<s>
In	O
statistics	O
,	O
restricted	B-General_Concept
randomization	I-General_Concept
occurs	O
in	O
the	O
design	O
of	O
experiments	O
and	O
in	O
particular	O
in	O
the	O
context	O
of	O
randomized	O
experiments	O
and	O
randomized	O
controlled	O
trials	O
.	O
</s>
<s>
Restricted	B-General_Concept
randomization	I-General_Concept
allows	O
intuitively	O
poor	O
allocations	O
of	O
treatments	O
to	O
experimental	O
units	O
to	O
be	O
avoided	O
,	O
while	O
retaining	O
the	O
theoretical	O
benefits	O
of	O
randomization	O
.	O
</s>
<s>
One	O
approach	O
to	O
analyzing	O
these	O
data	O
would	O
be	O
to	O
compute	O
the	O
mean	O
of	O
all	O
these	O
points	O
as	O
well	O
as	O
their	O
standard	B-General_Concept
deviation	I-General_Concept
and	O
use	O
those	O
results	O
as	O
responses	O
for	O
each	O
run	O
.	O
</s>
<s>
This	O
kind	O
of	O
restricted	B-General_Concept
randomization	I-General_Concept
always	O
produces	O
nested	O
sources	O
of	O
variation	O
.	O
</s>
<s>
Examples	O
of	O
nested	B-General_Concept
variation	I-General_Concept
or	O
restricted	B-General_Concept
randomization	I-General_Concept
discussed	O
on	O
this	O
page	O
are	O
split-plot	B-General_Concept
and	O
strip-plot	B-General_Concept
designs	I-General_Concept
.	O
</s>
<s>
Because	O
the	O
wafers	O
and	O
the	O
sites	O
represent	O
unwanted	O
sources	O
of	O
variation	O
and	O
because	O
one	O
of	O
the	O
objectives	O
is	O
to	O
reduce	O
the	O
process	O
sensitivity	O
to	O
these	O
sources	O
of	O
variation	O
,	O
treating	O
wafers	O
and	O
sites	O
as	O
random	B-General_Concept
effects	I-General_Concept
in	O
the	O
analysis	O
of	O
the	O
data	O
is	O
a	O
reasonable	O
approach	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
nested	B-General_Concept
variation	I-General_Concept
is	O
often	O
another	O
way	O
of	O
saying	O
nested	O
random	B-General_Concept
effects	I-General_Concept
or	O
nested	O
sources	O
of	O
noise	O
.	O
</s>
<s>
If	O
the	O
factors	O
"	O
wafers	O
"	O
and	O
"	O
sites	O
"	O
are	O
treated	O
as	O
random	B-General_Concept
effects	I-General_Concept
,	O
then	O
it	O
is	O
possible	O
to	O
estimate	O
a	O
variance	B-General_Concept
component	I-General_Concept
due	O
to	O
each	O
source	O
of	O
variation	O
through	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
techniques	O
.	O
</s>
<s>
Once	O
estimates	O
of	O
the	O
variance	B-General_Concept
components	I-General_Concept
have	O
been	O
obtained	O
,	O
an	O
investigator	O
is	O
then	O
able	O
to	O
determine	O
the	O
largest	O
source	O
of	O
variation	O
in	O
the	O
process	O
under	O
experimentation	O
,	O
and	O
also	O
determine	O
the	O
magnitudes	O
of	O
the	O
other	O
sources	O
of	O
variation	O
in	O
relation	O
to	O
the	O
largest	O
source	O
.	O
</s>
<s>
If	O
an	O
experiment	O
or	O
process	O
has	O
nested	B-General_Concept
variation	I-General_Concept
,	O
the	O
experiment	O
or	O
process	O
has	O
multiple	O
sources	O
of	O
random	O
error	O
that	O
affect	O
its	O
output	O
.	O
</s>
<s>
Having	O
nested	O
random	B-General_Concept
effects	I-General_Concept
in	O
a	O
model	O
is	O
the	O
same	O
thing	O
as	O
having	O
nested	B-General_Concept
variation	I-General_Concept
in	O
a	O
model	O
.	O
</s>
<s>
Split-plot	B-General_Concept
designs	O
result	O
when	O
a	O
particular	O
type	O
of	O
restricted	B-General_Concept
randomization	I-General_Concept
has	O
occurred	O
during	O
the	O
experiment	O
.	O
</s>
<s>
A	O
simple	O
factorial	O
experiment	O
can	O
result	O
in	O
a	O
split-plot	B-General_Concept
type	O
of	O
design	O
because	O
of	O
the	O
way	O
the	O
experiment	O
was	O
actually	O
executed	O
.	O
</s>
<s>
An	O
experiment	O
run	O
under	O
one	O
of	O
the	O
above	O
three	O
situations	O
usually	O
results	O
in	O
a	O
split-plot	B-General_Concept
type	O
of	O
design	O
.	O
</s>
<s>
Running	O
the	O
experiment	O
in	O
this	O
way	O
results	O
in	O
a	O
split-plot	B-General_Concept
design	O
.	O
</s>
<s>
A	O
split-plot	B-General_Concept
design	O
has	O
more	O
than	O
one	O
size	O
experimental	O
unit	O
.	O
</s>
<s>
The	O
ANOVA	B-General_Concept
table	O
for	O
this	O
experiment	O
would	O
look	O
,	O
in	O
part	O
,	O
as	O
follows	O
:	O
</s>
<s>
A	O
normal	B-Application
probability	I-Application
plot	I-Application
of	O
the	O
12	O
subplot	O
term	O
estimates	O
could	O
be	O
used	O
to	O
look	O
for	O
statistically	B-General_Concept
significant	I-General_Concept
terms	O
.	O
</s>
<s>
Running	O
the	O
experiment	O
in	O
this	O
way	O
also	O
results	O
in	O
a	O
split-plot	B-General_Concept
design	O
in	O
which	O
the	O
whole-plot	O
factors	O
are	O
now	O
solution	O
concentration	O
and	O
solution	O
temperature	O
,	O
and	O
the	O
subplot	O
factor	O
is	O
current	O
.	O
</s>
<s>
The	O
ANOVA	B-General_Concept
for	O
this	O
experiment	O
looks	O
,	O
in	O
part	O
,	O
as	O
follows	O
:	O
</s>
<s>
Since	O
there	O
are	O
8	O
degrees	O
of	O
freedom	O
for	O
the	O
subplot	O
error	O
term	O
,	O
this	O
MSE	B-Algorithm
can	O
be	O
used	O
to	O
test	O
each	O
effect	O
that	O
involves	O
current	O
.	O
</s>
<s>
Running	O
the	O
experiment	O
in	O
this	O
way	O
also	O
results	O
in	O
a	O
split-plot	B-General_Concept
design	O
in	O
which	O
the	O
whole-plot	O
factors	O
are	O
again	O
solution	O
concentration	O
and	O
solution	O
temperature	O
and	O
the	O
subplot	O
factor	O
is	O
current	O
.	O
</s>
<s>
The	O
ANOVA	B-General_Concept
for	O
this	O
(	O
third	O
)	O
approach	O
is	O
,	O
in	O
part	O
,	O
as	O
follows	O
:	O
</s>
<s>
As	O
can	O
be	O
seen	O
from	O
these	O
three	O
scenarios	O
,	O
one	O
of	O
the	O
major	O
differences	O
in	O
split-plot	B-General_Concept
designs	O
versus	O
simple	O
factorial	O
designs	O
is	O
the	O
number	O
of	O
different	O
sizes	O
of	O
experimental	O
units	O
in	O
the	O
experiment	O
.	O
</s>
<s>
Split-plot	B-General_Concept
designs	O
have	O
more	O
than	O
one	O
size	O
experimental	O
unit	O
,	O
i.e.	O
,	O
more	O
than	O
one	O
error	O
term	O
.	O
</s>
<s>
Since	O
these	O
designs	O
involve	O
different	O
sizes	O
of	O
experimental	O
units	O
and	O
different	O
variances	O
,	O
the	O
standard	B-General_Concept
errors	I-General_Concept
of	O
the	O
various	O
mean	O
comparisons	O
involve	O
one	O
or	O
more	O
of	O
the	O
variances	O
.	O
</s>
<s>
Specifying	O
the	O
appropriate	O
model	O
for	O
a	O
split-plot	B-General_Concept
design	O
involves	O
being	O
able	O
to	O
identify	O
each	O
size	O
of	O
experimental	O
unit	O
.	O
</s>
<s>
As	O
a	O
result	O
of	O
having	O
greater	O
than	O
one	O
size	O
experimental	O
unit	O
,	O
the	O
appropriate	O
model	O
used	O
to	O
analyze	O
split-plot	B-General_Concept
designs	O
is	O
a	O
mixed	B-General_Concept
model	I-General_Concept
.	O
</s>
<s>
Similar	O
to	O
a	O
split-plot	B-General_Concept
design	O
,	O
a	O
strip-plot	B-General_Concept
design	I-General_Concept
can	O
result	O
when	O
some	O
type	O
of	O
restricted	B-General_Concept
randomization	I-General_Concept
has	O
occurred	O
during	O
the	O
experiment	O
.	O
</s>
<s>
A	O
simple	O
factorial	O
design	O
can	O
result	O
in	O
a	O
strip-plot	B-General_Concept
design	I-General_Concept
depending	O
on	O
how	O
the	O
experiment	O
was	O
conducted	O
.	O
</s>
<s>
Strip-plot	B-General_Concept
designs	I-General_Concept
often	O
result	O
from	O
experiments	O
that	O
are	O
conducted	O
over	O
two	O
or	O
more	O
process	O
steps	O
in	O
which	O
each	O
process	O
step	O
is	O
a	O
batch	O
process	O
,	O
i.e.	O
,	O
completing	O
each	O
treatment	O
combination	O
of	O
the	O
experiment	O
requires	O
more	O
than	O
one	O
processing	O
step	O
with	O
experimental	O
units	O
processed	O
together	O
at	O
each	O
process	O
step	O
.	O
</s>
<s>
As	O
in	O
the	O
split-plot	B-General_Concept
design	O
,	O
strip-plot	B-General_Concept
designs	I-General_Concept
result	O
when	O
the	O
randomization	O
in	O
the	O
experiment	O
has	O
been	O
restricted	O
in	O
some	O
way	O
.	O
</s>
<s>
As	O
a	O
result	O
of	O
the	O
restricted	B-General_Concept
randomization	I-General_Concept
that	O
occurs	O
in	O
strip-plot	B-General_Concept
designs	I-General_Concept
,	O
there	O
are	O
multiple	O
sizes	O
of	O
experimental	O
units	O
.	O
</s>
<s>
A	O
traditional	O
strip-plot	B-General_Concept
design	I-General_Concept
has	O
three	O
sizes	O
of	O
experimental	O
units	O
.	O
</s>
<s>
Running	O
the	O
experiment	O
in	O
this	O
way	O
results	O
in	O
a	O
strip-plot	B-General_Concept
design	I-General_Concept
with	O
three	O
sizes	O
of	O
experimental	O
units	O
.	O
</s>
<s>
Actually	O
,	O
the	O
above	O
description	O
of	O
the	O
strip-plot	B-General_Concept
design	I-General_Concept
represents	O
one	O
block	O
or	O
one	O
replicate	O
of	O
this	O
experiment	O
.	O
</s>
<s>
If	O
the	O
experiment	O
contains	O
no	O
replication	O
and	O
the	O
model	O
for	O
the	O
implant	O
contains	O
only	O
the	O
main	B-General_Concept
effects	I-General_Concept
and	O
two-factor	O
interactions	O
,	O
the	O
three-factor	O
interaction	O
term	O
A*B*C	O
(	O
1	O
degree	O
of	O
freedom	O
)	O
provides	O
the	O
error	O
term	O
for	O
the	O
estimation	O
of	O
effects	O
within	O
the	O
implant	O
experimental	O
unit	O
.	O
</s>
