<s>
The	O
NM-method	B-General_Concept
or	O
Naszodi	B-General_Concept
–	I-General_Concept
Mendonca	I-General_Concept
method	I-General_Concept
is	O
the	O
operation	O
that	O
can	O
be	O
applied	O
in	O
statistics	O
,	O
econometrics	O
,	O
economics	O
,	O
sociology	O
,	O
and	O
demography	O
to	O
construct	O
counterfactual	O
contingency	B-Application
tables	I-Application
.	O
</s>
<s>
Since	O
the	O
solution	O
for	O
matrix	O
is	O
unique	O
,	O
the	O
NM-method	B-General_Concept
is	O
a	O
function	O
:	O
,	O
where	O
is	O
an	O
all	O
one	O
row	O
vector	O
of	O
size	O
,	O
while	O
is	O
an	O
all	O
one	O
column	O
vector	O
of	O
size	O
.	O
</s>
<s>
The	O
NM-method	B-General_Concept
is	O
consistent	O
with	O
a	O
definition	O
relying	O
on	O
the	O
ordinal	B-General_Concept
Liu-Lu	O
index	O
which	O
is	O
the	O
slightly	O
modified	O
version	O
of	O
the	O
Coleman-index	O
defined	O
by	O
Eq	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
if	O
they	O
are	O
ranked	O
the	O
same	O
by	O
the	O
ordinal	B-General_Concept
Liu-Lu	O
index	O
.	O
</s>
<s>
So	O
,	O
this	O
is	O
how	O
the	O
NM-method	B-General_Concept
works	O
for	O
2-by-2	O
seed	O
tables	O
.	O
</s>
<s>
Next	O
,	O
let	O
us	O
see	O
how	O
the	O
NM-method	B-General_Concept
works	O
if	O
matrix	O
is	O
such	O
that	O
the	O
second	O
precondition	O
of	O
is	O
not	O
met	O
for	O
.	O
</s>
<s>
Finally	O
,	O
neither	O
the	O
NM-method	B-General_Concept
,	O
nor	O
is	O
defined	O
if	O
pair	O
such	O
that	O
,	O
while	O
for	O
another	O
pair	O
of	O
.	O
</s>
<s>
As	O
a	O
first	O
step	O
of	O
the	O
NM-method	B-General_Concept
,	O
is	O
multiplied	O
by	O
the	O
,	O
and	O
matrices	O
for	O
each	O
pair	O
of	O
(	O
,	O
and	O
)	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
NM-method	B-General_Concept
is	O
defined	O
.	O
</s>
<s>
As	O
a	O
first	O
step	O
of	O
the	O
NM-method	B-General_Concept
,	O
is	O
multiplied	O
by	O
the	O
,	O
and	O
matrices	O
for	O
each	O
pair	O
of	O
(	O
,	O
and	O
)	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
NM-method	B-General_Concept
is	O
defined	O
.	O
</s>
<s>
The	O
NM-method	B-General_Concept
is	O
implemented	O
in	O
Excel	O
,	O
Visual	O
Basic	O
,	O
R	O
,	O
and	O
also	O
in	O
Stata	O
.	O
</s>
<s>
The	O
NM-method	B-General_Concept
can	O
be	O
applied	O
to	O
study	O
various	O
phenomena	O
including	O
assortative	O
mating	O
,	O
intergenerational	O
mobility	O
as	O
a	O
type	O
of	O
social	O
mobility	O
,	O
residential	O
segregation	O
,	O
recruitment	O
and	O
talent	O
management	O
.	O
</s>
<s>
Although	O
the	O
NM-method	B-General_Concept
has	O
a	O
wide	O
range	O
of	O
applicability	O
,	O
all	O
the	O
examples	O
to	O
be	O
presented	O
next	O
are	O
about	O
assortative	O
mating	O
along	O
the	O
education	O
level	O
.	O
</s>
<s>
Matrix	O
to	O
be	O
constructed	O
with	O
the	O
NM-method	B-General_Concept
tells	O
us	O
what	O
would	O
be	O
the	O
joint	O
educational	O
distribution	O
of	O
couples	O
in	O
Zimbabwe	O
,	O
if	O
the	O
educational	O
distributions	O
of	O
husbands	O
and	O
wives	O
were	O
the	O
same	O
as	O
in	O
Yemen	O
,	O
while	O
the	O
overall	O
desire	O
for	O
homogamy	O
(	O
also	O
called	O
as	O
aggregate	O
marital	O
preferences	O
in	O
economics	O
,	O
or	O
marital	O
matching	O
social	O
norms/social	O
barriers	O
in	O
sociology	O
)	O
were	O
unchanged	O
.	O
</s>
<s>
First	O
,	O
the	O
NM-method	B-General_Concept
does	O
not	O
yield	O
a	O
meaningful	O
solution	O
if	O
it	O
reaches	O
the	O
limit	O
of	O
its	O
applicability	O
.	O
</s>
<s>
For	O
instance	O
,	O
in	O
the	O
third	O
application	O
,	O
the	O
NM-method	B-General_Concept
signals	O
with	O
a	O
negative	O
entry	O
in	O
matrix	O
that	O
the	O
counterfactual	O
is	O
impossible	O
(	O
see	O
:	O
AlternativeMethod_US_1980s_2010s_age3035_main.xls	O
Sheet	O
PT_A1981_P2011_Not_meaningful	O
)	O
.	O
</s>
<s>
In	O
this	O
respect	O
,	O
the	O
NM-method	B-General_Concept
is	O
similar	O
to	O
the	O
linear	O
probability	O
model	O
that	O
signals	O
the	O
same	O
with	O
a	O
predicted	O
probabiity	O
outside	O
the	O
unit	O
interval	O
.	O
</s>
<s>
Second	O
,	O
the	O
NM-method	B-General_Concept
commutes	O
with	O
merging	O
neighboring	O
categories	O
of	O
the	O
row	O
variable	O
and	O
that	O
of	O
the	O
column	O
variable	O
:	O
,	O
where	O
is	O
the	O
row	O
merging	O
matrix	O
of	O
size	O
;	O
and	O
,	O
where	O
is	O
the	O
column	O
merging	O
matrix	O
of	O
size	O
.	O
</s>
<s>
Third	O
,	O
the	O
NM-method	B-General_Concept
works	O
even	O
if	O
there	O
are	O
zero	O
entries	O
in	O
matrix	O
.	O
</s>
<s>
The	O
iterative	B-General_Concept
proportional	I-General_Concept
fitting	I-General_Concept
procedure	O
(	O
IPF	O
)	O
is	O
also	O
a	O
function	O
:	O
.	O
</s>
<s>
It	O
is	O
the	O
operation	O
of	O
finding	O
the	O
fitted	O
matrix	O
(	O
)	O
which	O
fulfills	O
a	O
set	O
of	O
conditions	O
similar	O
to	O
those	O
met	O
by	O
matrix	O
constructed	O
with	O
the	O
NM-method	B-General_Concept
.	O
</s>
<s>
However	O
,	O
there	O
are	O
differences	O
between	O
the	O
IPF	O
and	O
the	O
NM-method	B-General_Concept
.	O
</s>
<s>
To	O
recall	O
,	O
the	O
NM-method	B-General_Concept
'	O
s	O
condition	O
for	O
equal	O
ranking	O
of	O
matrices	O
and	O
is	O
.	O
</s>
<s>
The	O
following	O
numerical	O
example	O
highlights	O
that	O
the	O
IPF	O
and	O
the	O
NM-method	B-General_Concept
are	O
not	O
identical	O
:	O
.	O
</s>
<s>
The	O
NM-method	B-General_Concept
yields	O
the	O
following	O
matrix	O
:	O
</s>
<s>
In	O
contrast	O
to	O
the	O
problem	O
solved	O
by	O
the	O
IPF	O
,	O
matrix	O
is	O
not	O
sampled	O
from	O
this	O
population	O
in	O
the	O
problem	O
that	O
the	O
NM-method	B-General_Concept
was	O
developed	O
to	O
solve	O
.	O
</s>
<s>
This	O
difference	O
facilitates	O
the	O
choice	O
between	O
the	O
NM-method	B-General_Concept
and	O
the	O
IPF	O
in	O
empirical	O
applications	O
.	O
</s>
<s>
In	O
addition	O
,	O
the	O
domains	O
are	O
different	O
for	O
which	O
the	O
IPF	O
and	O
the	O
NM-method	B-General_Concept
yield	O
solutions	O
.	O
</s>
<s>
First	O
,	O
unlike	O
the	O
NM-method	B-General_Concept
,	O
the	O
IPF	O
does	O
not	O
provide	O
a	O
solution	O
for	O
all	O
seed	O
tables	O
with	O
zero	O
entries	O
(	O
Csiszár	O
(	O
1975	O
)	O
found	O
necessary	O
and	O
sufficient	O
conditions	O
for	O
applying	O
the	O
IPF	O
with	O
general	O
tables	O
having	O
zero	O
entries	O
)	O
.	O
</s>
<s>
Second	O
,	O
unlike	O
the	O
IPF	O
,	O
the	O
NM-method	B-General_Concept
does	O
not	O
provide	O
a	O
meaningful	O
solution	O
for	O
pairs	O
of	O
matrices	O
and	O
defining	O
impossible	O
counterfactuals	O
.	O
</s>
<s>
Third	O
,	O
the	O
precondition	O
of	O
the	O
NM-method	B-General_Concept
(	O
of	O
either	O
or	O
)	O
is	O
not	O
a	O
precondition	O
for	O
the	O
applicability	O
of	O
the	O
IPF	O
.	O
</s>
<s>
For	O
,	O
and	O
matrices	O
of	O
size	O
the	O
two	O
methods	O
produces	O
the	O
same	O
transformed	O
table	O
provided	O
ranks	O
the	O
contingency	B-Application
tables	I-Application
the	O
same	O
as	O
the	O
scalar-valued	O
Liu-Lu	O
index	O
does	O
.	O
</s>
