cna-solutions: Extract solutions from an object of class “cna”

Description

Given a solution object x produced by cna, msc(x) extracts all minimally sufficient conditions, asf(x) all atomic solution formulas, and csf(x, n.init) builds approximately n.init complex solution formulas. All solution attributes (details) available in showMeasures can be computed. The three functions return a data frame with the additional class attribute “condTbl”.

Usage

msc(x, details = x$details, cases = FALSE)
asf(x, details = x$details)
csf(x, n.init = 1000, details = x$details, asfx = NULL, 
    inus.only = x$control$inus.only, inus.def = x$control$inus.def,  
    minimalizeCsf = inus.only, 
    acyclic.only = x$acyclic.only, cycle.type = x$cycle.type, verbose = FALSE)

Value

msc, asf and csf return objects of class “condTbl”, an object similar to a data.frame, which features the following components:

`outcome`:	the outcomes
`condition`:	the relevant conditions or solutions
`con`:	the scores on the sufficiency measure (e.g. consistency)
`cov`:	the scores on the necessity measure (e.g. coverage)
`complexity`:	the complexity scores (number of factor value appearances to the left of “`<->`”)
`...`:	scores on additional evaluation measures and solution attributes as specified in
	`details`

Arguments

x: Object of class “cna”.
details: A character vector specifying the evaluation measures and additional solution attributes to be computed. Possible elements are all the measures in
showMeasures(). Defaults to the details specification stored in x. Can also be TRUE/FALSE. If FALSE, no additional attributes are returned; TRUE resolves to c("inus", "cyclic", "exhaustiveness", "faithfulness", "coherence") (cf. detailMeasures).
cases: Logical; if TRUE, an additional column listing the cases where the msc is instantiated in combination with the outcome is added to the output.
n.init: Integer capping the amount of initial asf combinations. Default at 1000. Serves to control the computational complexity of the csf building process.
asfx: Object of class “condTbl” produced by the asf function. Internal parameter not meant to be set by users.
inus.only: Logical; if TRUE, csf are freed of structural redundancies and only csf not featuring partial structural redundancies are retained (cf. vignette("cna"); cf. also is.inus and cnaControl).
inus.def: Character string specifying the definition of partial structural redundancy to be applied. Possible values are "implication" or "equivalence". The strings can be abbreviated. Cf. also is.inus and cnaControl.
minimalizeCsf: Logical; if TRUE, csf are freed of structural redundancies (cf. vignette("cna")).
acyclic.only: Logical; if TRUE, csf featuring a cyclic substructure are not returned. FALSE by default.
cycle.type: Character string specifying what type of cycles to be detected: "factor" (the default) or "value" (cf. cyclic).
verbose: Logical; if TRUE, some details on the csf building process are printed. FALSE by default.

Details

Depending on the processed data, the solutions (models) output by cna are often ambiguous, to the effect that many atomic and complex solutions fit the data equally well. To facilitate the inspection of the cna output, however, cna standardly returns only 5 minimally sufficient conditions (msc) and 5 atomic solution formulas (asf) for each outcome as well as 5 complex solution formulas (csf). msc can be used to extract all msc from an object x of class “cna”, asf to extract all asf, and csf to build approximately n.init csf from the asf stored in x. All solution attributes (details) that are saved in x are recovered as well. Moreover, all evaluation measures and solution attributes available in showMeasures---irrespective of whether they are saved in x---can be computed by specifying them in the details argument. The outputs of msc, asf, and csf can be further processed by the condition function.

While msc and asf merely extract information stored in x, csf builds csf from the inventory of asf recovered at the end of the third stage of the cna algorithm (cf. vignette("cna"), section 4). That is, the csf function implements the fourth stage of that algorithm. It proceeds in a stepwise manner as follows.

n.init possible conjunctions featuring one asf of every outcome are built.
If inus.only = TRUE or minimalizeCsf = TRUE, the solutions resulting from step 1 are freed of structural redundancies (cf. Baumgartner and Falk 2023).
If inus.only = TRUE, tautologous and contradictory solutions as well as solutions with partial structural redundancies (as defined in inus.def) and constant factors are eliminated. [If inus.only = FALSE and minimalizeCsf = TRUE, only structural redundancies are eliminated, meaning only step 2, but not step 3, is executed.]
If acyclic.only = TRUE, solutions with cyclic substructures are eliminated.
Solutions that are a submodel of another solution are removed.
For those solutions that were modified in the previous steps, the scores on the selected evaluation measures are re-calculated and solutions that no longer reach con or cov are eliminated (cf. cna).
The remaining solutions are returned as csf, ordered by complexity and the product of their scores on the evaluation measures.

References

Lam, Wai Fung, and Elinor Ostrom. 2010. “Analyzing the Dynamic Complexity of Development Interventions: Lessons from an Irrigation Experiment in Nepal.” Policy Sciences 43 (2):1-25.

Examples

Run this code

# Crisp-set data from Lam and Ostrom (2010) on the impact of development interventions
# ------------------------------------------------------------------------------------
# CNA with causal ordering that corresponds to the ordering in Lam & Ostrom (2010); coverage 
# cut-off at 0.9 (consistency cut-off at 1).
cna.irrigate <- cna(d.irrigate, ordering = "A, R, F, L, C < W", cov = .9,
                    maxstep = c(4, 4, 12), details = TRUE)
cna.irrigate

# The previous function call yields a total of 12 complex solution formulas, only
# 5 of which are returned in the default output. 
# Here is how to extract all 12 complex solution formulas along with all 
# solution attributes.
csf(cna.irrigate)
# With only the used evaluation measures and complexity plus exhaustiveness and faithfulness.
csf(cna.irrigate, details = c("e", "f"))
# Calculate additional evaluation measures from showCovCovMeasures().
csf(cna.irrigate, details = c("e", "f", "PAcon", "PACcov", "AACcon", "AAcov"))

# Extract all atomic solution formulas.
asf(cna.irrigate, details = c("e", "f"))

# Extract all minimally sufficient conditions.
msc(cna.irrigate) # capped at 20 rows
print(msc(cna.irrigate), n = Inf) # prints all rows
# Add cases featuring the minimally sufficient conditions combined 
# with the outcome.
(msc.table <- msc(cna.irrigate, cases = TRUE))
# Render as data frame.
as.data.frame(msc.table)

# Extract only the conditions (solutions).
csf(cna.irrigate)$condition
asf(cna.irrigate)$condition
msc(cna.irrigate)$condition

# A CNA of d.irrigate without outcome specification and ordering is even more
# ambiguous.
cna2.irrigate <- cna(d.irrigate, cov = .9, maxstep = c(4,4,12), 
                     details = c("e", "f", "ccon", "ccov"))

# Reduce the initial asf combinations to 50.
csf(cna2.irrigate, n.init = 50)
# Print the first 20 csf.
csf(cna2.irrigate, n.init = 50)[1:20, ]

# Print details about the csf building process.
csf(cna.irrigate, verbose = TRUE)

# Return evaluation measures and solution attributes with 5 digits.
print(asf(cna2.irrigate), digits = 5)
 

# Further examples
# ----------------
# An example generating structural redundancies.
target <- "(A*B + C <-> D)*(c + a <-> E)"
dat1 <- selectCases(target)
ana1 <- cna(dat1, maxstep = c(3, 4, 10))
# Run csf with elimination of structural redundancies.
csf(ana1, verbose = TRUE)
# Run csf without elimination of structural redundancies. 
csf(ana1, verbose = TRUE, inus.only = FALSE)

# An example generating partial structural redundancies.
dat2 <- data.frame(
  A=c(0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0, 1),
  B=c(0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1),
  C=c(1,1,0,0,0,1,0,0,1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,1,0,1,0),
  D=c(0,1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0,1,0,0,0,1,1,0,0,1,0),
  E=c(1,0,0,0,0,1,1,1,1,1,1,0,0,1,0,0,0,1,1,1,1,0,0,0,0,1,1),
  F=c(1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0),
  G=c(1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1))
ana2 <- cna(dat2, con = .8, cov = .8, maxstep = c(3, 3, 10))
# Run csf without elimination of partial structural redundancies.
csf(ana2, inus.only = FALSE, verbose = TRUE)
# Run csf with elimination of partial structural redundancies.
csf(ana2, verbose = TRUE)
# Prior to version 3.6.0, the "equivalence" definition of partial structural
# redundancy was used by default (see ?is.inus() for details). Now, the 
# "implication" definition is used. To replicate old behavior
# set inus.def to "equivalence".
csf(ana2, verbose = TRUE, inus.def = "equivalence")
# The two definitions only come apart in case of cyclic structures.
# Build only acyclic models.
csf(ana2, verbose = TRUE, acyclic.only = TRUE)
# Add further details.
csf(ana2, verbose = TRUE, acyclic.only = TRUE, details = c("PAcon", "PACcov"))

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