Fractional factorial Chapter 8

Hand outs

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 what’s up• Useful to pursue any items? More data?• Preliminary ANOVA’s • Fancy ANOVAs as you find out more:

– Many Factors– Fractional Factorial

Notes

• ‘Fractional’ factorial is last on the list

• You should know your problem pretty well by now.

• You see that you can’t take LOADS of data and a fraction of the total number of runs will be EASIER/CHEAPER

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 won’t mathematically match the model!

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

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].

Chart in book

• The book’s 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 can’t have a 2(3-2). Which would be a quarter!

Half fraction[of 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 ab=c, using either fraction. You will use only the PLUS or MINUS values of the confounded value

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/interactions 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

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.

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

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]

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

Chop the chart into halves

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

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 N2^k]

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.

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 = [contrast][contrast]/n2k• SST = same as always, SSE likewise• F = MSx/MSE, and F[lookup] same

procedure

If you don’t 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 don’t cancel out]

• EXAMPLE: Five variable, choose ABCE, and BCDF

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, F’s etc!

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!!!!!

More deciphering

• 8-29, Sixteen runs, nine variables [2*2*2*2*2*2*2*2*2] 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 book’s page 673!!!