOVA: By default the 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

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Diagnostics

 

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 assumptions

If the residuals lie roughly on a straight line, then the noise is approximately 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 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)

 

 

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Model Graphs

 

Use the Factors Tool to choose the effect you wish to display. You can view One Factor plots of the main effects of each factor as it moves from its low to high setting.

 

 

An Interaction plot shows whether the effect of moving from a low to a high setting of one factor depends on the setting of a second factor.

 

 

Example: Increasing A: Magnesium Stearate from 3.5 to 14.0 mg/tab decreases Hardness. However, the drop in Hardness due to increasing Magnesium Stearate also depends on the setting of D: Granulation Blend Time. A steeper drop in hardness at high blend time (red line) is seen compared to low Granulation Blend Time (black line). Similarly, at low Magnesium Stearate there is no statistical difference in Hardness using short vs. long Blend Time – 95% least significant difference (LSD) bars overlap – compared with high Magnesium Stearate where there is a statistical difference in hardness – 95% LSD bars for short (black) and long (red) Granulation Blend Time do not overlap.

Independently of Magnesium Stearate and Granulation Blend Time, B: Granulation Paste Type statistically significantly decreases Hardness as the ratio changes from 0.8 to 1.2 eq. Therefore, to increase Hardness we must use low Magnesium Stearate and low Granulation Paste Type and, at low Magnesium Stearate, Hardness is robust or insensitive to changes in Granulation Blend Time.

Choose alternative ways to plot the effects of the factors using the View menu

Cube plot displays the effects of three factors – we see confirmation that the combination of low Magnesium Stearate & low Granulation Paste Type (bottom left-hand cube edge) yields the best results for Hardness. There is little change in Hardness as Granulation Blend Time varies at low Magnesium Stearate
 

Contour plot displays the gradations of effect on Hardness as two factors vary

 

 

3D Surface displays the predicted response as a 3rd dimension and is rotatable using the Rotation tool

 

 

Model Graphs: 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

Optimization: Multiple Results

 

Following the steps above to also analyse the Degradation & Dissolution responses suggests ideal settings for the four factors:

A: Magnesium Stearate low to achieve criteria for Hardness (>20lb) & Dissolution (>95%)
B: Granulation Paste Type low to help meet the targets for Hardness & Dissolution
C: Dibasic Sodium Phosphate at low setting provides a wider operating window for Magnesium Stearate to meet criteria for Degradation (<4.5mg/h), but needs to be set high in order to also meet Dissolution goal
D: only affects Hardness, but at low Magnesium Stearate Hardness is robust to changes in Granulation Blend Time so set it at its most practical/economical setting

Example: Dissolution

 

 

 

Example: Hardness

 

 

 

Once you have arrived at models for all your responses, Design Expert provides optimization tools to help locate settings and ranges to simultaneously meet the Criteria and Goals you set for 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 or 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).

Here the solutions are displayed as a report.
 

Here the solutions are presented as ramps.

 

 

 

Click Graphs to locate the most desirable solution or region and by how much desirability falls off as you deviate from this ideal (c.f. Taguchi loss function).

Choose 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 (c.f. all or nothing loss function).

On this graph and on the Overlay Plot below, 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

Overlay Plot

 

 

 

Point Prediction presents the predicted values of each your responses for specified factor settings together with the reliability surrounding a predicted average result (SE Mean & 95% Confidence Interval), or a predicted individual result (SE Prediction & 95% Prediction Interval).

To choose settings for the parameters on which to base predictions:

• click on the Numerical Solutions, or
• use the Factors Tool to move the slider bars (Guages), or
• enter values on the Sheet

Optimization: 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

Optimization: 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

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