Compute All Correctness-Preserving Submodels of a QCA Reference Model
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).
submodels(expression, noflevels = c(), test = TRUE)
expression |
A string representing a csQCA or an fsQCA model, or a csQCA
or fsQCA solution object of class 'qca' (created by the
|
noflevels |
A numeric vector specifying the number of levels for each factor (experimental, can be ignored). |
test |
Logical, test whether |
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").
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.
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. |
| Baumgartner, Michael | : development, testing |
| Thiem, Alrik | : development, documentation, programming, testing |
Alrik Thiem (Personal Website; ResearchGate Website)
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.
## 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
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