submodels: Compute All Correctness-Preserving Submodels of a QCA...

In QCApro: Advanced Functionality for Performing and Evaluating Qualitative Comparative Analysis

Description

This evaluation function computes all correctness-preserving submodels of a QCA reference model. It has initially been programmed for Baumgartner and Thiem (2015) to test the correctness of QCA's three search strategies (conservative/complex, intermediate, parsimonious).

Usage

submodels(expression, noflevels = c(), test = TRUE)

Arguments

expression	A string representing a csQCA or an fsQCA model, or a csQCA or fsQCA solution object of class 'qca' (created by the eQMC function).
noflevels	A numeric vector specifying the number of levels for each factor (experimental, can be ignored).
test	Logical, test whether expression is a causal structure.

Details

This function has initially been programmed for Baumgartner and Thiem (2015) to test the correctness of QCA's three solution types (conservative/complex, intermediate, parsimonious). It computes all submodels of a csQCA or an fsQCA reference model that do not violate the criterion of correctness (mvQCA models are not yet supported). The following expression structures can be used: "A*B + C*D <=> Y" or "AB + CD <=> Y". Empty spaces and the type of conditional operator (<->/<=>/->/=>) are irrelevant, but only single letters are allowed for exogenous factors. The full model need not be provided; the antecedent also suffices (e.g., "AB + CD").

Objects of class 'qca', which are returned by the eQMC function, are also accepted, provided that all exogenous factors have a single-letter label (set the argument use.letters to TRUE in the function call to eQMC if original factor labels are not single letters).

The argument noflevels expects a numeric vector of the number of factor levels with a names attribute. Currently, this argument is experimental and can be ignored.

The argument test specifies whether expression should be pre-tested for its causal interpretability before forming submodels. The value to this argument does not affect whether basic tests for likely typos in expressions such as "abb <-> C" or "abB <-> C" are performed. If expression is an object of class 'qca', test will be set to FALSE because QCA models generated by the eQMC function at default argument settings are always causally interpretable.

Note that for highly complex models containing many conjuncts within many disjuncts, computing times tend to increase considerably.

Value

A list with the following four main components:

model	The reference model.
noflevels	The number of levels for each factor in the factor frame of the model.
outcome	The outcome specified as part of the expression or a pseudo outcome if only an antecedent was specified.
submodels	A character vector of all correctness-preserving submodels.

Contributors (alphabetical)

Baumgartner, Michael	: development, testing
Thiem, Alrik	: development, documentation, programming, testing

Author(s)

Alrik Thiem (Personal Website; ResearchGate Website)

References

Baumgartner, Michael, and Alrik Thiem. 2015. Often Trusted but Never (Properly) Tested: Evaluating Qualitative Comparative Analysis. Paper presented at the 12th Conference of the European Sociological Association, 25-28 August, Czech Technical University, Prague (Czech Republic). Link.

See Also

eQMC, limitedDiversity

Examples

## Not run: 
# provide a) a full model as an equivalence and inspect its submodels
models1 <- submodels("a*B + B*c + D <-> Z") 
models1$submodels

# ... b) a full model with a negated outcome
# submodels
models2 <- submodels("AcD + BCD + abcd <=> e")
length(models2$submodels)

# ... c) or only an antecedent
models3 <- submodels("aB + Bc + D")
models3$submodels

# directly provide an object of class 'qca' generated by the 'eQMC' function,
# even when the solution comprises multiple models; specify 
# 'use.letters = TRUE' when the original exogenous factors have multi-letter 
# labels; for example:
data(d.represent)
sol1 <- eQMC(d.represent, outcome = "WNP", neg.out = TRUE, use.letters = TRUE)
sol1
# M1: ae + cde + (bdE) <=> wnp 
# M2: ae + cde + (bcd) <=> wnp 
# M3: ae + cde + (Abc) <=> wnp
# M1 has 138 submodels, M2 has 129, and M3 has 139 submodels
models4 <- submodels(sol1)
sapply(models4, "[")

# when original labels of exogenous factors already consist of single 
# letters only, 'use.letters = TRUE' need not be specified
data(d.napoleon)
sol2 <- eQMC(d.napoleon, outcome = "O")
sol2
models5 <- submodels(sol2)
sapply(models5, "[")

# prior testing is recommended because non-causal models can sometimes only
# be identified computationally
submodels("aB + Ac + Ad + bc + bd + CD")

# can a + AbC => Y be an acceptable QCA solution as Schneider and Wagemann 
# (2012, p. 108) argue? No, because in Boolean algebra, it holds that
# F + fG = (F + f) * (F + G) = 1*(F + G) = F + G by the laws of distribution,
# complementarity, and identity
submodels("a + AbC => Y", test = TRUE)

# proof that the conservative/complex solution type of QCA is incorrect, 
# using model 3 from above (see Baumgartner and Thiem (2015) for more details)

# 1. build saturated truth table on the basis of model 3: aB + Bc + D
tt <- data.frame(mintermMatrix(rep(2, 5)))
dimnames(tt) <- list(as.character(1:32), c(LETTERS[1:4], "OUT"))
tt <- tt[pmax(pmin(1 - tt$A, tt$B), pmin(tt$B, 1 - tt$C), tt$D) == tt$OUT, ]

# 2. use function 'limitedDiversity' to generate all conservative/complex
# solutions for all 16 + 120 scenarios of one/two dropped minterm/s
sollist.cs <- vector("list", 2)
sollist.cs <- lapply(1:2, function (x) {
  limitedDiversity(tt, outcome = "OUT", sol.type = "cs", n.drop = x)
  }
)

# 3. compute in how many scenarios a correctness-preserving submodel of 
# model 3 was part of the solution (43.75% for one dropped minterm and 
# 16.67% for two dropped minterms)
cs.correct <- numeric(2)
cs.correct <- sapply(1:2, function (x) {round((sum(unlist(lapply(
  sollist.cs[[x]][[2]], function (y) {any(models3$submodels %in% y)}
  ))) / choose(16, x))*100, 2)}
)
cs.correct

## End(Not run)

QCApro documentation built on May 1, 2019, 10:09 p.m.