superSubset, findSubsets, findSupersets: Functions to find subsets or supersets

Description

Functions to find a list of implicants that satisfy some restrictions (see details), or to find the corresponding row numbers in the implicant matrix, for all subsets, or supersets, of a (prime) implicant or an initial causal configuration.

Usage

superSubset(data, outcome = "", conditions = "", relation = "necessity",
    incl.cut = 1, cov.cut = 0, ron.cut = 0, pri.cut = 0, depth = NULL,
    use.letters = FALSE, use.labels = FALSE, add = NULL, ...)
findSubsets(input, noflevels = NULL, stop = NULL, ...)
findSupersets(input, noflevels = NULL, ...)

Value

The result of the superSubset() function is an object of class "ss", which is a list with the following components:

incl.cov: A data frame with the parameters of fit.
coms: A data frame with the (m)embersip (s)cores of the resulting (co)mbinations.

For findSubsets() and findSupersets(), a vector with the row numbers corresponding to all possible subsets, or supersets, of a (prime) implicant.

Arguments

data: A data frame with crisp (binary and multi-value) or fuzzy causal conditions
outcome: The name of the outcome.
conditions: A string containing the conditions' names, separated by commas.
relation: The set relation to outcome, either "necessity", "sufficiency", "necsuf" or "sufnec". Partial words like "suf" are accepted.
incl.cut: The minimal inclusion score of the set relation.
cov.cut: The minimal coverage score of the set relation.
ron.cut: The minimal score for the RoN - relevance of necessity.
pri.cut: The minimal score for the PRI - proportional reduction in inconsistency.
use.letters: Logical, use simple letters instead of original conditions' names.
use.labels: Logical, use category labels if present.
noflevels: A vector containing the number of levels for each causal condition plus 1 (all subsets are located in the higher dimension, implicant matrix)
input: A vector of row numbers where the (prime) implicants are located, or a matrix of configurations (only for supersets).
stop: The maximum line number (subset) to stop at, and return
depth: Integer, an upper number of causal conditions to form expressions with.
add: A function, or a list containing functions, to add more parameters of fit.
...: Other arguments, mainly for backward compatibility.

Author

Adrian Dusa

Details

The function superSubset() finds a list of implicants that satisfy some restrictions referring to the inclusion and coverage with respect to the outcome, under given assumptions of necessity and/or sufficiency.

Ragin (2000) posits that under the necessity relation, instances of the outcome constitute a subset of the instances of the cause(s). Conversely, under the sufficiency relation, instances of the outcome constitute a superset of the instances of the cause(s).

When relation = "necessity" the function finds all implicants which are supersets of the outcome, then eliminates the redundant ones and returns the surviving (minimal) supersets, provided they pass the inclusion and coverage thresholds. If none of the surviving supersets pass these thresholds, the function will find disjunctions of causal conditions, instead of conjunctions.

When relation = "sufficiency" it finds all implicants which are subsets of the outcome, and similarly eliminates the redundant ones and return the surviving (minimal) subsets.

When relation = "necsuf", the relation is interpreted as necessity, and cov.cut is automatically set equal to the inclusion cutoff incl.cut. The same automatic equality is made for relation = "sufnec", when relation is interpreted as sufficiency.

The argument outcome specifies the name of the outcome, and if multi-value the argument can also specify the level to explain, using square brackets notation.

Outcomes can be negated using a tilde operator ~X. The logical argument neg.out is now deprecated, but still backwards compatible. Replaced by the tilde in front of the outcome name, it controls whether outcome is to be explained or its negation. If outcome is from a multivalent variable, it has the effect that the disjunction of all remaining values becomes the new outcome to be explained. neg.out = TRUE and a tilde ~ in the outcome name don't cancel each other out, either one (or even both) signaling if the outcome should be negated.

If the argument conditions is not specified, all other columns in data are used.

Along with the standard measures of inclusion and coverage, the function also returns PRI for sufficiency and RoN (relevance of necessity, see Schneider & Wagemann, 2012) for the necessity relation.

A subset is a conjunction (an intersection) of causal conditions, with respect to a larger (super)set, which is another (but more parsimonious) conjunction of causal conditions.

All subsets of a given set can be found in the so called “implicant matrix”, which is a \(n^k\) space, understood as all possible combinations of values in any combination of bases \(n\), each causal condition having three or more levels (Dusa, 2007, 2010).

For every two levels of a binary causal conditions (values 0 and 1), there are three levels in the implicants matrix:

0	to mark a minimized literal
1	to replace the value of 0 in the original binary condition
-1	to replace the value of 1 in the original binary condition

A prime implicant is a superset of an initial combination of causal conditions, and the reverse is also true: the initial combination is a subset of a prime implicant.

Any normal implicant (not prime) is a subset of a prime implicant, and in the same time a superset of some initial causal combinations.

Functions findSubsets() and findSupersets() find:

	- all possible such subsets for a given (prime) implicant, or
	- all possible supersets of an implicant or initial causal combination

in the implicant matrix.

The argument depth can be used to impose an upper number of causal conditions to form expressions with, it is the complexity level where the search is stopped. Depth is set to a maximum by default, and the algorithm will always stop at the maximum complexity level where no new, non-redundant prime implicants are found. Reducing the depth below that maximum will also reduce computation time.

For examples on how to add more parameters of fit via argument add, see the function pof().

References

Cebotari, V.; Vink, M.P. (2013) “A Configurational Analysis of Ethnic Protest in Europe”. International Journal of Comparative Sociology vol.54, no.4, pp.298-324, tools:::Rd_expr_doi("10.1177/0020715213508567").

Cebotari, Victor; Vink, Maarten Peter (2015) Replication Data for: A configurational analysis of ethnic protest in Europe, Harvard Dataverse, V2, tools:::Rd_expr_doi("10.7910/DVN/PT2IB9").

Dusa, A. (2007b) Enhancing Quine-McCluskey. WP 2007-49, COMPASSS Working Papers series.

Dusa, Adrian (2010) “A Mathematical Approach to the Boolean Minimization Problem.” Quality & Quantity vol.44, no.1, pp.99-113, tools:::Rd_expr_doi("10.1007/s11135-008-9183-x").

Lipset, S. M. (1959) “Some Social Requisites of Democracy: Economic Development and Political Legitimacy”, American Political Science Review vol.53, pp.69-105.

Schneider, Carsten Q.; Wagemann, Claudius (2012) Set-Theoretic Methods for the Social Sciences: A Guide to Qualitative Comparative Analysis (QCA). Cambridge: Cambridge University Press.

Examples

Run this code

    
# Lipset binary crisp sets
ssLC <- superSubset(LC, "SURV")

library(venn)
x = list("SURV" = which(LC$SURV == 1),
         "STB" = which(ssLC$coms[, 1] == 1),
         "LIT" = which(ssLC$coms[, 2] == 1))
venn(x, cexil = 0.7)

# Lipset multi-value sets
superSubset(LM, "SURV")

# Cebotari & Vink (2013) fuzzy data
# all necessary combinations with at least 0.9 inclusion and 0.6 coverage cut-offs
ssCVF <- superSubset(CVF, outcome = "PROTEST", incl.cut = 0.90, cov.cut = 0.6)
ssCVF

# the membership scores for the first minimal combination (GEOCON)
ssCVF$coms$GEOCON

# same restrictions, for the negation of the outcome
superSubset(CVF, outcome = "~PROTEST", incl.cut = 0.90, cov.cut = 0.6)

# to find supersets or supersets, a hypothetical example using
# three binary causal conditions, having two levels each: 0 and 1
noflevels <- c(2, 2, 2)

# second row of the implicant matrix: 0 0 1
# which in the "normal" base is:      - - 0
# the prime implicant being: ~C
(sub <- findSubsets(input = 2, noflevels + 1))
#  5  8 11 14 17 20 23 26 


getRow(sub, noflevels + 1)

# implicant matrix   normal values
#      a  b  c    |       a  b  c       
#   5  0  1  1    |    5  -  0  0      ~b~c    
#   8  0  2  1    |    8  -  1  0      b~c
#  11  1  0  1    |   11  0  -  0      ~a~c
#  14  1  1  1    |   14  0  0  0      ~a~b~c
#  17  1  2  1    |   17  0  1  0      ~ab~c
#  20  2  0  1    |   20  1  -  0      a~c
#  23  2  1  1    |   23  1  0  0      a~b~c               
#  26  2  2  1    |   26  1  1  0      ab~c 


# stopping at maximum row number 20
findSubsets(input = 2, noflevels + 1, stop = 20)
#  5  8 11 14 17 20


# -----
# for supersets
findSupersets(input = 14, noflevels + 1)
#  2  4  5 10 11 13 14

findSupersets(input = 17, noflevels + 1)
#  2  7  8 10 11 16 17

# input as a matrix
(im <- getRow(c(14, 17), noflevels + 1))

# implicant matrix   normal values
#  14  1  1  1    |   14  0  0  0       ~a~b~c
#  17  1  2  1    |   17  0  1  0       ~ab~c


sup <- findSupersets(input = im, noflevels + 1)
sup
#  2  4  5  7  8 10 11 13 14 16 17


getRow(sup, noflevels + 1)

# implicant matrix   normal values
#      a  b  c    |       a  b  c       
#   2  0  0  1    |    2  -  -  0       ~c      
#   4  0  1  0    |    4  -  0  -       ~b
#   5  0  1  1    |    5  -  0  0       ~b~c
#   7  0  2  0    |    7  -  1  -       b
#   8  0  2  1    |    8  -  1  0       b~c
#  10  1  0  0    |   10  0  -  -       ~a  
#  11  1  0  1    |   11  0  -  0       ~a~c                 
#  13  1  1  0    |   13  0  0  -       ~a~b   
#  14  1  1  1    |   14  0  0  0       ~a~b~c
#  16  1  2  0    |   16  0  1  -       ~ab
#  17  1  2  1    |   17  0  1  0       ~ab~c

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