Two factor factorial_design_pdf

1. TWO-­‐FACTOR
FACTORIAL
DESIGN
PREPARED
BY:
SITI
AISYAH
BT
NAWAWI

2. Basic
Definition
and
Principles
▪ Factorial
designs
➢
most
efficient
in
experiments
that
involve
the
study
of
the
effects
of
two
or
more
factors.
➢2k
means
there
are
k
factors
in
the
experiment
and
each
factor
has
two
levels
➢Factor
levels:
❖ Quantitative
❖ All
combinations
of
factor
levels
will
be
investigated
❖ Number
of
treatment
combinations
=
2k
!
E.g.:
a
levels
of
factor
A,
b
levels
of
factor
B;
each
replicate
contains
all
ab
treatment
combinations.
!!!!!!!!!
!!!
32
=2
factors
with
each
factor
has
3
levels

3. THE
ADVANTAGE
OF
FACTORIALS
!
• The
factorial
designs
can
be
easily
illustrated.
• More
efficient
than
one-­‐factor-­‐at-­‐a-­‐time
experiments.
• A
factorial
design
is
necessary
when
interactions
may
be
present
to
avoid
misleading
conclusions.
• Factorial
designs
allow
the
effects
of
a
factor
to
be
estimated
at
several
levels
of
the
other
factors,
yielding
conclusions
that
are
valid
over
a
range
of
experimental
conditions.
35
26
18
9
0
no alch. one alch
no barb.
one barb.
Interaction
exist

4. THE
TWO-­‐FACTOR
FACTORIAL
DESIGN
!
• The
effects
model

5. General
arrangement
for
a
Two-­‐Factor
Factorial
Design
!
• General
arrangement
for
a
Two-­‐Factor
Factorial
Design
table
it
comes
from
yijk
where
i=1,2,3,…,a
(level
of
factor
A)
j=1,2,3,…,b
(level
of
factor
B)
k=1,2,3,…,n
(replication) See
table
on
next
page…

6. General
arrangement
for
a
Two-­‐Factor
Factorial
Design
!
!
!
yijk

7. ANOVA
Table
Source
of
Variation
Sum
of
Squares Degrees
of
Freedom
Mean
Square F
A
treatments SS a
–
1
B
treatments SS b
–
1
Interaction SS (a
–
1)
(b
–
1)
Error SS ab
(n
–
1)
Total SS abn
–
1
MS SSA
−1
=
a
A
A
E
F MS 0 =
MS
MS SSB
−1
=
b
B
B
E
F MS 0 =
MS
MS SSAB
=
a b
( −1)( −1)
AB
AB
MS
E
F MS 0 =
MS SSE
=
ab n
( −1)
E

8. Cont..
a
ΣΣΣ
− = =
= − ⋅⋅⋅
Total ijk abn
i
b
j
n
k
y
SS y
1 1 1
2
2
y
1 ⋅⋅⋅
abn
y
= Σ −
A i
bn
SS
a
i
2
1
2.
.
=
y
1 ⋅⋅⋅
abn
y
= Σ −
B j
an
SS
b
j
2
1
2
. .
=
AB total A B SS = SS − SS − SS
E Total A B AB SS = SS − SS − SS − SS

9. Example

10. Cont..

11. cont..
• 2)
ANOVA
Now,
calculate
ANOVA
table
using
formula
given
in
previous
slide

12. Cont..
Source
of
Variation Sum
of
Squares
Degrees
of
Freedom
Mean
Square F
Material
types 10683.72 2 5,341.86 7.91
Temperature 39118.72 2 19,559.36 28.97
Interaction
(Material*Temperature)
9613.78 4 2,403.44 3.56
Error 18230.75 27 675.21
Total 77646.97 35

13. cont..

14. Cont..
Do
you
get
the
same
answer?
If
YES,
lets
continue..

15. Cont..

18. !
• Conclusion
:
This
analysis
indicates
that
at
the
temperature
level
700F,
the
mean
battery
life
is
the
same
for
material
types
2
and
3