Uses one of three methods to decompose the difference between two M indices: (1) "shapley" / "shapley_detailed": a method based on the Shapley decomposition with a few advantages over the Karmel-Maclachlan method (recommended and the default, Deutsch et al. 2006), (2) "km": the method based on Karmel-Maclachlan (1988), (3) "mrc": the method developed by Mora and Ruiz-Castillo (2009). All methods have been extended to account for missing units/groups in either data input.

mutual_difference(
data1,
data2,
group,
unit,
weight = NULL,
method = "shapley",
se = FALSE,
CI = 0.95,
n_bootstrap = 100,
base = exp(1),
...
)

## Arguments

data1

A data frame with same structure as data2.

data2

A data frame with same structure as data1.

group

A categorical variable or a vector of variables contained in data. Defines the first dimension over which segregation is computed.

unit

A categorical variable or a vector of variables contained in data. Defines the second dimension over which segregation is computed.

weight

Numeric. (Default NULL)

method

Either "shapley" (the default), "km" (Karmel and Maclachlan method), or "mrc" (Mora and Ruiz-Castillo method).

se

If TRUE, the segregation estimates are bootstrapped to provide standard errors and to apply bias correction. The bias that is reported has already been applied to the estimates (i.e. the reported estimates are "debiased") (Default FALSE)

CI

If se = TRUE, compute the confidence (CI*100) in addition to the bootstrap standard error. This is based on percentiles of the bootstrap distribution, and a valid interpretation relies on a larger number of bootstrap iterations. (Default 0.95)

n_bootstrap

Number of bootstrap iterations. (Default 100)

base

Base of the logarithm that is used in the calculation. Defaults to the natural logarithm.

...

Only used for additional arguments when when method is set to shapley or km. See ipf for details.

## Value

Returns a data.table with columns stat and est. The data frame contains the following rows defined by stat:

M1 contains the M for data1.

M2 contains the M for data2.

diff is the difference between M2 and M1. The sum of the five rows following diff equal diff.

additions contains the change in M induces by unit and group categories present in data2 but not data1, and removals the reverse.

All methods return the following three terms:

unit_marginal is the contribution of unit composition differences.

group_marginal is the contribution of group composition differences.

structural is the contribution unexplained by the marginal changes, i.e. the structural difference. Note that the interpretation of these terms depend on the exact method used.

When using "km", one additional row is returned:

interaction is the contribution of differences in the joint marginal distribution of unit and group.

When "shapley_detailed" is used, an additional column "unit" is returned, along with six additional rows for each unit that is present in both data1 and data2. The five rows have the following meaning:

p1 (p2) is the proportion of the unit in data1 (data2) once non-intersecting units/groups have been removed. The changes in local linkage are given by ls_diff1 and ls_diff2, and their average is given by

ls_diff_mean. The row named total

summarizes the contribution of the unit towards structural change using the formula .5 * p1 * ls_diff1 + .5 * p2 * ls_diff2. The sum of all "total" components equals structural change.

If se is set to TRUE, an additional column se contains the associated bootstrapped standard errors, an additional column CI contains the estimate confidence interval as a list column, an additional column bias contains the estimated bias, and the column est contains the bias-corrected estimates.

## Details

The Shapley method is an improvement over the Karmel-Maclachlan method (Deutsch et al. 2006). It is based on several margins-adjusted data inputs and yields symmetrical results (i.e. data1 and data2 can be switched). When "shapley_detailed" is used, the structural component is further decomposed into the contributions of individuals units.

The Karmel-Maclachlan method (Karmel and Maclachlan 1988) adjusts the margins of data1 to be similar to the margins of data2. This process is not symmetrical.

The Shapley and Karmel-Maclachlan methods are based on iterative proportional fitting (IPF), first introduced by Deming and Stephan (1940). Depending on the size of the dataset, this may take a few seconds (see ipf for details).

The method developed by Mora and Ruiz-Castillo (2009) uses an algebraic approach to estimate the size of the components. This will often yield substantively different results from the Shapley and Karmel-Maclachlan methods. Note that this method is not symmetric in terms of what is defined as group and unit categories, which may yield contradictory results.

A problem arises when there are group and/or unit categories in data1 that are not present in data2 (or vice versa). All methods estimate the difference only for categories that are present in both datasets, and report additionally the change in M that is induced by these cases as additions (present in data2, but not in data1) and removals (present in data1, but not in data2).

## References

W. E. Deming, F. F. Stephan. 1940. "On a Least Squares Adjustment of a Sampled Frequency Table When the Expected Marginal Totals are Known." The Annals of Mathematical Statistics 11(4): 427-444.

T. Karmel and M. Maclachlan. 1988. "Occupational Sex Segregation — Increasing or Decreasing?" Economic Record 64: 187-195.

R. Mora and J. Ruiz-Castillo. 2009. "The Invariance Properties of the Mutual Information Index of Multigroup Segregation." Research on Economic Inequality 17: 33-53.

J. Deutsch, Y. Flückiger, and J. Silber. 2009. "Analyzing Changes in Occupational Segregation: The Case of Switzerland (1970–2000)." Research on Economic Inequality 17: 171–202.

## Examples

# decompose the difference in school segregation between 2000 and 2005,
# using the Shapley method
mutual_difference(schools00, schools05, group = "race", unit = "school",
weight = "n", method = "shapley", precision = .1)
#>              stat          est
#> 1:             M1  0.425538976
#> 2:             M2  0.413385092
#> 3:           diff -0.012153884
#> 5:       removals -0.011405093
#> 6: group_marginal  0.012989172
#> 7:  unit_marginal -0.004240311
#> 8:     structural -0.006084876
# => the structural component is close to zero, thus most change is in the marginals.
# This method gives identical results when we switch the unit and group definitions,
# and when we switch the data inputs.

# the Karmel-Maclachlan method is similar, but only adjust the data in the forward direction...
mutual_difference(schools00, schools05, group = "school", unit = "race",
weight = "n", method = "km", precision = .1)
#>              stat          est
#> 1:             M1  0.425538976
#> 2:             M2  0.413385092
#> 3:           diff -0.012153884
#> 5:       removals -0.011405093
#> 6: group_marginal -0.020609129
#> 7:  unit_marginal  0.017497722
#> 8:    interaction -0.017497722
#> 9:     structural  0.023273114

# ...this means that the results won't be identical when we switch the data inputs
mutual_difference(schools05, schools00, group = "school", unit = "race",
weight = "n", method = "km", precision = .1)
#>              stat          est
#> 1:             M1  0.413385092
#> 2:             M2  0.425538976
#> 3:           diff  0.012153884
#> 5:       removals  0.003412776
#> 6: group_marginal  0.019784513
#> 7:  unit_marginal -0.016961244
#> 8:    interaction -0.002823269
#> 9:     structural -0.002663986

# the MRC method indicates a much higher structural change...
mutual_difference(schools00, schools05, group = "race", unit = "school",
weight = "n", method = "mrc")
#>              stat          est
#> 1:             M1  0.425538976
#> 2:             M2  0.413385092
#> 3:           diff -0.012153884
#> 5:       removals -0.011405093
#> 6:  unit_marginal -0.004207757
#> 7: group_marginal  0.062334448
#> 8:     structural -0.055462706

# ...and is not symmetric
mutual_difference(schools00, schools05, group = "school", unit = "race",
weight = "n", method = "mrc")
#>              stat          est
#> 1:             M1  0.425538976
#> 2:             M2  0.413385092
#> 3:           diff -0.012153884