2/22/03 Factorial designs.ppt Steve Brainerd 1

Factorial Designs :Design of Experiments Steve Brainerd

• Designs

• Complete or full factorials: Chapter 5, 6, 7

• Fractional: Chapter 8

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Designs and resolution

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Designs and resolution from Design Expert Stat-Ease, Inc.Factorial Design display. Expect the following consequences:

Red – Stop and Think: A resolution III design indicates that main effects may be aliased with two factor interactions. Resolution III designs can be misleading when significant two factor interactions affect the response.Yellow – Proceed with Caution: A resolution IV design indicates that main effects may be aliased with three factor interactions. And two factor interactions may be aliased with other two factor interactions. Resolution IV designs are a good choice for a screening design since the main effects will be clear of two-factor interactions.Green – Go Ahead: Resolution V (or higher) designs are just about as good as a full factorial, generally at a great savings in the number of runs to perform. Assuming that no three factor (and higher) interactions occur, all the main effects and two factor interactions can be estimated.

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Designs and resolution from Design Expert Stat-Ease, Inc.Factorial Design display. 2 level factorial design menu

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Designs and resolution from Design Expert2 Level Factorial designs (2-15 factors) – Full and fractional designs are available to explore many factors, setting each factor to only two levels. Fractional factorials are efficient designs used to screen many factors to find the few that are significant. Full factorials can be used to estimate the effects of all interactions.

·Irregular Fraction designs (4-9 factors) – These are a special set of Resolution V designs, usually with fewer runs than a normal fractional factorial, that allow clean estimation of the main effects and two-factor interactions.

·General Factorial designs (1-12 factors) – The General Factorials can be used to design an experiment where each factor has a different number of levels (2 to 20). The design generated will include all possible combinations of the factor levels. The factors will be analyzed as categorical rather than numeric.

2/22/03 Factorial designs.ppt Steve Brainerd 6

Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Designs and resolution from Design ExpertD-Optimal designs (2-14 categorical factors) – This is offered as an alternative to the General Factorial designs, which may produce a design with too many runs. The D-optimality criteria can be used to select a subset of all possible combinations of the categorical factors. The number of runs generated depends on the level of interaction you want to estimate. These designs should be used carefully, taking into account subject matter knowledge to decide if the design is acceptable.

·Plackett-Burman designs (up to 31 factors) – These are highly confounded designs that are useful if you can safely assume that interactions are not significant. Another useful application is ruggedness testing where you are testing factor levels that you hope will NOT affect the response.

·Taguchi designs (up to 63 factors) – A set of classic designs from Taguchi teachings. These may be used as a base to build a particular design. Note that all analyses will be completed using standardized ANOVA reports and interaction graphs.

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Designs and Replicates from Design ExpertReplicates: By specifying the number of replicates you tell Design-Expert how many times each design point (set of experimental conditions) should be run. The default is 1 replicate, which means that each design point will be run once. If you ask for two replicates, then each experimental condition will be repeated. Repeats provide you with the ability to compute estimates of pure error.It is important to remember the difference between replicates and repeated measurements. A true replicate of a design point is the result of physically re-creating all the conditions for that experiment. This will give a more accurate estimate of the overall process error.

If you take several samples out of the same run, this is considered a repeated measurement and the variation observed in the response does not reflect the complete process error. If you specify this as your replicate, the pure error will be too small. Instead, you may want to simply use the average measurement from the samples as a single response data point.

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Designs and BLOCKS from Design ExpertBlocking is a technique used to remove the expected variation caused by some change during the course of the experiment. For example, you may need to use two different raw material batches to complete the experiment, or the experiment may take place over the course of several shifts or days. Design-Expert’s default of 1 block really means "no blocking."For example, in experiments with 16 runs, you may choose to carry out the experiment in 2 or 4 blocks. Two blocks might be helpful if, for some reason, you must do half the runs on one day and the other half the next day. In this case, day to day variation may be removed from the analysis by blocking.

Important: If you try to block on a factor, that factor will be aliased with the block and you will not get any statistical details on the effect of that factor. Only block on things that you are NOT interested in studying.

2/22/03 Factorial designs.ppt Steve Brainerd 9

Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Designs and BLOCKS from Design ExpertBlocking: Consider this example: say you are trying to determine the effects of factors in a coating process such as speed, temperature, and pressure on your product’s tensile and elongation properties. Due to the number of runs involved, you will need to use two different batches of raw material. You expect that variations in the raw material may have an effect on these properties, but you are not interested in studying that effect at this time. Therefore, raw material is NOT a factor and you should block on it instead. This will remove the effect of raw material on tensile and elongation from the ANOVA and allow you to better identify the other factor effects.

On the other hand, if you want to study the effect of raw material batch variation, then it should be included as a factor and you should NOT set up blocks on this factor.

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Designs from Design Expert

Factorial Model: Factorial model is composed of a list of coefficients multiplied by associated factor levels. That model is in the form of

Y = β0 + β1A + β2B+ β3C + β12AB + β13AC + ...

where βn is the coefficient associated with factor, n, and the letters, A, B, C, ... represent the factors in the model. Combinations of factors (such as AB) represent an interaction between the individual factors in that term.

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Designs from Design ExpertFactorial Model:

Note that the regression is done in coded units and the coefficients are based on that coding. The actual equation that is reported is derived from the coded equation.

Order Description Zero InterceptFirst Main effects: A, B, C, D...Second 2-factor interactions (2FI) - AB, AC, BC, AD, ...Third 3-factor interactions (3FI) - ABC, ABD, ACD, ...

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42

• Plasma Etch Example: Desire to optimize across the wafer etch rate uniformity for PCVD oxide. i.e. Minimum %uniformity is desired.

• The Pressure and He flow rates were varied across an experimental space desired to be evaluated. (reasonable design space). 2 factors at 4 levels

• FACTORS:

• Pressure: Levels: 2200, 2300, 2400, and 2500 mTorr• Power: 500/375 watts

• CHF3 flow rate: 10 SCCM

• C2F6 flow rate: 15 SCCM

• He Flow rate: Levels :45, 50, 55, and 60 SCCM

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert

Continue >>

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert• Factor 1: He Flow rate SCCM

Continue >>

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert • Factor 2: Pressure mTorr

Continue >>

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert • # Replicates and responses

Continue >>

Continue >>

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert • Run experiment in random order and measure responses

• Note missing data

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis• Click on Analysis Oxide etch Rate Uniformity

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis• Click on Effects

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis• Click on Effects

M = model and e = error

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis• Click on ANOVA

So Only factor B (pressure) has a significant effects on across the wafer PCVD oxide etch rate uniformity.

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis• Click on Diagnostics Normal

Residuals look normally distributed

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis• Click on Diagnostics Predicted

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis• Click on Diagnostics RUN

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis• Click on Diagnostics Factor

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis• Click on Model Graphs

Interaction Plot

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis• Click on Effects No missing data

Now both A and B are significant!

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis• Click on ANOVA No missing data

A and B are significant!

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis• Lets Model this experiment and get a relationship for Oxide Etch

Uniformity as a function of Pressure and He flow rate

1st : make both Factor 1 ( He flow rate and factor 2: Pressure numeric

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis Plasma Etch Model

2nd : Fit Summary and then click on Model

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis Plasma Etch Model

3rd : click on ANOVA

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis Plasma Etch Model

4th : Diagnostics

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis Plasma Etch Model

5th : Graphics RSM plot

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis Plasma Etch Model

6th : Graphics Interaction plot

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis Plasma Etch Model

6th : Graphics Factor 1 and 2 Effects plots

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis Plasma Etch Model

Quadratic fit

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis Plasma Etch Model

Quadratic fit

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis Plasma Etch Model

Cubic fit

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis Plasma Etch Model

Cubic fit

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Factorial Designs :Design of Experiments Steve Brainerd

• Factorial Example 42 from Design Expert Analysis Plasma Etch Model

Cubic fit