<s>
In	O
statistics	O
,	O
a	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	I-General_Concept
to	I-General_Concept
lack	I-General_Concept
of	I-General_Concept
fit	I-General_Concept
,	O
or	O
more	O
tersely	O
a	O
lack-of-fit	B-General_Concept
sum	I-General_Concept
of	I-General_Concept
squares	I-General_Concept
,	O
is	O
one	O
of	O
the	O
components	O
of	O
a	O
partition	B-General_Concept
of	I-General_Concept
the	I-General_Concept
sum	I-General_Concept
of	I-General_Concept
squares	I-General_Concept
of	O
residuals	O
in	O
an	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
,	O
used	O
in	O
the	O
numerator	O
in	O
an	O
F-test	B-General_Concept
of	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
says	O
that	O
a	O
proposed	O
model	O
fits	O
well	O
.	O
</s>
<s>
The	O
other	O
component	O
is	O
the	O
pure-error	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
.	O
</s>
<s>
The	O
pure-error	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
is	O
the	O
sum	O
of	O
squared	O
deviations	O
of	O
each	O
value	O
of	O
the	O
dependent	O
variable	O
from	O
the	O
average	O
value	O
over	O
all	O
observations	O
sharing	O
its	O
independent	O
variable	O
value(s )	O
.	O
</s>
<s>
The	O
remainder	O
of	O
the	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
is	O
attributed	O
to	O
lack	O
of	O
fit	O
of	O
the	O
model	O
since	O
it	O
would	O
be	O
mathematically	O
possible	O
to	O
eliminate	O
these	O
errors	O
entirely	O
.	O
</s>
<s>
In	O
order	O
for	O
the	O
lack-of-fit	B-General_Concept
sum	I-General_Concept
of	I-General_Concept
squares	I-General_Concept
to	O
differ	O
from	O
the	O
sum	B-Algorithm
of	I-Algorithm
squares	I-Algorithm
of	I-Algorithm
residuals	I-Algorithm
,	O
there	O
must	O
be	O
more	O
than	O
one	O
value	O
of	O
the	O
response	O
variable	O
for	O
at	O
least	O
one	O
of	O
the	O
values	O
of	O
the	O
set	O
of	O
predictor	O
variables	O
.	O
</s>
<s>
by	O
the	O
method	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
One	O
takes	O
as	O
estimates	O
of	O
α	O
and	O
β	O
the	O
values	O
that	O
minimize	O
the	O
sum	B-Algorithm
of	I-Algorithm
squares	I-Algorithm
of	I-Algorithm
residuals	I-Algorithm
,	O
i.e.	O
,	O
the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
of	O
the	O
differences	O
between	O
the	O
observed	O
y-value	O
and	O
the	O
fitted	O
y-value	O
.	O
</s>
<s>
To	O
have	O
a	O
lack-of-fit	B-General_Concept
sum	I-General_Concept
of	I-General_Concept
squares	I-General_Concept
that	O
differs	O
from	O
the	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
,	O
one	O
must	O
observe	O
more	O
than	O
one	O
y-value	O
for	O
each	O
of	O
one	O
or	O
more	O
of	O
the	O
x-values	O
.	O
</s>
<s>
One	O
then	O
partitions	O
the	O
"	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	O
to	O
error	O
"	O
,	O
i.e.	O
,	O
the	O
sum	B-Algorithm
of	I-Algorithm
squares	I-Algorithm
of	I-Algorithm
residuals	I-Algorithm
,	O
into	O
two	O
components	O
:	O
</s>
<s>
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	O
to	O
error	O
=	O
(	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	O
to	O
"	O
pure	O
"	O
error	O
)	O
+	O
(	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	I-General_Concept
to	I-General_Concept
lack	I-General_Concept
of	I-General_Concept
fit	I-General_Concept
)	O
.	O
</s>
<s>
The	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	O
to	O
"	O
pure	O
"	O
error	O
is	O
the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
of	O
the	O
differences	O
between	O
each	O
observed	O
y-value	O
and	O
the	O
average	O
of	O
all	O
y-values	O
corresponding	O
to	O
the	O
same	O
x-value	O
.	O
</s>
<s>
The	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	I-General_Concept
to	I-General_Concept
lack	I-General_Concept
of	I-General_Concept
fit	I-General_Concept
is	O
the	O
weighted	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
of	O
differences	O
between	O
each	O
average	O
of	O
y-values	O
corresponding	O
to	O
the	O
same	O
x-value	O
and	O
the	O
corresponding	O
fitted	O
y-value	O
,	O
the	O
weight	O
in	O
each	O
case	O
being	O
simply	O
the	O
number	O
of	O
observed	O
y-values	O
for	O
that	O
x-value	O
.	O
</s>
<s>
Because	O
it	O
is	O
a	O
property	O
of	O
least	B-Algorithm
squares	I-Algorithm
regression	O
that	O
the	O
vector	O
whose	O
components	O
are	O
"	O
pure	O
errors	O
"	O
and	O
the	O
vector	O
of	O
lack-of-fit	O
components	O
are	O
orthogonal	O
to	O
each	O
other	O
,	O
the	O
following	O
equality	O
holds	O
:	O
</s>
<s>
Hence	O
the	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
has	O
been	O
completely	O
decomposed	O
into	O
two	O
components	O
.	O
</s>
<s>
be	O
the	O
least	B-Algorithm
squares	I-Algorithm
estimates	O
of	O
the	O
unobservable	O
parameters	O
α	O
and	O
β	O
based	O
on	O
the	O
observed	O
values	O
of	O
x	O
i	O
and	O
Y	O
i	O
j	O
.	O
</s>
<s>
We	O
partition	O
the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	O
to	O
error	O
into	O
two	O
components	O
:	O
</s>
<s>
It	O
can	O
be	O
shown	O
to	O
follow	O
that	O
if	O
the	O
straight-line	O
model	O
is	O
correct	O
,	O
then	O
the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	O
to	O
error	O
divided	O
by	O
the	O
error	O
variance	O
,	O
</s>
<s>
The	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	O
to	O
pure	O
error	O
,	O
divided	O
by	O
the	O
error	O
variance	O
σ2	O
,	O
has	O
a	O
chi-squared	O
distribution	O
with	O
Nn	O
degrees	O
of	O
freedom	O
;	O
</s>
<s>
The	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	I-General_Concept
to	I-General_Concept
lack	I-General_Concept
of	I-General_Concept
fit	I-General_Concept
,	O
divided	O
by	O
the	O
error	O
variance	O
σ2	O
,	O
has	O
a	O
chi-squared	O
distribution	O
with	O
np	O
degrees	O
of	O
freedom	O
(	O
here	O
p	O
=	O
2	O
as	O
there	O
are	O
two	O
parameters	O
in	O
the	O
straight-line	O
model	O
)	O
;	O
</s>
<s>
has	O
an	O
F-distribution	B-General_Concept
with	O
the	O
corresponding	O
number	O
of	O
degrees	O
of	O
freedom	O
in	O
the	O
numerator	O
and	O
the	O
denominator	O
,	O
provided	O
that	O
the	O
model	O
is	O
correct	O
.	O
</s>
<s>
One	O
uses	O
this	O
F-statistic	O
to	O
test	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
the	O
linear	O
model	O
is	O
correct	O
.	O
</s>
<s>
Since	O
the	O
non-central	O
F-distribution	B-General_Concept
is	O
stochastically	O
larger	O
than	O
the	O
(	O
central	O
)	O
F-distribution	B-General_Concept
,	O
one	O
rejects	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
if	O
the	O
F-statistic	O
is	O
larger	O
than	O
the	O
critical	O
F	B-General_Concept
value	I-General_Concept
.	O
</s>
<s>
The	O
critical	O
value	O
corresponds	O
to	O
the	O
cumulative	O
distribution	O
function	O
of	O
the	O
F	B-General_Concept
distribution	I-General_Concept
with	O
x	O
equal	O
to	O
the	O
desired	O
confidence	O
level	O
,	O
and	O
degrees	O
of	O
freedom	O
d1	O
=(	O
np	O
)	O
and	O
d2	O
=(	O
Nn	O
)	O
.	O
</s>
<s>
The	O
assumptions	O
of	O
normal	O
distribution	O
of	O
errors	O
and	O
independence	O
can	O
be	O
shown	O
to	O
entail	O
that	O
this	O
lack-of-fit	O
test	O
is	O
the	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
of	O
this	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
