<s>
In	O
statistics	O
and	O
in	O
particular	O
in	O
regression	O
analysis	O
,	O
a	O
design	B-Algorithm
matrix	I-Algorithm
,	O
also	O
known	O
as	O
model	B-Algorithm
matrix	I-Algorithm
or	O
regressor	B-Algorithm
matrix	I-Algorithm
and	O
often	O
denoted	O
by	O
X	O
,	O
is	O
a	O
matrix	B-Architecture
of	O
values	O
of	O
explanatory	O
variables	O
of	O
a	O
set	O
of	O
objects	O
.	O
</s>
<s>
The	O
design	B-Algorithm
matrix	I-Algorithm
is	O
used	O
in	O
certain	O
statistical	O
models	O
,	O
e.g.	O
,	O
the	O
general	O
linear	O
model	O
.	O
</s>
<s>
It	O
can	O
contain	O
indicator	O
variables	O
(	O
ones	O
and	O
zeros	O
)	O
that	O
indicate	O
group	O
membership	O
in	O
an	O
ANOVA	B-General_Concept
,	O
or	O
it	O
can	O
contain	O
values	O
of	O
continuous	O
variables	O
.	O
</s>
<s>
The	O
design	B-Algorithm
matrix	I-Algorithm
contains	O
data	O
on	O
the	O
independent	O
variables	O
(	O
also	O
called	O
explanatory	O
variables	O
)	O
in	O
statistical	O
models	O
which	O
attempt	O
to	O
explain	O
observed	O
data	O
on	O
a	O
response	O
variable	O
(	O
often	O
called	O
a	O
dependent	O
variable	O
)	O
in	O
terms	O
of	O
the	O
explanatory	O
variables	O
.	O
</s>
<s>
The	O
theory	O
relating	O
to	O
such	O
models	O
makes	O
substantial	O
use	O
of	O
matrix	B-Architecture
manipulations	O
involving	O
the	O
design	B-Algorithm
matrix	I-Algorithm
:	O
see	O
for	O
example	O
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
A	O
notable	O
feature	O
of	O
the	O
concept	O
of	O
a	O
design	B-Algorithm
matrix	I-Algorithm
is	O
that	O
it	O
is	O
able	O
to	O
represent	O
a	O
number	O
of	O
different	O
experimental	O
designs	O
and	O
statistical	O
models	O
,	O
e.g.	O
,	O
ANOVA	B-General_Concept
,	O
ANCOVA	B-General_Concept
,	O
and	O
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
The	O
design	B-Algorithm
matrix	I-Algorithm
is	O
defined	O
to	O
be	O
a	O
matrix	B-Architecture
such	O
that	O
(	O
the	O
jth	O
column	O
of	O
the	O
ith	O
row	O
of	O
)	O
represents	O
the	O
value	O
of	O
the	O
jth	O
variable	O
associated	O
with	O
the	O
ith	O
object	O
.	O
</s>
<s>
where	O
X	O
is	O
the	O
design	B-Algorithm
matrix	I-Algorithm
,	O
is	O
a	O
vector	O
of	O
the	O
model	O
's	O
coefficients	O
(	O
one	O
for	O
each	O
variable	O
)	O
,	O
is	O
a	O
vector	O
of	O
random	O
errors	O
with	O
mean	O
zero	O
,	O
and	O
y	O
is	O
the	O
vector	O
of	O
predicted	O
outputs	O
for	O
each	O
object	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
of	O
data	O
has	O
dimension	O
n-by-p	O
,	O
where	O
n	O
is	O
the	O
number	O
of	O
samples	O
observed	O
,	O
and	O
p	O
is	O
the	O
number	O
of	O
variables	O
(	O
features	B-Algorithm
)	O
measured	O
in	O
all	O
samples	O
.	O
</s>
<s>
The	O
data	B-Algorithm
matrix	I-Algorithm
M	O
would	O
be	O
a	O
10×4	O
matrix	B-Architecture
(	O
meaning	O
10	O
rows	O
and	O
4	O
columns	O
)	O
.	O
</s>
<s>
The	O
datum	O
in	O
row	O
i	O
and	O
column	O
j	O
of	O
this	O
matrix	B-Architecture
would	O
be	O
the	O
answer	O
of	O
the	O
i	O
th	O
person	O
to	O
the	O
j	O
th	O
question	O
.	O
</s>
<s>
The	O
design	B-Algorithm
matrix	I-Algorithm
for	O
an	O
arithmetic	O
mean	O
is	O
a	O
column	O
vector	O
of	O
ones	O
.	O
</s>
<s>
This	O
section	O
gives	O
an	O
example	O
of	O
simple	B-General_Concept
linear	I-General_Concept
regression	I-General_Concept
—	O
that	O
is	O
,	O
regression	O
with	O
only	O
a	O
single	O
explanatory	O
variable	O
—	O
with	O
seven	O
observations	O
.	O
</s>
<s>
where	O
is	O
the	O
y-intercept	O
and	O
is	O
the	O
slope	O
of	O
the	O
regression	B-General_Concept
line	I-General_Concept
.	O
</s>
<s>
where	O
the	O
first	O
column	O
of	O
1s	O
in	O
the	O
design	B-Algorithm
matrix	I-Algorithm
allows	O
estimation	O
of	O
the	O
y-intercept	O
while	O
the	O
second	O
column	O
contains	O
the	O
x-values	O
associated	O
with	O
the	O
corresponding	O
y-values	O
.	O
</s>
<s>
This	O
section	O
contains	O
an	O
example	O
of	O
multiple	B-General_Concept
regression	I-General_Concept
with	O
two	O
covariates	O
(	O
explanatory	O
variables	O
)	O
:	O
w	O
and	O
x	O
.	O
</s>
<s>
Here	O
the	O
7×3	O
matrix	B-Architecture
on	O
the	O
right	O
side	O
is	O
the	O
design	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
This	O
section	O
contains	O
an	O
example	O
with	O
a	O
one-way	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
(	O
ANOVA	B-General_Concept
)	O
with	O
three	O
groups	O
and	O
seven	O
observations	O
.	O
</s>
<s>
The	O
ANOVA	B-General_Concept
model	O
could	O
be	O
equivalently	O
written	O
as	O
each	O
group	O
parameter	O
being	O
an	O
offset	O
from	O
some	O
overall	O
reference	O
.	O
</s>
<s>
is	O
not	O
included	O
in	O
the	O
matrix	B-Architecture
because	O
its	O
difference	O
from	O
the	O
reference	O
group	O
(	O
itself	O
)	O
is	O
necessarily	O
zero	O
.	O
</s>
