QUALITY CONTROL
ANNEBERG
Fractional Factorial Experiments --
Chapter 8
MOST CONFUSING ITEM….
When you are breaking the design into two halves…
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
example
000
001
010
011
100
101
110
111
a
+
+
+
+
b
+
+
+
+
So now the chart is
a
b
00
01
+
10
+
11
+
+
abc’s
+
+
+
+
ab=c
+
+
So the values of the 3 go into the 2:
Ssa [for the two] has -000 [or 001], -010 [011], +100[101], +111[110]
The parentheses indicate the alternate fraction values
HOW TO DO THE SIGNS IN EACH OF THE ROWS????
Binarily expand the five main factors, a,b,c,d,e and then multiply across to get the signs for all the columns.
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 d…
a 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
HOW TO GET ROW VALUES:
1. count in binary from 0000000 to 1111111
2. the 0’s correspond to the LOW value, the 1’s correspond to the HIGH value
DO F TESTS ON ALL COLUMNS
FRACTIONAL NOTATION
Qk-p
Where Q is the number of levels, k is the number of factors, and p is the fractionated amount.
FRACTION DEFINITIONs
½, ¼ 1/8, 1/16 for 2 levels
1/3, 1/9, 1/27 for 3 levels
THE HALF FRACTION [the plan is to run HALF of the combinations, and still make valid F tests]
1. take the highest order interaction:
abc, abcd, abcde, abcdef, or whatever
2. make that one equal to the last variable [e, f, etc]. This clearly reduces the degrees of freedom by 1.
3. 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
4. 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.
5. ALIAS refers to two effects that will have the same F test. The Boolean expression says it all.
So in the 3 factor case:
000
lo
times ABC=abc
001
c
times ABC=ab [C and AB will be in different halfs, and will be aliases]
010
b
times ABC=ac
011
bc
times ABC=a
100
a
times ABC=bc
101
ac
times ABC=b
110
ab
times ABC=c
111
abc
times ABC=lo
and ABC’s value = + for a,b,c,abc and is minus for ab.ac.bc, and lo
so the + part of the fraction is the PRINCIPAL fraction and the - is the alternate part of the fraction.
DESIGN resolution…
You can look this up in Appendix 12 for ‘certified’ defining relations.
Resolution basically means ‘how much stuff is aliased with other stuff’
Another definition: resolution is equal to the smallest number of letters in the defining relation
a. resolution 3, no main effects are aliased with other main effects, but with other 2 factor interactions and 2 factor interactions
with each other.
resolution 4, no main effect is aliased with other main effect, or with any 2 factor interactions, but 2 factor interactions are
aliased with each other
c. resolution 5, no main effect or 2 factor interaction is aliased with other main or 2 factor interactions, but 2 factor interactions
with 3 factor interactions.
QUARTER FRACTION
Look up in appendix 12 to get the generators [what you will multiply the rows by to get aliases].
Here, we make d = ab, and e = ac [not e = abcd like before….]
b.
For example, in the 25-2 design, the generators are ABD=ACE=BCDE, or D=ab and E=ac. The fractions will be : 00, 01, 10, 11 for
DE [RESOLUTION 3 because the generators are ABD and ACE].
To figure out which fraction the alias lies, take the signs 00, 01, 10, and 11 of abd and ace
The principal fraction is where they are both positive.
Keep in mind, there will be 3 other quarter fractions [that will be aliases of the principal]. What is kind of neat is that the F tests for
the aliases are identical, so you really only DO have a QUARTER of the problem! But the results will be confounded [you can’t tell
the real difference]. BUT we argue that WHO CARES?!? Homework THAT WE WILL DO IN CLASS…..Chapter 7, problem 1
and 3.