Fractional factorial Chapter 8

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Fractional factorial Chapter 8

• Hand outs

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Initial Problem analysis

• Eyeball, statistics, few graphs
• Note what problems are and what direction would
  be useful to investigate
• Couple T/Z tests to see whats up
• Useful to pursue any items? More data?
• Preliminary ANOVAs
• Fancy ANOVAs as you find out more
• Many Factors
• Fractional Factorial

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Notes

• Fractional factorial is last on the list
• You should know your problem pretty well by now.
• You see that you cant take LOADS of data and a
  fraction of the total number of runs will be
  EASIER/CHEAPER

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Overlooked

• The item that is overlooked
• You will DESIGN the experiment before you run it.
• If you wait until after you take data, the data
  probably wont mathematically match the model!

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Factorial idea

• Simplify math make all levels for each factor
  the same.
• TWO levels is BEST/EASIEST!!
• Once you have that Designed, you can cut down on
  the combos you must test
• look at chart in book for reduction
• design your own

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Terminology

• W(Y-x)
• W is the number of levels
• Y is the number of factors
• X refers to the size of the fraction
• 1 is a half, 2 is a quarter, 3 is an eighth, 4
  is a sixteenth.

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Chart in book

• The books charts for fractional factorials are
  on pages 663-679.
• The charts tell you which fractions are
  allowable, math-wise
• For example, you can have a 2(3-1) but not a
  2(2-1). You also cant have a 2(3-2). Which
  would be a quarter!

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Half fractionof the fractions, this is easiest

• When you are breaking the design into two halves
  abc HIGH abc LO there will be two fractions
  you can test a b c abc and ac bc ab l You can
  test only a b abc, using either fraction. You
  will use only the PLUS or MINUS values of the
  confounded value

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THE HALF FRACTION the plan is to run HALF of
the combinations, and still make valid F tests

• Take the highest order interaction abc, abcd,
  abcde, abcdef, or whatever
• Make that one equal to the last variable e, f,
  etc. This clearly reduces the degrees of
  freedom by 1.
• Modularly multiply each of the effects/interaction
  s by the highest order interaction. If you get a
  squared term a2, b2 it will cancel to one. So a
  X abc bc
• b X abcd acd and so on

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How to partition

• Take where ABC and where ABC - to get the
  two fractions. In other words, you are dividing
  the entire set of combinations of runs into two
  parts, and ABC will not have an F test possible
  because all the plus ABC combinations are in ½
  and all the minus ABC combinations are in the
  other ½.
• . ALIAS refers to two effects that will have the
  same F test. The Boolean expression says it all.

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HOW TO GET THE COLUMN HEADINGS????

• Start with two factors a b ab next, add c to
  each of these a b ab c ac bc abc Next throw in
  da b ab c ac bc abc d ad bd abd cd acd bcd abcd
  Next throw in e a b ab c ac bc abc d ad bd abd
  cd acd bcd abcd e ae be abe ce ace bce abce de
  ade bde abde cde acde bcde abcde

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Rest of matrix

• HOW TO GET ROW VALUES 1. count in binary from
  0000000 to 1111111 2. the low values of the
  highest order interaction correspond to the LOW
  value, the high correspond to the HIGH value
  Fill in PLUSES hi value of factor and MINUSES
  lo value of factor

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• Example - part of 3 factor chart...
• a b c abc
• 000 - - - -
• 001 - -
• 010 - -
• 011 - -
• 100 - -
• 101 - -
• 110 - -
• 111

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Chop the chart into halves

• a b c ab ac bc
• 001 - - - -
• 010 - - - -
• 100 - - - -
• 111

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Notice

• The columns have similarities
• a bc, b ac, and c ab
• These are called aliases, the F tests will be the
  same!
• TO MAKE ESTIMATES OF EFFECTS
• divide contrasts by 2 or N2k

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Now what happened?

• Select the columns of ABC and the corresponding
  rows.
• 001,010,100,111 for ABC positive 001,
  101,100,111/for ABC negative, you have 0,3,5,6
• You now do the F tests, which are derived from
  the rows.

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F tests

• Add the values for each column contrast.
  Effects are half the contrast
• The BIGGEST absolute value for these columns is
  the most important
• SSx contrastcontrast/n2k
• SST same as always, SSE likewise
• F MSx/MSE, and Flookup same procedure

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If you dont have halves!

• The half fraction is the easiest, it divides
  according to the largest interaction
• If you have a quarter fraction, use TWO N-1
  interactions that each have the last two
  variables, and dont cancel out
• EXAMPLE Five variable, choose ABCE, and BCDF

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Quarter fractions are interesting

• You select two lower order interactions to GET
  RID OF From page 664,
• Say ABC and BCD
• there are four fractions ABC/BCD 00, ABC/BCD
  01, ABC/BCD 10, ABD/BCE 11
• Choose one, calculate contrasts, Fs etc!

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Example Problems

• Just get into deciphering the FRACTION for some
  problems
• 8-27 Five Variables 32, 16 runs, so this is a
  HALF FRACTION
• 8-26, 8 Variables 256, 16 runs, so a SIXTEENTH
  FRACTION!!!!!

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More deciphering

• 8-29, Sixteen runs, nine variables
  222222222 16/512 1/32 or ONE
  THIRTY-SECOND FRACTION
• 8-28 Sixteen runs, 10 variables, for a fraction
  of 16/1024 or a Sixty-Fourth, your books page
  673!!!