Designs for Experiments with More Than One Factor

1
Designs for Experiments with More Than One Factor

• When the experimenter is interested in the effect
  of multiple factors on a response a factorial
  design should be used.
• A factorial experiment means that all factor
  level combinations are included in each replicate
  of the experiment.
• For example, if the experimenter wanted to test 4
  factors at 2 levels each, then all 24 16
  combinations of factor levels would be included
  in each replicate.
• The effect of a factor is defined as the change
  in the response produced by a change in the level
  of a factor. This is often termed a main effect.

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• Consider the experiment depicted above. Here we
  have 2 factors each investigated at 2 levels.
• The main effect of each factor is calculated as
  the difference between the average response at
  the first level of the factor and the average
  response at the second level of the factor.

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• It is often useful to graphically display the
  results of the experiment. Note that we can
  easily see the effects calculated earlier.

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• This are the results from a similar experiment
  where factors interact. When the interaction
  effects are large the main effects have little
  meaning

20
0
B2
B1
10
30
A1
A2
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General 2k Designs

• 2k designs are popular in industry particularly
  in the exploratory phase of process and product
  improvement. They consist of k factors each
  studied at two levels (usually denoted as high
  and low levels). Our first example was a 22
  design.
• A simple notation has been developed to represent
  the replicates.
• A run is represented by a series of lowercase
  letters. If a letter is present it indicates
  that the corresponding factor is set at its high
  level. The run with all factors set at their low
  levels is denoted as (1)

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• This notation applied to our first example is as
  follows
• The main effect is then just the difference
  between the average of the observations on the
  right side of the square and the average of the
  observations on left side of the square, if n
  the number of replicates under each factor
  combination then the main effect of factor A is

b
ab
High()
Low(-)
(1)
a
Low(-)
High ()
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• The main effect of B is the difference between
  the average of the observations at the top of the
  square and the average at the bottom of the
  square
• The interaction effect is found by taking the
  difference in the diagonal averages
• The terms in the brackets in each of these
  equations are called contrasts. For example
• ContrastA a ab - b - (1)
• Note that the coefficients in these contrasts are
  always to either -1 or 1

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• A table of and - signs is helpful in
  determining the sign on each run for developing
  the contrasts.
• The sum of squares for each effect is then as
  follows
• The total sum of squares is obtained as usual and
  the error sum of squares can be obtained by
  subtraction

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An Example

• A router is used to cut notches in printed
  circuit boards. The process is in statistical
  control, the average dimension is satisfactory,
  but there is too much variability in the process
  which leads to problems in assembly. The quality
  improvement team identified two factors which may
  have an impact bit size (A) (tested at 1/8 inch
  and 1/16 inch) and the speed (B) (tested at 40
  rpm and 80 rpm). It was felt that vibration of
  the boards during the process was responsible for
  the excess variation. An experiment was
  conducted using four boards at each treatment
  level. The treatment levels were randomly
  assigned to the 16 boards and the results are as
  follows

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• Calculate the main and interaction effects and
  the sum of squares.

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Residual Analysis

• The residuals from a 2k design are easily
  obtained through fitting a regression model to
  the data. For our experiment the appropriate
  model is as follows
• This model can also be used for obtaining
  predicted values for the four points in our
  experimental design.

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• Minitab Ouput
• General Linear Model Vibration versus A, B
• Factor Type Levels Values
• A fixed 2 1 -1
• B fixed 2 1 -1
• Analysis of Variance for Vibration, using
  Adjusted SS for Tests
• Source DF Seq SS Adj SS Adj MS
  F P
• A 1 1107.23 1107.23 1107.23
  185.25 0.000
• B 1 227.26 227.26 227.26
  38.02 0.000
• AB 1 303.63 303.63 303.63
  50.80 0.000
• Error 12 71.72 71.72 5.98
• Total 15 1709.83
• Term Coef SE Coef T P

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Designs for K ? 3

• The methods for 22 designs discussed can be
  easily extended to designs involving more than
  two factors. The effects are calculated
  similarly,
• As are the sum of squares,
• An article in Solid State Technology describes an
  experiment for improving the etch rate on a wafer
  plasma etcher. The factors in the experiment
  were gas flow (A), power applied to the cathode
  (B), gap between the cathode and the anode (C),
  and pressure (D). Each factor was tested at two
  levels. We will assume a single replicate was
  performed with the following results
• Use Minitab to analyze the results

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Fractional Factorial Designs

• Since the number of runs increases exponentially
  with the number of factors investigated in a 2k
  design it is desirable to limit the number of
  runs while maintaining the ability to obtain
  information on the factors of interest. If we
  can assume that the higher order interactions are
  negligible, then a fractional factorial design
  involving fewer than 2k runs may be used.
  Consider the 23 design matrix depicted below
• Note that in the full factorial design, the sum
  of the products of any two columns 0. This
  indicates that the columns are orthogonal, e.g,
  their associated effects are statistically
  independent.
• If we were to assume that the high order
  interaction, ABC were insignificant we might
  conduct the experiment using just the top half of
  the design matrix

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• Notice that the 2(3-1) design is formed by
  selecting only those runs that yield a on the
  ABC effect. The interaction ABC is termed the
  generator of this fraction.
• The estimates of the main effects from this
  fractional design are as follows
• A 1/2 a - b - c abc
• B 1/2-a b - c abc
• C 1/2-a - b c abc
• also,
• BC 1/2a - b - c abc
• AC 1/2-a b - c abc
• AB 1/2-a - b c abc
• Note that the linear combination of observations
  in column A estimates A BC. Therefore, if the
  contrast is significant, we cannot tell whether
  it is due to the main effect of A or the
  interaction effect of B or a mixture of both.
  That is, the two columns are no longer
  orthogonal. Two or more effects that have this
  property are termed aliases.
• The alias for any factor can be found by
  multiplying the factor by the generator

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• The alias of A is
• A x ABC A2BC BC
• likewise,
• the alias of B AC
• the alias of C AB
• If we had chosen the other half fraction, the
  generator would have been -ABC and the aliases
  would be
• A -BC
• B -AC
• C -AB
• That is the column associated with A really
  estimates A - BC, the column associated with B
  estimates B - AC and the column associated with C
  estimates C - AB
• The fraction with the sign is sometimes
  referred to as the principle fraction while the
  fraction with the - sign is termed the alternate
  fraction.

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Using sequences of fractional designs to estimate
effects

• If we had chosen the first design and were
  convinced that the two-way interactions were
  insignificant then the design will produce
  estimates of the main effects of the three
  factors. If, however, after running the principle
  fraction we believe are uncertain as to the
  interaction effects we can estimate them by
  running the alternate fraction.
• It is easy to construct a 2 k-1 design. Simply
  write down the treatment levels for the full
  factorial experiment in k-1 factors. Then equate
  the column associated with the kth factor with
  the product of the signs of the k-1 factors. For
  example we would construct a 24-1 fractional
  factorial for our plasma etching experiment as
  follows

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• The minitab output below agrees substantially
  with the output generated from the full factorial
  design.
• Factor Type Levels Values
• A fixed 2 -1 1
• D fixed 2 -1 1
• Analysis of Variance for response, using Adjusted
  SS for Tests
• Source DF Seq SS Adj SS Adj MS
  F P
• A 1 32258 32258 32258
  71.80 0.001
• D 1 168780 168780 168780
  375.69 0.000
• AD 1 78012 78012 78012
  173.65 0.000
• Error 4 1797 1797 449
• Total 7 280848
• Term Coef SE Coef T P
• Constant 756.000 7.494 100.88 0.000
• A
• -1 63.500 7.494 8.47 0.001