ThSQCA for beginners, part 3: solution types and multiple solutions

Complex, parsimonious, and intermediate solutions explained with a small example, and what to do when a threshold yields more than one minimal solution.
R
ThSQCA
tutorial
Author

Yuki Toyoda

Published

October 5, 2026

In part 2 we ran all four sweeps. Two questions remain for a careful analysis. First, QCA can produce three kinds of solution (complex, parsimonious, and intermediate); which should you use? Second, what if a cell has more than one minimal solution? This post answers both with a second, smaller practice data set.

A second practice data set

We simulate 60 business software accounts. Each has a renewal score (RNW) and three evaluations: trust (TRU), price (PRC), and support (SUP), all from 0 to 10. In this simulated world, renewal is high when trust is high and either price or support is high. The three evaluations are strongly correlated, which will leave some combinations of conditions unobserved. That is deliberate, because unobserved combinations are exactly what the three solution types treat differently.

library(ThSQCA)
library(QCA)
Loading required package: admisc

To cite package QCA in publications, please use:
  Dusa, Adrian (2019) QCA with R. A Comprehensive Resource.
  Springer International Publishing.
Practice data set 2 
make_demo_data2 <- function(n = 60, seed = 89, rho = 0.85) {
  set.seed(seed)
  clip <- function(x) pmin(10, pmax(0, round(x)))
  L   <- rnorm(n)
  mk  <- function() clip(6 + 2 * (rho * L + sqrt(1 - rho^2) * rnorm(n)))
  TRU <- mk(); PRC <- mk(); SUP <- mk()
  RNW <- clip(pmin(TRU, pmax(PRC, SUP)) + rnorm(n, 0, 0.7))
  data.frame(RNW, TRU, PRC, SUP)
}

demo2 <- make_demo_data2()
head(demo2)
  RNW TRU PRC SUP
1   3   3   4   4
2   6   8   7   7
3   9   9   9   7
4   5   5   4   6
5  10  10   8   9
6   1   2   4   5

Unobserved combinations (“logical remainders”)

With three conditions, each either high or low, there are 2 x 2 x 2 = 8 possible combinations. QCA lists them in a truth table. Let us build one by hand with the QCA package, using renewal of 5 or more as the outcome and 7 or more as “high” for each condition.

bin2 <- data.frame(
  RNW = as.integer(demo2$RNW >= 5),
  TRU = as.integer(demo2$TRU >= 7),
  PRC = as.integer(demo2$PRC >= 7),
  SUP = as.integer(demo2$SUP >= 7)
)

tt2 <- truthTable(bin2, outcome = "RNW", conditions = c("TRU", "PRC", "SUP"),
                  incl.cut = 0.8, n.cut = 1, show.cases = FALSE)
tt2

  OUT: output value
    n: number of cases in configuration
 incl: sufficiency inclusion score
  PRI: proportional reduction in inconsistency

    TRU PRC SUP   OUT    n   incl  PRI  
1    0   0   0     0     30  0.433 0.433
2    0   0   1     1     7   0.857 0.857
4    0   1   1     1     3   1.000 1.000
6    1   0   1     1     3   1.000 1.000
7    1   1   0     1     4   1.000 1.000
8    1   1   1     1     13  1.000 1.000

Look at the row numbers on the left. Only 6 of the 8 possible combinations appear (rows 3 and 5 are missing), because the data contain no accounts with, for example, high trust alone, or high price evaluation alone. The combinations that are missing are the logical remainders. The column n is the number of accounts in each combination, incl is how consistently they renew, and OUT is 1 when the combination is consistent enough to count as sufficient.

A bar chart of the counts makes the gaps easy to see. Each bar is one of the 8 possible combinations, in the same order as the truth table rows.

cells <- expand.grid(SUP = 0:1, PRC = 0:1, TRU = 0:1)[, c("TRU", "PRC", "SUP")]
cells$n <- mapply(function(a, b, c) sum(bin2$TRU == a & bin2$PRC == b & bin2$SUP == c),
                  cells$TRU, cells$PRC, cells$SUP)
cells$label <- paste0(ifelse(cells$TRU == 1, "TRU", "~TRU"), "*",
                      ifelse(cells$PRC == 1, "PRC", "~PRC"), "*",
                      ifelse(cells$SUP == 1, "SUP", "~SUP"))

op <- par(mar = c(4.5, 9, 3, 1))
n_rev <- rev(cells$n)
bp <- barplot(n_rev, horiz = TRUE, names.arg = rev(cells$label), las = 1,
              col = "#233A9A", xlim = c(0, max(cells$n) * 1.35),
              xlab = "Number of accounts",
              main = "Accounts per combination")
for (i in seq_along(n_rev)) {
  if (n_rev[i] == 0) {
    text(0, bp[i], "not observed (remainder)", pos = 4, col = "#B45309", cex = 0.9, font = 2)
  } else {
    text(n_rev[i], bp[i], n_rev[i], pos = 4, cex = 0.9)
  }
}

Horizontal bar chart of 8 combinations. Not high on any of the three has 30 accounts; support only has 7; price and support without trust has 3; trust and support without price has 3; trust and price without support has 4; all three high has 13. Two combinations, price only and trust only, have no accounts and are labeled not observed.

Number of accounts in each of the 8 possible combinations of high and low trust, price, and support. Two combinations are not observed.
par(op)

The two combinations with no accounts are the remainders. Boolean minimization has no cases to check them against, so each solution type has to decide what to do with them.

Remainders matter because Boolean minimization can use them to make a formula shorter, and the three solution types differ in whether, and how, they do that.

Solution type Remainders used include dir.exp
Complex none "" (default) NULL
Parsimonious any remainder that shortens the formula "?" NULL
Intermediate only remainders that agree with your stated expectations "?" e.g. c(1, 1, 1)

Here dir.exp = c(1, 1, 1) says: we expect high trust, high price evaluation, and high support each to favor renewal.

The three solution types, side by side

We run the same outcome threshold sweep (renewal of 5, 6, 7, and 8) three times, changing only the solution type.

thr2  <- c(TRU = 7, PRC = 7, SUP = 7)
cond2 <- c("TRU", "PRC", "SUP")

run2 <- function(...) {
  otSweep(dat = demo2, outcome = "RNW", conditions = cond2, sweep_range = 5:8,
          thrX = thr2, incl.cut = 0.8, n.cut = 1, ...)
}

res_cx <- run2()                                     # complex
res_ps <- run2(include = "?")                        # parsimonious
Warning: Multiple equivalent solutions exist for thrY = 5 (n_solutions > 1).
Only the first solution (M1) and its fit metrics are shown. Use
generate_report() for full analysis.
res_im <- run2(include = "?", dir.exp = c(1, 1, 1))  # intermediate

data.frame(
  thrY         = 5:8,
  complex      = res_cx$summary$expression,
  parsimonious = res_ps$summary$expression,
  intermediate = res_im$summary$expression
)
  thrY            complex   parsimonious       intermediate
1    5      SUP + TRU*PRC      TRU + SUP      SUP + TRU*PRC
2    6 TRU*PRC + ~PRC*SUP TRU + ~PRC*SUP TRU*PRC + ~PRC*SUP
3    7            TRU*SUP        TRU*SUP            TRU*SUP
4    8        No solution    No solution        No solution

The warning shown with the output is a note from ThSQCA: at renewal of 5, the parsimonious analysis has more than one equally short solution. We come back to this below.

At renewal of 7, all three agree (TRU*SUP), and at 8 none finds a solution. At 5 and 6 the parsimonious solution is shorter. For example, at renewal of 5 it reports TRU + SUP, whereas the complex solution is SUP + TRU*PRC. The term TRU alone covers the combination “high trust, but low price and support evaluations”, which does not appear in the data. The parsimonious solution is willing to assume that this unobserved combination would also be sufficient. The complex solution makes no such assumption. In this particular data set the intermediate solution happens to coincide with the complex one.

How to choose:

Solution When it is a good choice Strength Caution
Complex Exploration; you want no assumptions about unobserved cases Every claim rests on observed cases Formulas can be long
Parsimonious You want to see the maximal simplification Shortest formulas May rely on implausible remainders
Intermediate Theory-driven reporting Uses only plausible remainders Needs a justified dir.exp

Whatever you choose, keep it the same across the whole sweep, and say which one you used.

When a cell has more than one minimal solution

Boolean minimization can return several equally short solutions for the same truth table. By default ThSQCA reports the first one (M1) and records the number of solutions in n_solutions. To see all of them, use extract_mode = "all".

res_all <- run2(include = "?", extract_mode = "all")
Warning: Multiple equivalent solutions exist for thrY = 5 (n_solutions > 1).
Only the first solution (M1) and its fit metrics are shown. Use
generate_report() for full analysis.
summary(res_all)
OTS Summary
=========== 

Analysis Parameters:
  Outcome: RNW 
  Conditions: TRU, PRC, SUP 
  Consistency cutoff: 0.8 
  Frequency cutoff: 1 

Results by Threshold:

 thrY                   expression inclS  covS n_solutions
    5 M1: TRU + SUP; M2: PRC + SUP 0.967 0.690           2
    6           M1: TRU + ~PRC*SUP 0.963 0.788           1
    7                  M1: TRU*SUP 0.875 0.667           1
    8                  No solution    NA    NA           0

Note: 1 threshold setting(s) had multiple solutions.
      Use generate_report() for full details.

At renewal of 5, the parsimonious analysis finds two minimal solutions of equal length (multiple minimal solutions), TRU + SUP and PRC + SUP. Each of them relies on a different unobserved combination, which is why neither can be preferred on the data alone. ThSQCA prints a note when this happens.

If you prefer a compact view, use extract_mode = "essential". It separates the terms that are present in every solution from the rest.

res_ess <- run2(include = "?", extract_mode = "essential")
Warning: Multiple equivalent solutions exist for thrY = 5 (n_solutions > 1).
Only the first solution (M1) and its fit metrics are shown. Use
generate_report() for full analysis.
res_ess$summary[, c("thrY", "expression", "selective_terms",
                    "unique_terms", "n_solutions")]
  thrY     expression selective_terms   unique_terms n_solutions
1    5            SUP       TRU + PRC M1:TRU; M2:PRC           2
2    6 TRU + ~PRC*SUP            <NA>           <NA>           1
3    7        TRU*SUP            <NA>           <NA>           1
4    8    No solution            <NA>           <NA>           0
Term type Meaning Here (renewal of 5)
Essential present in every solution SUP
Selective present in some solutions TRU and PRC
Unique present in only one solution TRU in M1, PRC in M2

ThSQCA records the alternatives, but deciding which one to prioritize is a theoretical judgment that stays with the researcher.

What comes next

In part 4, the last part, we turn the results into something you can report: a Markdown report, a configuration chart, and a check against the QCA package.

Software used: R 4.6.1, QCA 3.25.5, and ThSQCA 2.0.9.