stat.alloc

stat.alloc is an R package with a set of functions which allow for an analysis of status allocation, or a process whereby individuals from different origin categories (e.g., educational levels) are allocated slots in a set of ranked positions (e.g., occupations ranked by the degree of prestige). Krauze and Słomczyński (1985) proposed elementary status allocation principles: meritocracy and lottery, while Karpiński and Skvoretz (2023), building on their work, proposed a set of formal status allocation models which define the outcome of the status allocation process as a mixture of the elementary principles. A crucial feature of the Karpiński-Skvoretz status allocation model is a parameter called the mixing coefficient which specifies the contribution of the principle of meritocracy to the mix. In the simples case, if the mixing coefficient equal 0.4, it means that 40 percent of the time status allocation is driven by meritocracy, while 60 percent of the time it is driven by lottery. Consult the original publications for further details about the models and their substantive justification.

Functions in the package stat.alloc estimate a family of status allocation models, namely:

• models which assume that the mixing coefficient is constant across the origin categories. The relevant functions have names which begin with cmm, which stands for “constant mixing model”;
• models which assume that the mixing coefficients vary across the origin categories. The relevant functions have names which begin with dmm, which stands for “differential mixing model”;
• models which assume that there is a single basis for status allocation (i.e., a merit characteristic) and ones which assume that there are two such bases. Functions to estimate models of the latter type include 2d in their names.

In addition, some functions in the package use an approach to estimating the parameters which consists in minimising the distance (defined in terms of the Frobenius norm of the difference) between the observed and model-predicted status allocations; these functions have names which end with _mde, which stands for minimum distance estimation. Other functions in the package estimate the model parameters by maximising the value of the relevant likelihood function; these functions have names which end with _mle, which stands for maximum likelihood estimation. In most cases, the two approaches to estimation yield estimates which are very similar, but not identical.

Installation

You can install the development version of stat.alloc from GitHub with:

# install.packages("devtools")
devtools::install_github("zbig-karp/stat.alloc")

Example

This example walks through estimation of the simplest status allocation model, namely, the constant mixing model. The outcome of the status allocation process is represented in the form of a matrix whose rows correspond to origin categories, while its columns correspond to destination statuses. The code below builds a status allocation table with 5 origin categories (i.e., education levels) and 4 destination statuses (i.e., occupational prestige categories). The rows are ordered top to bottom from the highest to the lowest education level, while the columns are ordered left to right from the highest prestige to the lowest prestige category. Each element of the matrix represents the number of individuals with a given level of education who end up in a given occupation. The matrix is reproduced from Krauze and Słomczyński (1985).

data(ks1985)
t1 <- xtabs(freq ~ degree + status, data = ks1985)
t1
#>       status
#> degree    S1    S2    S3    S4
#>     E1 11883  2385  1144   277
#>     E2  4567  4960  4474   709
#>     E3  5155 11143 16131  2599
#>     E4  1081  2299  8686  2031
#>     E5   550   577  5731  1859

Function refall() is used to calculate meritocratic and lottery allocations from a given one:

library(stat.alloc)

refall(t1)
#> $Actual
#>       status
#> degree    S1    S2    S3    S4
#>     E1 11883  2385  1144   277
#>     E2  4567  4960  4474   709
#>     E3  5155 11143 16131  2599
#>     E4  1081  2299  8686  2031
#>     E5   550   577  5731  1859
#> 
#> $Meritocratic
#>       status
#> degree    S1    S2    S3   S4
#>     E1 15689     0     0    0
#>     E2  7547  7163     0    0
#>     E3     0 14201 20827    0
#>     E4     0     0 14097    0
#>     E5     0     0  1242 7475
#> 
#> $Lottery
#>       status
#> degree   S1   S2    S3   S4
#>     E1 4131 3798  6430 1329
#>     E2 3874 3561  6029 1246
#>     E3 9224 8481 14356 2967
#>     E4 3712 3413  5778 1194
#>     E5 2295 2110  3573  738

Function cmm_mde() estimates a model which treats the observed status allocation as a mixture of two “pure” status allocation. It returns a list of three elements:

• mixing coefficient, or the proportion of meritocracy in the mix,
• model-predicted status allocation,
• the index of dissimilarity between the observed and model-predicted allocations

cmm_mde(dat = refall(t1))
#> $`Mixing coefficient`
#>   Estimate     S.E. z test p value
#> 1    0.448 2.93e-05  15251 < 0.001
#> 
#> $`Model-predicted allocation`
#>       status
#> degree   S1    S2    S3   S4
#>     E1 9304  2098  3552  734
#>     E2 5518  5173  3331  688
#>     E3 5096 11041 17252 1639
#>     E4 2051  1885  9501  660
#>     E5 1268  1166  2530 3753
#> 
#> $`Dissimilarity index`
#> [1] 0.115

Other functions in the package are used in a similar way.

References

Karpiński, Zbigniew, and John Skvoretz. 2023. “Status Allocation from Elementary Allocation Principles.” Research in Social Stratification and Mobility 83 (February): 100769. https://doi.org/10.1016/j.rssm.2023.100769.

Krauze, Tadeusz, and Kazimierz M. Słomczyński. 1985. “How Far to Meritocracy? Empirical Tests of a Controversial Thesis.” Social Forces 63: 623–42.