NM-method
The NM-method or Naszodi–Mendonca method is the operation that can be applied in statistics, econometrics, economics, sociology, and demography to construct counterfactual contingency tables. The method finds the matrix  () which is "closest" to matrix  ( called the seed table) in the sense of being ranked the same but with the row and column totals of a target matrix  . While the row totals and column totals of  are known, matrix  itself may not be known.

Since the solution for matrix  is unique, the NM-method is a function: , where  is a row vector of size , while  is a column vector of size .

The NM-method was developed by Naszodi and Mendonca (2021) (and first applied by Naszodi and Mendonca (2019) to solve for matrix  in problems, where matrix  is not a sample from the population characterized by the row totals and column totals of matrix , but represents another population.

Their application aimed at quantifying intergenerational changes in the strength of educational homophily and thus measuring the historical change in social inequality between different educational groups in the US between 1980 and 2010. The trend in inequality was found to be U-shaped, supporting the view that with appropriate social and economic policies inequality can be reduced.

Definition of matrix ranking
The closeness between two matrices of the same size can be defined in several ways. The Euclidean distance, and the Kullback-Leibler divergence are two well-known examples.

The NM-method is consistent with a definition relying on the ordinal Liu-Lu index  which is the slightly modified version of the Coleman-index defined by Eq. (15) in Coleman (1958). According to this definition, matrix  is "closest" to matrix , if their Liu-Lu values are the same. In other words, if they are ranked the same by the ordinal Liu-Lu index.

If  is a 2×2 matrix, its  scalar-valued Liu-Lu index is defined as

 , where
;
;
;
;
.

Following Coleman (1958), this index is interpreted as the “actual minus expected over maximum minus minimum”, where  is the actual value of the  entry of the seed matrix ;  is its expected (integer) value under the counterfactual assumptions that the corresponding row total and column total of  are predetermined, while its interior is random. Also,  is its minimum value if the association between the row variable and the column variable of  is non-negative. Finally,  is the maximum value of  () for given row total  and column total .

For matrix  of size n×m (, ), the Liu-Lu index was generalized by Naszodi and Mendonca (2021) to a matrix-valued index. One of the  preconditions for the generalization is that the row variable and the column variable of matrix  have to be ordered. Equating the generalized, matrix-valued Liu-Lu index of  with that of matrix  is equivalent to dichotomizing their ordered row variable and ordered column variable  in  ways by explointing the ordered nature of the row and column variables. Than, equating the original, scalar-valued Liu-Lu indices of the 2×2 matrices obtained with the dichotomizations. I.e., for any pair of  (, and ) the restriction  is imposed, where  is the  matrix  with its  being of size , and its  being of size . Similarly,  is the  matrix given by the transpose of  with its  being of size , and its  being of size .

Constraints on the row totals and column totals
Matrix  should satisfy not only  but also the pair of constraints on its row totals and column totals:  and .

Solution
Assuming that  for all pairs of  (where , and ), the solution for  is unique, deterministic, and given by a closed-form formula.

For matrices  and  of size , the  solution is

.

The other 3 cells of  are uniquely determined by the row totals and column totals. So, this is how the NM-method works for 2×2 seed tables.

For , and  matrices of size  (, ), the solution is obtained by dichotomizing their ordered row variable and ordered column variable in all possible meaningful ways before solving  number of problems of 2×2 form. Each problem is defined for an  pair ( and ) with , and the target row totals and column totals: , and , respectively. Each problem is to be solved separately by the formula for . The set of solutions determine  number of entries of matrix . Its remaining  elements are uniquely determined by the target row totals and column totals.

Next, let us see how the NM-method works if matrix  is such that the second precondition of  is not met for .

If  for all pairs of , the solution for  is also unique, deterministic, and given by a closed-form formula. However, the corresponding concept of matrix ranking is slightly different from the one discussed above. Liu and Lu (2006) define it as  , where ;  is the smallest integer being larger than or equal to .

Finally, neither the NM-method, nor  is defined if  pair such that , while for another pair of  .

A numerical example
Consider the following   complemented with its row totals and column totals and the targets, i.e., the  and  :

As a first step of the NM-method,   is multiplied by the , and  matrices for each pair of  (, and ). It yields the following 9 matrices of size 2×2 with their target row totals and column totals:

The next step is to calculate the generalized matrix-valued Liu-Lu index , (where ) by applying the formula of the original scalar-valued Liu-Lu index to each of the 9 matrices:

Apparently, matrix  is positive. Therefore, the NM-method is defined. Solving each of the 9 problems of the 2×2 form yields 9 entries of the  matrix. Its other 7 entries are uniquely determined by the target row totals and column totals. The solution for  is:

Another numerical example taken from Abbott et al.(2019)
Consider the following   complemented with its row totals and column totals and the targets, i.e., the  and  :

As a first step of the NM-method,   is multiplied by the , and  matrices for each pair of  (, and ). It yields the following 4 matrices of size 2×2 with their target row totals and column totals:

The next step is to calculate the generalized matrix-valued Liu-Lu index , (where ) by applying the formula of the original scalar-valued Liu-Lu index to each of the 4 matrices:

Apparently, matrix  is positive. Therefore, the NM-method is defined. Solving each of the 4 problems of the 2×2 form yields 4 entries of the  matrix. Its other 5 entries are uniquely determined by the target row totals and column totals.  The solution for  is:

Implementation
The NM-method is implemented in Excel, Visual Basic, R, and also in Stata.

Applications
The NM-method can be applied to study various phenomena including assortative mating, intergenerational mobility as a type of social mobility, residential segregation, recruitment and talent management.

In all of these applications, matrices , , and  represent joint distributions of one-to-one matched entities (e.g. husbands and wives, or first born children and mothers, or dwellings and main tenants, or CEOs and companies, or chess instructors and their most talended students) characterized either by a dichotomous categorical variable (e.g. taking values vegetarian/non-vegetarian, Grandmaster/or not), or an ordered multinomial categorical variable (e.g. level of final educational attainment, skiers' ability level, income bracket, category of rental fee, credit rating, FIDE titles). Although the NM-method has a wide range of applicability, all the examples to be presented next are about assortative mating along the education level. In these applications, the two preconditions (of ordered trait variable, and positive assortative mating in all educational groups) are not debated to be met.

Assume that  matrix  characterizes the joint educational distribution of husbands and wives in Zimbabwe, while matrix  characterizes the same in Yemen. Matrix  to be constructed with the NM-method tells us what would be the joint educational distribution of couples in Zimbabwe, if the educational distributions of husbands and wives were the same as in Yemen, while the overall desire for homogamy (also called as aggregate marital preferences in economics, or marital matching social norms/social barriers in sociology) were unchanged.

In a  second application, matrices  and  characterize the same country in two different years. Matrix  is the joint educational distribution of American newlyweds in 2040, where the husbands are from Generation Z and being young adults when observed. Matrix  is the same but for Generation Y observed in year 2024. By constructing matrix , one can study in the future what would be the educational distribution among the just married American young couples if they sorted into marriages the same way as the males in Generation Z and their partners do, while the education level were the same as among the males in Generation Y and their partners.

In a  third application, matrices  and  characterize again the same country in two different years. In this application, matrix  is the joint educational distribution of Portuguese young couples (where the male partners' age is between 30 and 34 years) in 2011. And  is the same but it is observed in year 1981. One may aim to construct matrix  in order to study what would have been the educational distribution of Portuguese young couples if they had sorted into marriages like their peers did in 2011, while their gender-specific educational distributions were the same as in 1981.

In each of the first two applications, matrix  represents a counterfactual joint distribution. It can be used to quantify certain ceteris paribus effects. More precisely, to quantify on a cardinal scale the difference between the directly unobservable degree of marital sorting in Zimbabwe and Yemen, or in Generation Z and Generation Y with a counterfactual decomposition. For the decomposition, the counterfactual table  is used to calculate the contribution of each of the driving forces (i.e., the observed structural availability of potential partners with various education levels determining the opportunities at the population level; and the unobservable non-structural drivers, e.g., aggregate matching preferences, desires, norms, barriers) and that of their interaction (i.e., the effect of changes in aggregate preferences/desires/norms/barriers due to changes in structural availability) to an observable cardinal scaled statistics (e.g. the share of educationally homogamous couples).

The third application was used by Naszodi and Mendonca (2021) as an example for a non-sense counterfactual: the education level has changed so drastically in Portugal over the three decades studied that this counterfactual is impossible to be obtained.

Some features of the NM-method
First, the NM-method does not yield a meaningful solution if it reaches the limit of its applicability. For instance, in the third application, the NM-method signals with a negative entry in matrix  that the counterfactual is impossible (see: AlternativeMethod_US_1980s_2010s_age3035_main.xls Sheet PT_A1981_P2011_Not_meaningful). In this respect, the NM-method is similar to the linear probability model that signals the same with a predicted probabiity outside the unit interval .

Second, the NM-method commutes with merging neighboring categories of the row variable and that of the column variable: , where  is the row merging matrix of size ; and , where  is the column merging matrix of size .

Third, the NM-method works even if there are zero entries in matrix .

Comparison with the IPF
The iterative proportional fitting procedure (IPF) is also a function:. It is the operation of finding the fitted matrix  () which fulfills a set of conditions similar to those met by matrix  constructed with the NM-method. E.g., matrix  is the closest to matrix  but with the row and column totals of the target matrix .

However, there are differences between the IPF and the NM-method. The IPF defines closeness of matrices of the same size by the cross-entropy, or the Kullback-Leibler divergence. Accordingly, the IPF compatible concept of distance between the 2×2 matrices  and  is zero, if their crossproduct ratios (also known as the odds ratio) are the same: . To recall, the NM-method's condition for equal ranking of matrices  and  is .

The following numerical example highlights that the IPF and the NM-method are not identical: . Consider the   with its :

The NM-method yields the following matrix :

Whereas the solution for matrix  obtained with the IPF is:

The IPF is equivalent to the maximum likelihood estimator of a joint population distribution, where matrix  (the estimate for the joint population distribution) is calculated from matrix , the observed joint distribution in a random sample taken from the population characterized by the row totals and column totals of matrix . In contrast to the problem solved by the IPF, matrix  is not sampled from this population in the problem that the NM-method was developed to solve. In fact, in the NM-problem, matrices  and  characterize two different populations (either observed simultaneously like in the application for Zimbabwe and Yemen, or observed in two different points in time like in its application for the populations of Generation Z and Generation Y). This difference facilitates the choice between the NM-method and the IPF in empirical applications.

Deming and Stephan(1940), the inventors of the IPF, illustrated the application of their method on a classic maximum likelihood estimation problem, where matrix  was sampled from the population characterized by the row totals and column totals of matrix . They were aware of the fact that in general, the IPF is not suitable for counterfactual predictions: they explicitly warned that their algorithm is “not by itself useful for prediction” (see Stephan and Deming 1940 p. 444).

In addition, the domains are different for which the IPF and the NM-method yield solutions. First, unlike the NM-method, the IPF does not provide a solution for all seed tables  with zero entries (Csiszár (1975) found necessary and sufficient conditions for applying the IPF with general tables having zero entries). Second, unlike the IPF, the NM-method does not provide a meaningful solution for pairs of matrices  and  defining impossible counterfactuals. Third, the precondition of the NM-method (of either  or ) is not a precondition for the applicability of the IPF.

Finally, unlike the NM, the IPF does not commute with the operation of merging neighboring categories of the row variable and that of the column variable as it is illustrated with a numerical example in Naszodi(2023) (see page 10).
For this reason, the transformed table obtained with the IPF can be sensitive to the choice of the number of trait categories.

Kenneth Macdonald (2023) 
is at ease with the conclusion by Naszodi (2023) 
that the IPF is suitable for sampling correction tasks, but not for generation of counterfactuals. Similarly to Naszodi, Macdonald also questions whether the row and column proportional transformations of the IPF preserve the structure of association within a contingency table that allows us to study social mobility.

Comparison with the Minimum Euclidean Distance Approach 
The Minimum Euclidean Distance Approach (MEDA) (defined by Abbott et al., 2019 following Fernández and Rogerson, 2001) is also a function:

.

First, MEDA assigns a scalar to matrix : it is the weight used for
constructing the convex combination of two extreme cases (random and perfectly assortative matching with the pair of marginals ) by minimizing the Eucledean distance with . E.g. this scalar is  in the
numerical example taken from Abbott et al.(2019).
Second, for any pair of counterfactual marginal distributions () the MEDA constructs the convex combination of the two extreme cases (random and perfectly assortative matches with the pair of marginals ()).

Differences between the NM and the MEDA:
while the NM holds the assortativeness unchanged by keeping the generalized matrix-valued Liu-Lu index  fixed, the MEDA does the same by keeping the scalar  fixed.
For , and  matrices of size  the two methods produces the same transformed table provided  ranks the contingency tables the same as the scalar-valued Liu-Lu index does.
However, for  matrices larger that 2×2, the
generalized Liu-Lu index is matrix-valued, so it is different from the scalar-valued .
Therefore, the NM-transformed table is also different from the MEDA-transformed table.

For instance,
in the numerical example taken from Abbott et al.(2019), the counterfactual table constructed by MEDA 
is the matrix :

The difference between matrix  and matrix  is not negligible.
E.g. the share of homogamous couples is 2 percentage points smaller in the MEDA-constructed counterfactual matrix  than in the observed matrix ,
whereas it is 3.4 percentage points smaller in the NM-constructed counterfactual matrix  relative to .

Because Abbott's example is not a fictional one,
but is based on the empirical educational distribution of American couples, therefore
the difference between 2 percentage points and 3.4 percentage points can be interpreted as the MEDA quantifies changes in inequality from one generation to another generation to be significantly smaller compared to the NM.

See also
 Iterative proportional fitting procedure

External links
Generalized Naszodi-Mendonca method (GNM-method)

References 

Statistics
Econometrics