Method and data (common to the series)

• Method: Threshold-Sweep QCA (ThS-QCA). I move the cut-off (threshold) on “how strongly would you recommend this brand to friends and acquaintances” (0 to 10; 0 = would not recommend at all, 10 = would definitely recommend) and list the sufficient conditions at each cut-off (combinations of conditions whose holders mostly end up recommending the brand; Toyoda, 2026, Quality & Quantity). It does not search for one optimal threshold.
• Data: A casual fashion survey (N = 300) that I ran in June 2026 for the lecture “Marketing 2” at the Graduate School of Innovation Management, Hosei University, fielded by Freeasy; part of it is released for these case studies. Respondents had not answered “not conscious at all” about choosing casual fashion brands in the screening survey, so the sample does not represent consumers in general. The analysis uses the respondents who rated each brand (“don’t know” is excluded).
• Cases: 251 respondents who rated MUJI.
• Conditions (X): the same four perceptions as for UNIQLO (“affordable price range,” “good quality,” “easy to mix and match,” and “offers products that downplay trends” (hereafter, basic)). For each brand, respondents ticked every item they thought applied, and I split the cases into ticked and not ticked. The four conditions used here are an example of how to use the method. I picked them after looking at results, because they suit an illustration. In a real analysis, conditions must be set in advance, based on hypotheses or prior research.
• Consistency: among people holding a combination, the share who recommend (threshold 0.80 or higher). Coverage: among people who recommend, the share explained by the combination.
• Settings (same at every level): minimum number of cases 4; intermediate solution (for logical remainders, I assume that each perception contributes toward recommending, except “basic”, for which I assume no direction).

Results: same conditions, same level moving

(“/” means “or”. “Not X” means X was not ticked. Consistency and coverage in the table are values for the whole solution joined by “/”.)

• For MUJI, sufficient conditions appear only down to 6 or higher. For UNIQLO they appeared from 8 or higher.
• For UNIQLO, at 7 or 8 or higher the sufficient condition was “price × quality × versatile × not basic.” For MUJI, this pattern first appears at 6 or higher.
• The other pattern at 6 or higher is “quality holds while neither price nor versatility is ticked.” Many people recommend on quality alone, even without a price perception.
• Perceptions are distributed differently too: 44% of UNIQLO raters ticked “affordable price range,” against 30% for MUJI.

In short, with the four shared conditions, no solution met the criterion for MUJI at 7 or higher. The share of respondents at 7 or higher also differs (58% for UNIQLO, 46% for MUJI). With the same conditions and settings, the solutions obtained differed by brand.

A marketing reading (as hypotheses)

What the results show is only which combinations of perceptions are held by people who recommend at each level. What follows is marketing interpretation drawn from that, and it remains a hypothesis.

1. Even with the same conditions and settings, how solutions emerge differs. For UNIQLO, solutions appeared from 8 or higher. For MUJI, they appear only from 6 or higher. A view centered on price, quality and versatility, which fit UNIQLO’s recommendations, cannot simply be carried over to MUJI.
2. The way price is perceived differs. The share who ticked “affordable price range” is 44% for UNIQLO and 30% for MUJI. At 6 or higher, a pattern also appears in which quality is enough when price is not ticked. However, this is a difference in shares. It does not mean that people who recommend MUJI ignore price.
3. How “basic” appears at 5 or higher differs between UNIQLO and MUJI. The UNIQLO solutions contained “not basic.” The MUJI solution at 5 or higher contains “basic” itself. However, few MUJI respondents ticked basic (8%; UNIQLO 11%). Part 3 looks at this point again.

Cautions

• These four conditions were chosen as an example for the UNIQLO analysis. With other conditions, the results for MUJI would change.
• “No sufficient condition” does not mean “no reason exists.”
• This is an online survey run as part of a lecture (N = 300), and the sample does not represent consumers in general.

Reproducible code (R)

Data (same folder): fashion_muji.csv

library(QCA); library(ThSQCA)
dat   <- read.csv("fashion_muji.csv")
conds <- c("PRICE", "QUALITY", "VERSATILE", "BASIC")
res <- otSweep(dat = dat, outcome = "REC", conditions = conds,
  sweep_range = 5:9,
  thrX = setNames(rep(1, length(conds)), conds),
  include = "?", dir.exp = ifelse(conds == "BASIC", "-", "1"),
  incl.cut = 0.80, n.cut = 4)
res$summary[, c("thrY", "expression", "inclS", "covS", "n_solutions")]

  thrY                                                 expression     inclS
1    5               BASIC + PRICE*VERSATILE + QUALITY*~VERSATILE 0.9302326
2    6 ~PRICE*QUALITY*~VERSATILE + PRICE*QUALITY*VERSATILE*~BASIC 0.8723404
3    7                                                No solution        NA
4    8                                                No solution        NA
5    9                                                No solution        NA
       covS n_solutions
1 0.4020101           1
2 0.2827586           1
3        NA           0
4        NA           0
5        NA           0

Paper: Toyoda, Y. (2026). Threshold-sweep QCA: threshold dependence as an analytical dimension. Quality & Quantity. https://doi.org/10.1007/s11135-026-03092-3

What to read next

• Part 1: UNIQLO
• Part 3: Swapping conditions with the same MUJI data
• Begin here: ThS-QCA Case Study

R environment for this page

sessionInfo()

R version 4.6.1 (2026-06-24 ucrt)
Platform: x86_64-w64-mingw32/x64
Running under: Windows 11 x64 (build 26300)

Matrix products: default
  LAPACK version 3.12.1

locale:
[1] LC_COLLATE=Japanese_Japan.utf8  LC_CTYPE=Japanese_Japan.utf8   
[3] LC_MONETARY=Japanese_Japan.utf8 LC_NUMERIC=C                   
[5] LC_TIME=Japanese_Japan.utf8    

time zone: Asia/Tokyo
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] ThSQCA_2.0.9 QCA_3.25.5   admisc_0.41 

loaded via a namespace (and not attached):
 [1] digest_0.6.39     fastmap_1.2.0     xfun_0.60         knitr_1.52       
 [5] htmltools_0.5.9   rmarkdown_2.32    cli_3.6.6         venn_1.13        
 [9] declared_0.27     compiler_4.6.1    rstudioapi_0.19.0 tools_4.6.1      
[13] evaluate_1.0.5    yaml_2.3.12       otel_0.2.0        rlang_1.3.0      
[17] jsonlite_2.0.0    htmlwidgets_1.6.4

Method and data (common to the series)

• Method: Threshold-Sweep QCA (ThS-QCA). I move the cut-off (threshold) on “how strongly would you recommend this brand to friends and acquaintances” (0 to 10; 0 = would not recommend at all, 10 = would definitely recommend) and list the sufficient conditions at each cut-off (combinations of conditions whose holders mostly end up recommending the brand; Toyoda, 2026, Quality & Quantity). It does not search for one optimal threshold.
• Data: A casual fashion survey (N = 300) that I ran in June 2026 for the lecture “Marketing 2” at the Graduate School of Innovation Management, Hosei University, fielded by Freeasy; part of it is released for these case studies. Respondents had not answered “not conscious at all” about choosing casual fashion brands in the screening survey, so the sample does not represent consumers in general. The analysis uses the respondents who rated each brand (“don’t know” is excluded).
• Cases: 251 respondents who rated MUJI.
• Conditions (X): brand perceptions (“good quality,” “easy to mix and match,” “easy to walk into a physical store,” and “offers products that downplay trends” (hereafter, basic)). I selected them, from combinations that yield a solution at 7 or higher in MUJI’s data. Compared with part 2, “price” is dropped and “easy to walk into a physical store” is added. For each brand, respondents ticked every item they thought applied, and I split the cases into ticked and not ticked. The four conditions used here are an example of how to use the method. I picked them after looking at results, because they suit an illustration. In a real analysis, conditions must be set in advance, based on hypotheses or prior research.
• Consistency: among people holding a combination, the share who recommend (threshold 0.80 or higher). Coverage: among people who recommend, the share explained by the combination.
• Settings (same at every level): minimum number of cases 4; intermediate solution (for logical remainders, I assume that each perception contributes toward recommending, except “basic”, for which I assume no direction).
• Note: Beyond the items used here, I tried other combinations of four conditions, using other survey items as well. At 9 or higher, no combination yielded a sufficient condition. At 8 or higher, some combinations have an overall solution consistency of 0.80 or higher, but they explain very few people (6 or fewer of 72). This search used items that are not in the public CSV, so it cannot be reproduced with the public CSV.

Results: sufficient conditions with the swapped conditions

(“/” means “or”. “Not X” means X was not ticked. Consistency and coverage in the table are values for the whole solution joined by “/”.) “Easy store” means “easy to walk into a physical store.”

• With the same 251 MUJI respondents, replacing “price” with “easy to walk into a physical store” produced a sufficient condition at 7 or higher, which part 2 could not. Note that this set of conditions was selected from the same data as one that yields a solution at 7 or higher.
• Every sufficient condition at 7 or higher contains quality: a route in which quality is joined by the basic selection, and a route in which quality is joined by versatility and easy to walk into.
• At 5 or higher, quality alone is sufficient.

A marketing reading (as hypotheses)

What the results show is only which combinations of perceptions are held by people who recommend at each level. What follows is marketing interpretation drawn from that, and it remains a hypothesis.

1. Swapping conditions changes the combinations we see. With part 2’s four conditions, no solution appeared at 7 or higher. After the swap, two routes containing quality appeared at 7 or higher. What we put into the analysis changes what we see.
2. How “basic” appears differs by solution. The UNIQLO solutions contained “not basic,” and part of the MUJI solutions contains “basic.” However, the conditions entered differ, so a direct comparison is not possible. Few respondents ticked basic even for MUJI (8%). Also, the survey asked whether the brand “offers products that downplay trends,” and this survey alone cannot tell what that says about the brand.
3. In practice, conditions are set in advance, based on your own hypotheses. Here I use conditions chosen after seeing results, to show that swapping conditions changes the results. In a real analysis, you would run the analysis with conditions set beforehand and show how results differ when conditions change, as a sensitivity analysis.
4. A sufficient condition covers only “part” of the routes to recommendation. 22 respondents hold one of the solution’s routes, and 18 of them score 7 or higher (consistency 0.82). On the other hand, of the 116 respondents who score 7 or higher, only 18 hold one of these routes (about 16%, coverage 0.16).

Cautions

• The conditions are an example chosen after seeing MUJI’s results. Until checked on other data, the findings remain hypotheses.
• “No sufficient condition” does not mean “no reason exists.”
• This is an online survey run as part of a lecture (N = 300), and the sample does not represent consumers in general.

Reproducible code (R)

Data (same folder): fashion_muji.csv

library(QCA); library(ThSQCA)
dat   <- read.csv("fashion_muji.csv")
conds <- c("QUALITY", "VERSATILE", "EASYSTORE", "BASIC")
res <- otSweep(dat = dat, outcome = "REC", conditions = conds,
  sweep_range = 5:9,
  thrX = setNames(rep(1, length(conds)), conds),
  include = "?", dir.exp = ifelse(conds == "BASIC", "-", "1"),
  incl.cut = 0.80, n.cut = 4)
res$summary[, c("thrY", "expression", "inclS", "covS", "n_solutions")]

  thrY                                                         expression
1    5                              QUALITY + BASIC + VERSATILE*EASYSTORE
2    6 QUALITY*~VERSATILE*~EASYSTORE + QUALITY*VERSATILE*EASYSTORE*~BASIC
3    7                        QUALITY*BASIC + QUALITY*VERSATILE*EASYSTORE
4    8                                                        No solution
5    9                                                        No solution
      inclS      covS n_solutions
1 0.9042553 0.4271357           1
2 0.8536585 0.2413793           1
3 0.8181818 0.1551724           1
4        NA        NA           0
5        NA        NA           0

Paper: Toyoda, Y. (2026). Threshold-sweep QCA: threshold dependence as an analytical dimension. Quality & Quantity. https://doi.org/10.1007/s11135-026-03092-3

What to read next

• Part 2: MUJI with UNIQLO’s conditions
• Begin here: ThS-QCA Case Study

R environment for this page

sessionInfo()

R version 4.6.1 (2026-06-24 ucrt)
Platform: x86_64-w64-mingw32/x64
Running under: Windows 11 x64 (build 26300)

Matrix products: default
  LAPACK version 3.12.1

locale:
[1] LC_COLLATE=Japanese_Japan.utf8  LC_CTYPE=Japanese_Japan.utf8   
[3] LC_MONETARY=Japanese_Japan.utf8 LC_NUMERIC=C                   
[5] LC_TIME=Japanese_Japan.utf8    

time zone: Asia/Tokyo
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] ThSQCA_2.0.9 QCA_3.25.5   admisc_0.41 

loaded via a namespace (and not attached):
 [1] digest_0.6.39     fastmap_1.2.0     xfun_0.60         knitr_1.52       
 [5] htmltools_0.5.9   rmarkdown_2.32    cli_3.6.6         venn_1.13        
 [9] declared_0.27     compiler_4.6.1    rstudioapi_0.19.0 tools_4.6.1      
[13] evaluate_1.0.5    yaml_2.3.12       otel_0.2.0        rlang_1.3.0      
[17] jsonlite_2.0.0    htmlwidgets_1.6.4

What is ThS-QCA?

Qualitative Comparative Analysis (QCA) asks which combinations of conditions are sufficient for an outcome. Threshold-Sweep QCA (ThS-QCA) moves the cut-off for the outcome (here, how strongly people would recommend the brand) step by step and lists the sufficient conditions at each step (Toyoda, 2026, Quality & Quantity). In this series, I use ThS-QCA to move the threshold on the outcome side. It does not look for one optimal threshold; it shows how the picture changes with the threshold. It runs with the R package ThSQCA (available on CRAN).

Data

• A casual fashion survey (N = 300) that I ran in June 2026 for the lecture “Marketing 2” at the Graduate School of Innovation Management, Hosei University, fielded by Freeasy. Part of it is released for these case studies (reproducible CSVs: fashion_uniqlo.csv, fashion_muji.csv).
• Respondents had not answered “not conscious at all” about choosing casual fashion brands in the screening survey. The sample does not represent consumers in general.
• Outcome (Y): “How strongly would you recommend this brand to friends and acquaintances?” (0 = would not recommend at all, 10 = would definitely recommend).
• Conditions (X): from items where respondents ticked every item they thought applied to each brand, I take five: “affordable price range,” “good quality,” “easy to mix and match,” “easy to walk into a physical store,” and “offers products that downplay trends” (hereafter, basic). Each analysis uses four of them. Parts 1 and 2 use price, quality, versatility and basic. Part 3 drops price and adds easy to walk into a physical store, so it uses quality, versatility, easy store and basic.

An important note on conditions

The survey has other items, but here I take five items as an example of how to use the method, and each analysis uses four of them. I picked them after looking at results, because they suit an illustration. In a real analysis, conditions must be set in advance, based on hypotheses or prior research.

The parts

1. Part 1, UNIQLO: move the “recommend” level from 9 down to 5 and list the sufficient conditions.
2. Part 2, MUJI with UNIQLO’s conditions: change only the brand, with the same conditions and settings.
3. Part 3, swapping conditions with the same MUJI data: fix the brand and swap the conditions entered into the analysis.

Two findings

1. Changing the recommendation level changes the solution. For UNIQLO, the solutions at 8 or higher contain price, quality and versatility together. When the level is lowered to 5 or higher, shorter combinations appear.
2. Changing the conditions entered into the analysis changes the solution. With the four shared conditions, no solution met the criterion for MUJI at 7 or higher. With the same 251 respondents, replacing price with “easy to walk into a physical store” produced two routes containing quality at 7 or higher.

At 9 or higher, the conditions tried here yield no sufficient condition. This does not mean nothing can be done; the chosen conditions may simply not include the drivers of the highest ratings.

Settings

Minimum number of cases 4; consistency threshold 0.80; intermediate solution (no directional assumption for “basic”).

Paper: Toyoda, Y. (2026). Threshold-sweep QCA: threshold dependence as an analytical dimension. Quality & Quantity. https://doi.org/10.1007/s11135-026-03092-3

Method and data (common to the series)

• Method: Threshold-Sweep QCA (ThS-QCA). I move the cut-off (threshold) on “how strongly would you recommend this brand to friends and acquaintances” (0 to 10; 0 = would not recommend at all, 10 = would definitely recommend) and list the sufficient conditions at each cut-off (combinations of conditions whose holders mostly end up recommending the brand; Toyoda, 2026, Quality & Quantity). It does not search for one optimal threshold.
• Data: A casual fashion survey (N = 300) that I ran in June 2026 for the lecture “Marketing 2” at the Graduate School of Innovation Management, Hosei University, fielded by Freeasy; part of it is released for these case studies. Respondents had not answered “not conscious at all” about choosing casual fashion brands in the screening survey, so the sample does not represent consumers in general. The analysis uses the respondents who rated each brand (“don’t know” is excluded).
• Cases: 267 respondents who rated UNIQLO.
• Conditions (X): brand perceptions (“affordable price range,” “good quality,” “easy to mix and match,” and “offers products that downplay trends” (hereafter, basic)). For each brand, respondents ticked every item they thought applied, and I split the cases into ticked and not ticked. The four conditions used here are an example of how to use the method. I picked them after looking at results, because they suit an illustration. In a real analysis, conditions must be set in advance, based on hypotheses or prior research.
• Consistency: among people holding a combination, the share who recommend (threshold 0.80 or higher). Coverage: among people who recommend, the share explained by the combination.
• Settings (same at every level): minimum number of cases 4; intermediate solution (for logical remainders, I assume that each perception contributes toward recommending, except “basic”, for which I assume no direction).

Results: sufficient conditions from 9 down to 5

(“/” means “or”. “Not X” means X was not ticked. Consistency and coverage in the table are values for the whole solution joined by “/”.) “Not basic” means that “offers products that downplay trends” was not ticked.

• At 5 or higher, as long as the brand is not seen as basic, any one of price, quality or versatility is enough (when all three hold together, basic does not matter). The conditions are loose and take many forms.
• At 7 or 8 or higher, the sufficient condition is that, with the brand not seen as basic, price, quality and versatility all hold.
• At 9 or higher, these four conditions yield no sufficient condition.

The higher the target, the stricter the sufficient condition becomes: from “any one of them” to “several together.”

A marketing reading (as hypotheses)

What the results show is only which combinations of perceptions are held by people who recommend at each level. What follows is marketing interpretation drawn from that, and it remains a hypothesis.

1. The lower the level, the shorter the combinations that appear in the solution. The solutions at 8 or higher (41% of respondents) contained price, quality and versatility together. When the level is defined as 5 or higher (86%), shorter combinations appear: with the brand not seen as basic, any one of price, quality or versatility is part of a solution. People who recommend at high scores appear to hold several perceptions at once.
2. “Not basic” appears in the solutions at every level. Not ticking “offers products that downplay trends” is part of every solution. However, this survey alone cannot tell what not ticking this item says about how people see the brand. It cannot be read as “they feel novelty or trendiness.”
3. A sufficient condition covers only “part” of the routes to recommendation. 26 respondents hold “price × quality × versatile × not basic,” and 23 of them score 7 or higher (consistency 0.88). On the other hand, of the 154 respondents (58%) who score 7 or higher, only 23 hold this combination (about 15%, coverage 0.15). Most people who score 7 or higher recommend through other routes. Actions that only satisfy this combination will not necessarily increase recommenders by much.

Cautions

• “No sufficient condition” does not mean “no reason exists.” The chosen conditions may simply not include the drivers of the highest ratings. With other conditions, the results would change.
• The four conditions are an example (in a real analysis they are set in advance, as noted above).
• This is an online survey run as part of a lecture (N = 300), and the sample does not represent consumers in general.

Reproducible code (R)

Data (same folder): fashion_uniqlo.csv

library(QCA); library(ThSQCA)
dat   <- read.csv("fashion_uniqlo.csv")
conds <- c("PRICE", "QUALITY", "VERSATILE", "BASIC")
res <- otSweep(dat = dat, outcome = "REC", conditions = conds,
  sweep_range = 5:9,
  thrX = setNames(rep(1, length(conds)), conds),
  include = "?", dir.exp = ifelse(conds == "BASIC", "-", "1"),
  incl.cut = 0.80, n.cut = 4)
res$summary[, c("thrY", "expression", "inclS", "covS", "n_solutions")]

  thrY
1    5
2    6
3    7
4    8
5    9
                                                                  expression
1 PRICE*~BASIC + QUALITY*~BASIC + VERSATILE*~BASIC + PRICE*QUALITY*VERSATILE
2                         PRICE*VERSATILE*~BASIC + QUALITY*~VERSATILE*~BASIC
3                                             PRICE*QUALITY*VERSATILE*~BASIC
4                                             PRICE*QUALITY*VERSATILE*~BASIC
5                                                                No solution
      inclS      covS n_solutions
1 0.9263804 0.6565217           1
2 0.8860759 0.3888889           1
3 0.8846154 0.1493506           1
4 0.8076923 0.1926606           1
5        NA        NA           0

Paper: Toyoda, Y. (2026). Threshold-sweep QCA: threshold dependence as an analytical dimension. Quality & Quantity. https://doi.org/10.1007/s11135-026-03092-3

What to read next

• Part 2: MUJI with UNIQLO’s conditions
• Begin here: ThS-QCA Case Study

R environment for this page

sessionInfo()

R version 4.6.1 (2026-06-24 ucrt)
Platform: x86_64-w64-mingw32/x64
Running under: Windows 11 x64 (build 26300)

Matrix products: default
  LAPACK version 3.12.1

locale:
[1] LC_COLLATE=Japanese_Japan.utf8  LC_CTYPE=Japanese_Japan.utf8   
[3] LC_MONETARY=Japanese_Japan.utf8 LC_NUMERIC=C                   
[5] LC_TIME=Japanese_Japan.utf8    

time zone: Asia/Tokyo
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] ThSQCA_2.0.9 QCA_3.25.5   admisc_0.41 

loaded via a namespace (and not attached):
 [1] digest_0.6.39     fastmap_1.2.0     xfun_0.60         knitr_1.52       
 [5] htmltools_0.5.9   rmarkdown_2.32    cli_3.6.6         venn_1.13        
 [9] declared_0.27     compiler_4.6.1    rstudioapi_0.19.0 tools_4.6.1      
[13] evaluate_1.0.5    yaml_2.3.12       otel_0.2.0        rlang_1.3.0      
[17] jsonlite_2.0.0    htmlwidgets_1.6.4

1. Terms: the issue is not “SEM or not”

First, let me align the terms. If they stay vague, the later discussion will not connect.

SEM (structural equation modeling) is the name of a broad framework that expresses and estimates the relations among several variables as one unified model. Several kinds of models belong to it.

• Models with observed variables only (path analysis): These use only variables that were actually measured (including subscale scores built from them) and estimate the relations among them. (Multiple regression can be seen as a special case of path analysis.) In other words, path analysis is also part of SEM in the broad sense.
• Models with latent variables: A construct that cannot be measured directly is placed in the model as a “latent variable,” and several observed items are assumed to measure it. The part of the model that links items to a latent variable is called the measurement model. When relations among latent variables are added, we also handle the structural model (the part that describes the relations among constructs).
• Confirmatory factor analysis (CFA) is one kind of latent-variable model. It contains latent variables, but it may have no structural model.

For this reason, I avoid the phrase “use SEM or not” in this article. Saying it invites the fair objection that path analysis is also a kind of SEM. The issue of this article is one point.

Whether to put latent variables into the model.

A method that builds scales in advance and then runs regression or path analysis with subscale scores does not put latent variables into the model. In this article I call it the “observed-variable model.” A method that does put latent variables in is called the “latent-variable model.”

Note that “not putting latent variables in” does not mean that factor analysis is not used when you examine a scale. Factor analysis itself is a latent-variable model. The article distinguishes two things. One is a method in which factor analysis and similar tools are used at the measurement stage to build a scale, and the resulting scores are then treated as observed variables in regression or path analysis. The other is a method in which the measurement model is built into the structural model, and the relations among latent variables are estimated.

In this article I also write “association” for the links between variables. Arrows in path analysis and regression coefficients do not show causal relations when the data are cross-sectional survey data (observational data). This is the same whether the model has latent variables or only observed variables.

2. Why does “latent variables for everything” happen so easily?

From what I have seen and heard, I think the reasons fall roughly into the following three (this is my observation, not the result of a survey).

1. Because you see it in papers. Academic papers often use models with latent variables. They look like “proper academic analysis,” so people want to imitate them.
2. Because you think that constructs exist. You are dealing with a concept such as “brand trust,” so making it a latent variable feels natural. But having a construct and using a latent variable in the analysis are different things. As the previous article explained, defining a construct first, measuring it with several items, checking reliability, and using subscale scores is also a way of taking constructs seriously.
3. Because software makes it easy. With software such as AMOS, you draw a diagram, press a button, and get results. (If you wrote out the structural equations, you would see the complexity, but drawing the model as a diagram makes it look simple.) Being able to run a model and being able to explain its results are different things.

None of these reasons comes from the research question. The method comes first. I would like the order to be the opposite: first the question, then the conditions of measurement, and only after that the choice of method.

3. What increases when latent variables are added?

Adding latent variables increases the expressive power of the model. At the same time, the number of things you must judge yourself increases.

4. When latent variables are needed and when they can be avoided

This section is the center of the article. It is also probably the part most open to debate. For that reason, I write the judgment in two stages.

Stage 1: Is there a need to use latent variables? Stage 2: Can the model be estimated and interpreted properly?

These are different questions. Stage 1 asks “is there a reason to use them?” and Stage 2 asks “are the conditions for using them in place?” For example, having a sufficient sample size is a Stage 2 condition. A sufficient sample size does not become a Stage 1 “reason to use.” Conversely, even if there is a reason at Stage 1, it is difficult to use the model if the Stage 2 conditions are not in place.

5. How I recommend beginners proceed (my view)

Let me summarize the discussion so far as advice for beginners.

1. First, build a scale and check it. Define the construct from theory, prepare the items, check them with factor analysis and alpha, and turn them into subscale scores. The details of this procedure are in the previous article.
2. Next, analyze with an observed-variable model. Use the subscales to run regression or path analysis. Because the results of each stage can be checked separately, this is a method in which mistakes are easy to notice and easy to explain. However, because subscale scores contain measurement error, the coefficients of the structural part can come out smaller than they should be, or biased (Question A in Section 4). Thinking about whether this effect could change the conclusion leads to the judgment in step 3.
3. Move to a latent-variable model once the need is visible. It is not too late to do so after a reason becomes clear in the research, such as wanting to take the effect of measurement error into account explicitly or wanting to check measurement invariance.

There is one thing I want to emphasize. “Being able to use a latent-variable model” and “being in a study that should use one” are different things.

There are also intermediate options between “estimate everything at once” and “do everything separately.” They are the two-step procedure that checks the measurement part first (Anderson & Gerbing, 1988) and methods that estimate measurement and structure separately (such as SAM; Rosseel & Loh, 2024).

6. Reference: for readers who want to know more

This section is a reference for readers who want to know more. It collects viewpoints for checking things by yourself before you use a latent-variable model.

The point of this section is not that “latent-variable models are bad.” It is that knowing what can happen when you use them without understanding their mathematical properties is useful.

Closing

To repeat, a latent-variable model is a powerful tool. Because it is powerful, I think it is good to use it in a state in which you can explain why you use it.

For beginners, I recommend the path of building and checking a scale and then analyzing with an observed-variable model. As you accumulate that experience, the situations in which a latent-variable model becomes necessary, such as “here I want to take measurement error into account explicitly” and “here I want to examine measurement itself,” will come into view naturally. That is the next step.

References cited in the text

• Anderson, J. C., & Gerbing, D. W. (1988). Structural equation modeling in practice: A review and recommended two-step approach. Psychological Bulletin, 103(3), 411–423. https://doi.org/10.1037/0033-2909.103.3.411
• Burt, R. S. (1976). Interpretational confounding of unobserved variables in structural equation models. Sociological Methods & Research, 5(1), 3–52. https://doi.org/10.1177/004912417600500101
• MacCallum, R. C., Wegener, D. T., Uchino, B. N., & Fabrigar, L. R. (1993). The problem of equivalent models in applications of covariance structure analysis. Psychological Bulletin, 114(1), 185–199. https://doi.org/10.1037/0033-2909.114.1.185
• Rosseel, Y., & Loh, W. W. (2024). A structural after measurement approach to structural equation modeling. Psychological Methods, 29(3), 561–588. https://doi.org/10.1037/met0000503

Recommended further reading

These are not works cited in the text. They are introductory books and guides that I think are worth reading if you use SEM. The level and suitability of each book are my own impressions.

Overview

Quantitative research, which is built on statistics and aims to learn “average effects,” and qualitative research (case study research), which aims to pin down “the cause of a particular event,” ask different questions in the first place. Relatively speaking, quantitative research asks how much a cause affects outcomes on average, whereas qualitative research asks why this outcome happened in this case. The paper shows that the practices of both traditions are reasonable given their respective goals.

Why I chose this paper

In MBA research and business research, you may run into this wall: A/B tests and regression analysis reveal the overall trend, yet you still cannot tell why your single most important customer left.

This paper offers a way to think about why that happens. Measuring the “average trend” across the data and tracing the “causal mechanism” of an individual event are two different ways of thinking. The paper can be read as a “map of thinking” for choosing an analytical method that fits the question you want to answer. I chose it because I believe it is one of the papers MBA students should read before starting their own research.

The structure of the paper

The paper contrasts the two traditions across 10 areas (approaches to explanation, conceptions of causation, multivariate explanations, equifinality, scope and causal generalization, case selection practices, weighting observations, substantively important cases, lack of fit, and concepts and measurement). In this memo, to make the overall picture easier to grasp, I narrow them down to three: the purpose of the question, how causation is understood, and how cases are selected. For the other areas, please check the paper itself.

Reading the table from left to right, you can see that the differences between the two cultures are connected in a single line. The paper likens qualitative research to a detective. Quantitative research, by contrast, is explained as statistical methods that take the logic of the controlled experiment as their model. The phrase “the experimenter’s view” in this article is simply a label I use for this contrast.

A detective wants to know the cause of a crime that has already happened, so the detective examines in depth the cases in which it happened. The detective also assumes that the cause is not one thing but a combination of several conditions.

An experimenter, on the other hand, wants to know the average effect of a factor, and therefore takes care not to select cases by their outcome, which would introduce bias. The experimenter also thinks of the effects of the factors as adding up.

If the goals differ, then both how cases are selected and how causation is understood differ as well.

The paper’s contributions

First, the paper makes the difference in the “goal of the question” clearly visible. The question “why did this case happen?” and the question “what effect does this factor have on average?” are different questions. Much of the reason why debates about analytical methods fail to connect lies not in which method is better, but in this difference in purpose.

Second, the paper shines a light on “combinations of conditions.” A factor may not work on its own; sometimes an outcome appears only “when conditions A, B, and C are all present.” Moreover, there may be more than one route to the same outcome (this is called equifinality). The paper organizes, in a logical way, this feature of the real world that is easy to overlook in models that simply add effects together.

From here on, these are my own views, based on the paper.

What I found noteworthy

What I consider especially important about this paper is that it lets us understand qualitative and quantitative research by contrasting them. Seeing the two side by side makes clear what each aims for and why each works the way it does.

Another point is that it can serve as a foundation for understanding QCA as a method. If you grasp the logic of qualitative research (combinations of conditions, multiple routes), you can see what QCA has inherited.

Thinking about business applications: the case of customer churn

Let us step away from the paper and think about applications. Consider an e-commerce company analyzing customer churn (the numbers are a made-up example to aid understanding).

Quantitative approaches such as statistics and A/B tests are well suited to revealing the overall trend, for example “issuing a coupon lowers the churn rate across all customers by 3% on average.” However, this alone does not reveal the individual reason why Customer A, who had spent the most with the company, suddenly stopped using the service.

This is where a qualitative, logic-based approach comes in. It can bring out candidate combinations: churn tends to occur when several conditions overlap, such as “a delivery delay,” “dissatisfaction with the support response,” and “a competitor’s discount campaign.” However, this does not establish the cause; it is material for hypotheses to be checked later.

Connection to QCA

This paper organized the differences between the qualitative and quantitative cultures. One method that connects the two is QCA (qualitative comparative analysis). It keeps the qualitative way of thinking, which emphasizes combinations of conditions and multiple routes, while analyzing a dataset systematically. I also discuss the ideas behind QCA in the article Why QCA suits decision-making on this blog.

Summary points for review

Points to revisit

• Before starting an analysis, decide whether your question is “why did this case happen?” or “how does this factor work on average?”
• Once the question is decided, how to select cases and how to understand causation follow naturally.
• When you study QCA, return to this paper’s ideas of “combinations of conditions” and “multiple routes.”

Glossary

• Causes-of-effects: A detective-style question that works backward from an outcome: “Why did this happen?”
• Effects-of-causes: An experimenter-style question that measures the average effect of adding a factor: “What effect does this factor have on average?”
• Equifinality: The fact that there is not just one route to the same outcome, but several.

References

Mahoney, J., & Goertz, G. (2006). A tale of two cultures: Contrasting quantitative and qualitative research. Political Analysis, 14(3), 227-249. https://doi.org/10.1093/pan/mpj017

Back to the paper guides: About this shelf

Overview

　Suppose that many successful firms turn out to be customer-oriented. Can we then say that being thoroughly customer-oriented leads to success? Ragin deals with the logical problem behind this question.

　Charles C. Ragin is a sociologist known as the originator of QCA. The starting point of this paper is a question: do the relations stated in a theory match the relations we actually check in an analysis?

　Take the claim that “democracies do not fight wars with each other.” This claim states a set relation: the pairs of democracies form a subset of the pairs of countries that do not go to war. On the other hand, there are many pairs of countries that avoid war without both being democracies, so the correlation between “being a pair of democracies” and “not going to war” may not be strong. That does not, however, refute the original claim.

　On this basis, Ragin points out that theories stated as set relations tend to be translated, in analysis, into hypotheses about correlation. He then presents simple descriptive measures for evaluating set relations in the form in which they were stated.

Why I chose this paper

　In MBA research, we sometimes want to analyze how a certain event comes about, for example the factors behind a company’s success. However, what successful companies have in common cannot always be treated as a law of success.

　In business cases, we come across explanations such as “successful firms are thoroughly customer-oriented” or “high-performing stores offer good service.” To judge what can be concluded from them, we need to distinguish the following three questions.

• Do firms that have the condition succeed?
• Is the condition indispensable for success?
• How much of the success we observe can this condition explain?

　The value of reading this paper is not limited to learning QCA’s measures. It also helps you clarify what a “success factor” means and look for evidence that fits the claim you are making. Whether or not you use QCA, I recommend it as a paper that conveys a perspective every MBA student should have.

The structure of the paper

　The paper discusses two measures for evaluating set relations (consistency and coverage), including how to use them. In this memo, I narrow the discussion to four points.

1. Correlation and set relations answer different questions

　Correlation looks at how two variables move together. A set relation looks at whether the set of cases that have a condition is contained in the set of cases that achieve the outcome (or the reverse). Let the condition be and the outcome be . Sufficient and necessary conditions can then be distinguished as follows.

Relation	Question	As sets
is sufficient for	If X is present, is Y present?
is necessary for	If Y is present, is X present?

　As an example, consider the claim that “stores that combine a good location with excellent service are high performers.” This is a claim of sufficiency: the set of stores with this combination is contained in the set of high-performing stores. The claim says only whether the stores with the combination are high performers. It does not say the reverse, that every high-performing store has this combination. For instance, some stores may have an ordinary location yet perform well thanks to a distinctive product mix and a strong brand. Such stores simply lie outside the claim; they do not contradict it. (The stores and numbers in this article are a made-up example for illustration. Only the figure 17/20 = 0.85 is chosen to match an example in the paper.)

　A case that contradicts the claim is “a store with a good location and excellent service that is not a high performer.” To check a claim of sufficiency, we look at how many such stores there are.

• Suppose 20 stores have the combination, and 17 of them are high performers while 3 are not. These 3 stores are the cases that contradict the claim.
• Suppose there are 50 high-performing stores in total, and 17 of them have the combination. The remaining 33 stores achieve high performance without this combination. They do not contradict the claim.

　The first ratio (17/20) leads to the idea of “consistency” explained next. The second (17/50) leads to “coverage” after that.

The paper’s contributions

　First, the paper presents simple descriptive measures for evaluating set relations. The paper itself says that consistency and coverage are neither complicated nor entirely new. What is new, it says, is that they are clearly tied to set-theoretic thinking, which shows how this differs from ordinary quantitative analysis built on correlation.

　Second, the paper argues that for necessary conditions, too, consistency alone is not enough. A condition that is present in most cases regardless of the outcome can have high consistency as a necessary condition and still contribute little to understanding the outcome. The coverage of a necessary condition is a clue for spotting such trivial conditions. This point builds on Goertz’s discussion.

　Third, the paper separates the coverage of multiple paths into overlapping and unique parts. This makes it possible to evaluate the diversity of patterns of success and the weight of each path.

From here on, these are my own views, based on the paper.

What I found noteworthy

　What I consider especially important is the order: check consistency first, then interpret coverage. When the consistency of a sufficient condition is weak, we cannot call a pattern a strong pattern of success merely because it covers the outcome broadly.

　Another point is the stance emphasized in the conclusion: the measures themselves should not become the goal of research. Ragin treats consistency and coverage as aids to judgment, and gives priority to whether the results help us understand individual cases and the processes and mechanisms at work in them. When studying business cases in an MBA program, I think this stance of moving back and forth between the numbers and the substance of the cases is important.

Thinking about business applications: analyzing success factors with care

　For example, when comparing high-performing stores, before settling on a single explanation such as “low prices matter most,” we can examine different combinations, such as “a good location and excellent service” and “a strong brand and a distinctive product mix” (a made-up example). For each combination, we then check how consistently it is linked to high performance, how many of the high-performing stores it covers, and which stores are the exceptions.

　However, high consistency does not guarantee a causal effect, in the sense that adopting the combination will raise performance. Large coverage does not directly translate into investment priorities either. To decide on a measure, we also need to consider cost, feasibility, the causal mechanism, and more.

Connection to QCA

　When reading the results of a QCA, we check not only the expression of the combinations of conditions but also consistency, coverage, and the specific cases behind them. Beyond “was the value high?”, being able to explain “consistency of what?”, “coverage with what as the denominator?”, and “what are the exceptions?” is what shows real understanding of the analysis. I also introduce the ideas behind QCA in the article Why QCA suits decision-making on this blog.

Summary points for review

Points to revisit

• Distinguish sufficient from necessary conditions. “If the condition is present, the outcome is present” and “if the outcome is present, the condition is present” are different claims.
• Distinguish consistency from coverage. For a sufficient condition, the former is how consistent the relation is, and the latter is how much of the outcome it covers. Check consistency first, then look at coverage.
• Do not judge by the consistency value alone. In its discussion of crisp sets, the paper says that below 0.75 it becomes hard even to claim a rough set relation. This does not mean that 0.75 or above is automatically adequate; you still need to check the number and the substance of the cases.
• For necessary conditions, the denominators switch (see the table below).
• The measures are descriptive evaluations. They do not prove causal effects.

　In the table below, let .

Relation evaluated	Consistency	Coverage
is sufficient for
is necessary for

　The coverage of a necessary condition reads differently from the coverage of a sufficient condition, which asks how much of the outcome is covered. It is a measure for checking that the set of X, the supposed necessary condition, is not too large compared with the set of the outcome Y, so that X does not turn out to be an empirically trivial condition.

Glossary

• QCA (qualitative comparative analysis): A method that treats each case as a combination of conditions and examines its relation to the outcome through set relations and logical comparison.
• Condition: In QCA, the term for the explanatory side ( in this article).
• Outcome: In QCA, the term for the result to be explained ( in this article). In this article I often write simply “outcome” or “result” for readability.
• Crisp set: A set in which membership is expressed as 0 or 1.
• Fuzzy set: A set in which membership is expressed as a degree from 0 to 1. The values are calibrated according to the meaning of the concept.
• Consistency: How consistently the cases follow the assumed set relation.
• Coverage: The empirical importance of a consistent relation. For a sufficient condition, how much of the outcome it covers.
• Equifinality: The fact that different combinations of conditions lead to the same outcome.
• Raw / unique coverage: The total range covered by a path, and the part of it that no other path covers.
• Trivial necessary condition: A condition that is widespread regardless of the outcome; it may be consistent as a necessary condition but does not help us understand the outcome.

What to check in the original

　This article is my own summary of the paper’s main points. When you read the original, the following three places make good entry points.

• Introduction (pp.291-292): What goes wrong when a theory stated as a set relation is translated into a correlation hypothesis?
• The explanation of Figures 1 and 2 (pp.294-297): How do set relations show up in a scatterplot? Why are cases that look like “errors” from a correlation viewpoint not counterexamples to the claim?
• Conclusion (pp.309-310): Why is understanding the cases considered more important than the height of the measures?

References

Ragin, C. C. (2006). Set relations in social research: Evaluating their consistency and coverage. Political Analysis, 14(3), 291-310. https://doi.org/10.1093/pan/mpj019

Back to the paper guides: About this shelf

Purpose of this shelf

The purpose is to let you grasp, in a short time, what a paper says and where its important points lie before you read it. The guides are also written so that I can use them later as my own reading notes.

Papers covered

I choose papers related to the topics of this blog: QCA methodology, the handling of thresholds, the foundations of data analysis, applications to marketing, and so on.

How each article is organized

Every article follows the same order.

• Reference and summary
• Overview
• Why I chose this paper
• The structure of the paper
• The paper’s contributions
• What I found noteworthy
• Points to revisit, glossary, and references

A section on business applications and a section on connections to related articles are added only when needed.

A note for readers

• Each article is written in my own words, without reproducing the text or the figures and tables of the paper.
• The interpretations are based on my own understanding. If you cite a paper or use it in your research, please always check the original.

Published paper guides

• Paper Guide: Quantitative and Qualitative Research Are Two Cultures, Not Rivals (Mahoney & Goertz, 2006)
• Paper Guide: How Should We Evaluate the “Conditions for Success”? (Ragin, 2006)

The question this article answers

Research in marketing and management deals with concepts such as brand trust, perceived quality, customer satisfaction, and price sensitivity, which are hard to measure with a single question. These concepts are called constructs, and we measure them with several questionnaire items.

Suppose we measure brand trust with these three items:

• Q1: This brand is sincere
• Q2: I think this brand keeps its promises
• Q3: I feel secure using this brand

What we want to know is not the effect of Q1, Q2, or Q3 separately. We want to know how brand trust, the three items taken together, relates to outcomes such as purchase intention. We also often want to use such constructs in later analyses: regression, logistic regression, QCA, cluster analysis, decision trees, and so on.

Structural equation modeling (SEM) is a wonderful method, but you do not have to start with it (and if you use it without understanding its characteristics, it can do more harm than good). What matters is not which method is more advanced, but how you measure the constructs of your research, and how you analyze them in the way that best fits your research question.

This article presents a procedure that works well for MBA research and empirical studies, with an example in R. The example data are generated with random numbers (they are not real survey data).

The overall flow

1. Define the construct from theory and prior research
2. Prepare several questionnaire items for each construct
3. Use exploratory factor analysis (EFA) to examine, in an exploratory way, whether the item groupings you expected are consistent with the data
4. Align the direction of reverse-keyed items
5. Check internal consistency (Cronbach’s alpha)
6. Build subscales by averaging items
7. Use the subscales in regression and QCA

Let us go through them in order.

Step 1: Define the construct from theory (before factor analysis)

This is the most important step. Before factor analysis, the researcher has to decide what to measure. Concepts such as “brand trust” and “perceived quality” are not things that statistical software finds for you automatically.

The approach to avoid looks like this:

“Let’s ask 20 questions for now, run a factor analysis, and name the factors that come out.”

This is a misunderstanding that beginners especially tend to fall into. The preferred order is as follows.

From prior research and theory, define what brand trust is, and think about the questionnaire items needed to observe it. When you prepared Q1 to Q3 above, you already held a theoretical expectation before the survey began: “these three items should be measuring one construct, brand trust.” You then run a factor analysis to check whether these three items actually move in the same direction in the real response data.

We do not build theory from factor analysis; we examine, in an exploratory way, whether the data are consistent with a measurement structure that we assumed from theory. This is why research on prior studies and theory beforehand is so important.

Step 2: Prepare several items for each construct

As a rule, prepare several items for each construct. Do not line up items that merely rephrase the same sentence. Items with the same meaning tend to raise Cronbach’s alpha, but this does not mean that the construct is covered adequately. It is desirable for the items to represent different aspects of the construct appropriately.

For example, compare these two ways of measuring brand trust.

An example of rephrasing

• This brand is trustworthy
• This brand is reliable
• This brand is dependable

All three ask almost the same thing. The answers will be similar, so alpha will be high. But if you duplicate content to raise alpha, you can no longer tell whether the necessary aspects of brand trust are covered. Whether they are covered has to be examined separately. Note that when the concept you want to measure is narrow, items with close meanings can be appropriate, and they can also help average out errors caused by differences in wording.

An example of different aspects

• I think this brand keeps its promises (the aspect of fulfilling promises)
• This brand is sincere (the aspect of caring about the other party)
• I feel secure using this brand (the aspect of reassurance in use)

All three are part of “brand trust,” but each looks at a different aspect, so the concept is captured more broadly. However, if you vary the aspects too much, the correlations among items fall and the items no longer form one factor. In light of the definition of the construct, choose items within the range where you can say they are “different aspects of the same concept.”

Disagree  1 - 2 - 3 - 4 - 5 - 6 - 7  Agree

If a scale has been used widely in prior research, using the existing scale is often better than writing items from scratch, because its validity has been examined and it is easy to compare with other studies. This does not mean, however, that “the existing study used three items, so we can use them as they are.” If the industry, culture, respondents, or purpose of the survey differ, the same factor structure may not be reproduced. Even with an existing scale, it is worth checking the structure in your own data.

Example data

From here we follow the steps in R. Suppose we measured nine items on a 7-point scale. The theoretical expectation is the following three constructs.

The wordings are examples (the data are generated with random numbers, and no actual survey corresponds to these wordings).

The outcome variable Y is purchase intention (one item on a 7-point scale).

The following code creates the example data (the random seed is fixed, so you get the same data).

set.seed(2026)
n   <- 300
Phi <- matrix(c(1.0, 0.5, -0.2,
                0.5, 1.0, -0.1,
               -0.2,-0.1,  1.0), 3, 3, byrow = TRUE)   # correlations among the 3 constructs
Fm  <- matrix(rnorm(n * 3), n) %*% chol(Phi)
lam <- c(.80,.75,.70,  .78,.72,.68,  .75,.70,.65)       # item loadings (the "true" values)
fac <- rep(1:3, each = 3)
to7 <- function(z) as.integer(cut(z, c(-Inf,-1.6,-1.0,-0.4,0.4,1.0,1.6,Inf)))  # make 7-point responses
X <- sapply(1:9, function(j) to7(lam[j] * Fm[, fac[j]] + sqrt(1 - lam[j]^2) * rnorm(n)))
X[, 6] <- 8L - X[, 6]                                    # record Q6 as a reverse-keyed item
colnames(X) <- paste0("Q", 1:9)
b <- c(.45, .30, -.20)                       # effects of the 3 constructs on purchase intention
v <- drop(t(b) %*% Phi %*% b)                # variance of the signal part (including correlations among the constructs)
Y <- to7(drop(Fm %*% b) + sqrt(1 - v) * rnorm(n))   # add error so that the total variance is 1
dat <- data.frame(X, Y = Y)
head(dat)

  Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Y
1  5  5  3  3  3  3  4  4  6 6
2  2  4  2  3  3  5  5  4  5 3
3  4  4  3  3  4  5  5  6  3 4
4  5  4  4  4  5  1  5  4  4 3
5  4  4  4  4  2  7  3  4  4 4
6  2  1  1  2  2  6  4  5  5 1

Step 3: Examine how the items group, with exploratory factor analysis

Once the items are ready, examine how they group by using exploratory factor analysis (EFA). The role we expect of EFA here is not “to invent constructs from a state of knowing nothing.” It is to examine, in an exploratory way, whether the item structure we assumed from theory is consistent with the actual data. EFA is different from confirmatory factor analysis (CFA), which specifies in advance as a model which item measures which factor and tests it. The result of EFA is material for judging consistency with your expectation; it does not mean that the measurement structure has been verified.

First, as a guide to the number of factors, look at the eigenvalues of the correlation matrix and at parallel analysis. The count of eigenvalues of 1 or more is only a rough guide. Parallel analysis compares the eigenvalues of the real data with those obtained from random data, and uses the comparison as a clue to the number of factors. Decide the number of factors together with your theoretical expectation. Then extract three factors by principal axis factoring with promax rotation.

library(psych)
items <- paste0("Q", 1:9)
round(eigen(cor(dat[, items]))$values, 3)

[1] 2.918 1.921 1.107 0.640 0.584 0.528 0.490 0.462 0.350

set.seed(2026)
fa.parallel(dat[, items], fa = "fa", fm = "pa")   # parallel analysis (a figure appears)

Parallel analysis suggests that the number of factors =  3  and the number of components =  NA 

fa3 <- fa(dat[, items], nfactors = 3, fm = "pa", rotate = "Promax",
          min.err = 1e-8, max.iter = 1000)   # strict convergence criteria for the iterations

Loading required namespace: GPArotation

print(fa3$loadings, cutoff = 0.3)

Loadings:
   PA1    PA3    PA2   
Q1         0.731       
Q2         0.724       
Q3         0.582       
Q4  0.769              
Q5  0.775              
Q6 -0.632              
Q7                0.695
Q8                0.674
Q9                0.639

                PA1   PA3   PA2
SS loadings    1.62 1.430 1.350
Proportion Var 0.18 0.159 0.150
Cumulative Var 0.18 0.339 0.489

round(fa3$Phi, 2)

     PA1   PA3   PA2
PA1 1.00  0.54  0.01
PA3 0.54  1.00 -0.20
PA2 0.01 -0.20  1.00

Here is what to look at.

• Number of factors: Look at the eigenvalues (the number of those at 1 or more) and the parallel analysis figure (the number of factors whose real-data eigenvalues exceed those from random data), and check whether they are consistent with the three factors we expected from theory.
• Item groupings: Check whether Q1 to Q3, Q4 to Q6, and Q7 to Q9 load on the same factor, as expected. (Labels such as PA1 only indicate the order. Read the meaning of a factor from the items that load on it.)
• The negative loading of Q6: Q6 has the opposite sign to Q4 and Q5 within the same factor. This is a clue that Q6 may be a reverse-keyed item. However, the sign is only a guide (the sign of a whole factor can be flipped). Whether an item is reverse-keyed is decided by the meaning of the wording.
• Whether any item loads strongly on several factors (cross-loading).

For example, if only Q3 loaded strongly on the perceived quality factor, consider the following before deleting it:

• Was the wording of Q3 appropriate?
• Are brand trust and perceived quality conceptually too close?
• Is there a problem with the translation or the phrasing?
• Did respondents read it with a meaning different from the one we expected?

Use the result of EFA not as a “keep or delete verdict,” but as material for understanding the gap between the measurement model assumed from theory and the actual response behavior. If you mechanically delete a content-wise important item only because its loading is a little low, you may damage content validity.

Step 4: Align the direction of reverse-keyed items

Q6 is a reverse-keyed item. In practice, it is easier to follow if you decide the reverse-keyed items from the meaning of the wording before the analysis and align their direction in advance. In this article, we ran the EFA first, to show that a negative loading can be a clue. Use the minimum and maximum of the scale (here 1 and 7) and compute (minimum + maximum) − original value, that is, 8 - Q6. For 1 to 7, use 8 - Q6; for 1 to 5, use 6 - Q6. The same formula also works for a scale whose minimum is not 1 (for example, 0 to 10). Use the ends of the scale on the questionnaire (1 and 7 for a 7-point scale), not the minimum and maximum observed in the data. Whether an item is reverse-keyed is decided by the meaning of the wording (theory).

dat$Q6r <- 8 - dat$Q6

Step 5: Check internal consistency (Cronbach’s alpha)

Once you have examined the factor structure, check the internal consistency of each item group. A typical index is Cronbach’s alpha. For example, indicates that the item group has relatively high internal consistency.

a_trust   <- alpha(dat[, c("Q1", "Q2", "Q3")],  check.keys = FALSE)
a_quality <- alpha(dat[, c("Q4", "Q5", "Q6r")], check.keys = FALSE)
a_price   <- alpha(dat[, c("Q7", "Q8", "Q9")],  check.keys = FALSE)
round(c(trust   = a_trust$total$raw_alpha,
        quality = a_quality$total$raw_alpha,
        price   = a_price$total$raw_alpha), 3)

  trust quality   price 
  0.731   0.774   0.705 

There are two points to note.

• A high alpha does not prove a one-factor structure. Alpha is an index of internal consistency, not an index for verifying factor structure. Also, alpha alone cannot tell you about reliability in general or about validity. The order is: examine the structure by factor analysis, then check the internal consistency of that item group.
• Alpha tends to rise as the number of items grows. If needed, you may use McDonald’s omega alongside it.

Step 6: Build the subscales

Once you have examined the structure and checked the internal consistency, build a subscale for each construct. Usually the simple mean of the items is the default.

dat$Trust   <- rowMeans(dat[, c("Q1", "Q2", "Q3")])
dat$Quality <- rowMeans(dat[, c("Q4", "Q5", "Q6r")])
dat$Price   <- rowMeans(dat[, c("Q7", "Q8", "Q9")])
round(cor(dat[, c("Trust", "Quality", "Price")]), 2)

        Trust Quality Price
Trust    1.00    0.43 -0.14
Quality  0.43    1.00  0.01
Price   -0.14    0.01  1.00

A scale built from a simple mean still contains measurement error. An ordinary regression that uses such a scale does not model the measurement error explicitly. Therefore, the coefficients you obtain are not necessarily the same as the relationships among the latent constructs themselves. With low reliability, this effect can be larger. In multiple regression especially, the effect of measurement error also depends on the correlations among the predictors, so you cannot judge the effect on the coefficients from reliability alone.

A simple mean has the following advantages:

• The calculation is clear and highly reproducible
• It depends relatively little on the sample
• The unit of the original response scale is kept, so it is easy to interpret
• It can be used in both regression and QCA

Step 7: Use the subscales in regression and QCA

summary(lm(Y ~ Trust + Quality + Price, data = dat))

Call:
lm(formula = Y ~ Trust + Quality + Price, data = dat)

Residuals:
     Min       1Q   Median       3Q      Max 
-3.11817 -0.87767  0.06223  0.76259  2.85718 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.42862    0.37338   3.826 0.000159 ***
Trust        0.57160    0.06622   8.632 3.78e-16 ***
Quality      0.36507    0.05991   6.093 3.44e-09 ***
Price       -0.27133    0.05812  -4.668 4.61e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.157 on 296 degrees of freedom
Multiple R-squared:  0.4399,    Adjusted R-squared:  0.4342 
F-statistic: 77.49 on 3 and 296 DF,  p-value: < 2.2e-16

What to report

In a paper or report, writing at least the following lets readers follow your procedure:

• The definition of each construct and the source of the items (an existing scale, or one you created)
• The sample size and how missing values were handled (for example, only complete cases were used)
• The estimation method of the factor analysis (such as principal axis factoring), the rotation (promax), and the basis for the number of factors
• The reverse-keyed items
• The number of items and the alpha of each subscale
• How the subscales were built (the simple mean of the items)

The whole procedure in a few words

A standard practical procedure has the following 11 stages. Steps 1 to 7 of this article rearrange these stages, except CFA (8) and SEM (11), in the order of building a scale (with a check of reverse-keyed items added).

1. Define the construct from prior research
2. Prepare several questionnaire items for each construct
3. Hold in advance a theoretical expectation of “which item measures which construct”
4. Use EFA to examine, in an exploratory way, whether that expectation is consistent with the actual data structure
5. Examine the items from both theoretical and statistical viewpoints
6. Check internal consistency with Cronbach’s alpha (and McDonald’s omega if needed)
7. Build subscales by simple means
8. Verify the measurement structure further with CFA, if needed
9. Use the subscales in regression and similar analyses
10. In QCA, calibrate the subscales and use them
11. If the structure among latent variables is itself the research aim, consider SEM

The most important point is not “collect data and then look for meaning by factor analysis,” but “define the meaning from theory, and use factor analysis to examine, in an exploratory way, whether that meaning also holds in the data.”

You do not need to decide mechanically that “we use constructs, so we use SEM.” What matters is that the research question, the theory, the measurement, the way of building scales, and the method of analysis are consistent with one another.

This article does not cover why you might not use SEM, or how SEM compares. I plan to write about that in a separate article (Notes).

What to read next

The subscales you built in this article can also be used in QCA. While regression looks at the average association when the other variables are held constant, QCA looks for which combinations of conditions lead to the outcome. ThSQCA is an R package that shows how those combinations change when you move the target level (the threshold) of the outcome.

• Why QCA suits decision-making: looking for “recipes” instead of single effects
• Introducing ThSQCA: threshold-sweep QCA in R
• ThSQCA for beginners, part 1: your first threshold sweep
• Home page: all articles, in reading order

R environment for this page

The results were checked in the following environment. A fixed seed produces the same data, but results can change slightly with different versions of R or of the packages.

R.version.string

[1] "R version 4.6.1 (2026-06-24 ucrt)"

packageVersion("psych")

[1] '2.6.5'

1. Applications in marketing and services

• Ordanini, A., Parasuraman, A., & Rubera, G. (2014). When the recipe is more important than the ingredients: A qualitative comparative analysis (QCA) of service innovation configurations. Journal of Service Research, 17(2), 134-149. https://doi.org/10.1177/1094670513513337 Studies luxury hotel services and uses QCA to examine which combinations of attributes lead to the adoption of a new service. An application that looks not only at individual attributes but at how they are combined.
• Frambach, R. T., Fiss, P. C., & Ingenbleek, P. T. M. (2016). How important is customer orientation for firm performance? A fuzzy set analysis of orientations, strategies, and environments. Journal of Business Research, 69(4), 1428-1436. https://doi.org/10.1016/j.jbusres.2015.10.120 Combines customer orientation with other strategic orientations, strategy types, and market environments, and uses fsQCA to find the configurations that go with high firm performance. An application that considers the role of customer orientation through its combinations with other conditions.

2. Theoretical background in management and organization research

• Fiss, P. C. (2007). A set-theoretic approach to organizational configurations. Academy of Management Review, 32(4), 1180-1198. https://doi.org/10.5465/amr.2007.26586092 Treats an organization as a combination of elements and argues for analyzing the link between those combinations and outcomes with set theory. A foundation for thinking about several combinations that lead to the same outcome.
• Fiss, P. C. (2011). Building better causal theories: A fuzzy set approach to typologies in organization research. Academy of Management Journal, 54(2), 393-420. https://doi.org/10.5465/amj.2011.60263120 Analyzes organizational typologies with fuzzy sets and examines the combinations of conditions that lead to an outcome. Important for understanding that the causes can be asymmetric between the presence and the absence of an outcome.
• Woodside, A. G. (2013). Moving beyond multiple regression analysis to algorithms: Calling for adoption of a paradigm shift from symmetric to asymmetric thinking in data analysis and crafting theory. Journal of Business Research, 66(4), 463-472. https://doi.org/10.1016/j.jbusres.2012.12.021 Alongside the effects of single variables in multiple regression, it proposes an analysis that looks at combinations of conditions and at causal asymmetry. It discusses how to analyze data and build theory, using regression and fsQCA on the same data.
• Misangyi, V. F., Greckhamer, T., Furnari, S., Fiss, P. C., Crilly, D., & Aguilera, R. (2017). Embracing causal complexity: The emergence of a neo-configurational perspective. Journal of Management, 43(1), 255-282. https://doi.org/10.1177/0149206316679252 Reviews how QCA has developed in management and organization research and sets out a “neo-configurational perspective” for capturing causal complexity. It links set membership, necessity and sufficiency, and counterfactual analysis to theoretical viewpoints.

3. How to run and report an analysis (practical guides)

• Greckhamer, T., Furnari, S., Fiss, P. C., & Aguilera, R. V. (2018). Studying configurations with qualitative comparative analysis: Best practices in strategy and organization research. Strategic Organization, 16(4), 482-495. https://doi.org/10.1177/1476127018786487 Sets out practical guidelines for QCA in strategy and organization research. It covers the whole course of a study, from model building and case selection to calibration, analysis, reporting, and interpretation.
• Pappas, I. O., & Woodside, A. G. (2021). Fuzzy-set qualitative comparative analysis (fsQCA): Guidelines for research practice in information systems and marketing. International Journal of Information Management, 58, Article 102310. https://doi.org/10.1016/j.ijinfomgt.2021.102310 A practical guide to using fsQCA in information systems and marketing research. It explains the procedure with data from earlier studies and also touches on checking predictive validity and combining fsQCA with methods such as SEM.
• Park, Y., Fiss, P. C., & El Sawy, O. A. (2020). Theorizing the multiplicity of digital phenomena: The ecology of configurations, causal recipes, and guidelines for applying QCA. MIS Quarterly, 44(4), 1493-1520. https://doi.org/10.25300/MISQ/2020/13879 For digital phenomena that involve several theoretical perspectives and combinations of conditions, it offers a framework that ties QCA to theory building. Through examples, it explains how to apply QCA to deductive theory testing and to inductive theory building.

4. Choosing the threshold

• Krogslund, C., Choi, D. D., & Poertner, M. (2015). Fuzzy sets on shaky ground: Parameter sensitivity and confirmation bias in fsQCA. Political Analysis, 23(1), 21-41. https://doi.org/10.1093/pan/mpu016 Argues that fsQCA results can change with small changes in the parameters a user chooses, and that combinations containing randomly generated variables are often found to be sufficient for an outcome. It re-analyzes three existing studies and uses sensitivity analyses and Monte Carlo simulations.

On how the placement of the threshold changes the results, see also Toyoda (2026), Quality & Quantity, https://doi.org/10.1007/s11135-026-03092-3.

1. Getting the idea and the measures (papers)

• Mahoney, J., & Goertz, G. (2006). A tale of two cultures: Contrasting quantitative and qualitative research. Political Analysis, 14(3), 227-249. https://doi.org/10.1093/pan/mpj017 Contrasts quantitative and qualitative research as two cultures with different values, beliefs, and norms, and compares them across ten dimensions, including explanation, causation, case selection, and measurement. A starting point for seeing which research tradition QCA belongs to. Paper guide available
• Ragin, C. C. (2006). Set relations in social research: Evaluating their consistency and coverage. Political Analysis, 14(3), 291-310. https://doi.org/10.1093/pan/mpj019 Introduces two measures for evaluating set relations, consistency and coverage. It shows that fuzzy sets overcome earlier limits on binary variables and on tolerance for error. Paper guide available
• Schneider, C. Q., & Wagemann, C. (2010). Standards of good practice in qualitative comparative analysis (QCA) and fuzzy-sets. Comparative Sociology, 9(3), 397-418. https://doi.org/10.1163/156913210X12493538729793 Sets out standards of good practice for QCA and fuzzy sets. Useful for checking the procedures to follow before you start an analysis.

2. Learning the method systematically (books)

• Ragin, C. C. (2008). Redesigning Social Inquiry: Fuzzy Sets and Beyond. University of Chicago Press. Contrasts set-theoretic analysis with conventional quantitative research and explains calibration, combinations of conditions, and causal complexity. A book for learning the way of thinking behind QCA, not only its procedure.
• Schneider, C. Q., & Wagemann, C. (2012). Set-Theoretic Methods for the Social Sciences: A Guide to Qualitative Comparative Analysis. Cambridge University Press. https://doi.org/10.1017/CBO9781139004244 A textbook that runs from the basics of set theory to the practice of QCA. It also covers cautions and common problems, which helps you understand the procedure together with the reasons behind each judgment.
• Oana, I.-E., Schneider, C. Q., & Thomann, E. (2021). Qualitative Comparative Analysis Using R: A Beginner’s Guide. Cambridge University Press. https://doi.org/10.1017/9781009006781 A beginner’s guide to running QCA in R. A good fit for readers who want to carry out the analysis in R and check the basic operations.

The next post introduces papers that use QCA in marketing and management, together with the theoretical background from management and organization research (applications).

The one idea you need

QCA looks for combinations of conditions that are sufficient for an outcome. Suppose we study customer loyalty. A combination such as “high quality AND high service” is sufficient for high loyalty if (almost) every customer who has both also has high loyalty.

To do this, QCA first has to decide who counts as “high”. With survey scores from 0 to 10, we pick a threshold: a customer is “high” on a variable when the score is greater than or equal to the threshold. In this series, each score is split at the threshold into “high” and “not high” (crisp-set QCA).

Here is the catch. A threshold of 7 and a threshold of 8 are both reasonable, but they can lead to different answers. A normal QCA analysis picks one threshold and reports one answer. ThSQCA reruns the same analysis for a whole range of thresholds and puts all the answers in one table, so you can see where the answer holds and where it changes.

ThSQCA does not replace the QCA package. For every threshold, it calls QCA’s truthTable() and minimize() for you, with identical settings each time, and collects the results.

Step 1: install the packages

Run this once. (If you already installed them, skip it.)

install.packages(c("QCA", "ThSQCA"))

Then load ThSQCA in every new R session:

library(ThSQCA)

Step 2: create a small practice data set

To keep things reproducible, we simulate 400 customers. Each has four scores from 0 to 10: loyalty (LOY), and evaluations of quality (QUA), service (SER), and store environment (ENV). In this simulated world, loyalty is high when quality and service are both high, or (a little less strongly) when quality and environment are both high. A good analysis should discover this.

make_demo_data <- function(n = 400, seed = 2026) {
  set.seed(seed)
  clip <- function(x) pmin(10, pmax(0, round(x)))
  QUA <- clip(rnorm(n, 6, 2))
  SER <- clip(rnorm(n, 6, 2))
  ENV <- clip(rnorm(n, 6, 2))
  core <- pmax(pmin(QUA, SER), pmin(QUA, ENV) - 1)
  LOY  <- clip(core + rnorm(n, 0, 0.7))
  data.frame(LOY, QUA, SER, ENV)
}

demo <- make_demo_data()
head(demo)

  LOY QUA SER ENV
1   7   7   9   3
2   5   4   7   5
3   4   6   4   4
4   6   6   6   4
5   5   5   4   7
6   0   1   2   5

Each row is one customer. You do not have to understand the function above; it only produces the practice data. With your own data, you would simply read your file into a data frame with these kinds of columns.

Step 3: run the sweep

We want to know: how does the answer change as we raise the bar for “high loyalty”? So we keep the thresholds of the three conditions fixed at 7, and vary the outcome threshold from 5 to 8. This is called an outcome threshold sweep, and the function is otSweep().

res <- otSweep(
  dat         = demo,
  outcome     = "LOY",
  conditions  = c("QUA", "SER", "ENV"),
  sweep_range = 5:8,
  thrX        = c(QUA = 7, SER = 7, ENV = 7)
)

Here is what each line means.

Two settings are not written here because they have defaults: the minimum consistency for a combination to count as sufficient (incl.cut, default 0.8) and the minimum number of cases per combination (n.cut, default 1).

Step 4: read the result

summary(res)

OTS Summary
=========== 

Analysis Parameters:
  Outcome: LOY 
  Conditions: QUA, SER, ENV 
  Consistency cutoff: 0.8 
  Frequency cutoff: 1 

Results by Threshold:

 thrY        expression inclS  covS n_solutions
    5 QUA*SER + QUA*ENV 0.981 0.403           1
    6 QUA*SER + QUA*ENV 0.926 0.565           1
    7           QUA*SER 0.894 0.656           1
    8       No solution    NA    NA           0

The table has one row per outcome threshold. The columns are:

• thrY: the outcome threshold used in that row. In the first row, “high loyalty” means LOY >= 5.
• expression: the sufficient combination found. * means AND, + means OR, and ~ means NOT.
• inclS: solution consistency. How reliably the cases covered by the solution are members of the outcome. Higher is better, and 1 is perfect.
• covS: solution coverage. The share of the outcome cases that the solution explains.
• n_solutions: how many minimal solutions were found (1 here).

Now read the expression column from top to bottom.

• At LOY >= 5 and LOY >= 6, two combinations are sufficient: QUA*SER (high quality and high service) and QUA*ENV (high quality and a high store environment).
• At LOY >= 7, only QUA*SER remains.
• At LOY >= 8, the result is No solution. At this level, no combination of the three conditions reached the required consistency.

This is exactly what the simulation was built to contain, and it shows why a sweep is useful. Had we analyzed only LOY >= 6, we would have reported two paths. Had we analyzed only LOY >= 7, we would have reported one. The sweep shows that the answer depends on how high the target is, and where it stops working altogether.

A note on “No solution”: it is a technical result, not a statement that no route to high loyalty exists. It means that, for these data and these settings, QCA could not produce a minimal solution that meets the consistency requirement.

Step 5: see the result as a picture

A table is precise, but a picture shows the trend at a glance. The summary table is an ordinary data frame (res$summary), so we can plot it with base R. The next chunk draws solution coverage and solution consistency for each outcome threshold.

tab <- res$summary

plot(tab$thrY, tab$covS, type = "b", pch = 19, lwd = 2, col = "#233A9A",
     xlim = c(4.8, 8.2), ylim = c(0, 1), xaxt = "n",
     xlab = "Outcome threshold (thrY)", ylab = "Score",
     main = "Coverage and consistency by threshold")
axis(1, at = tab$thrY)
lines(tab$thrY, tab$inclS, type = "b", pch = 17, lwd = 2, col = "#F28018")
abline(v = 8, lty = 3, col = "grey60")
text(7.9, 0.3, "No solution\nat thrY = 8", adj = 1, col = "grey40")
legend("bottomleft", legend = c("Coverage (covS)", "Consistency (inclS)"),
       col = c("#233A9A", "#F28018"), pch = c(19, 17), lwd = 2, bty = "n")

As the target gets higher, coverage rises (the solution explains a larger share of the high-loyalty customers) while consistency slowly falls. At thrY = 8 the line simply stops, because there is no solution to plot.

What comes next

In the next part we will look more closely at the four kinds of sweeps (outcome, one condition, several conditions, and both together), and at how to choose the range of thresholds in a way you can defend.

The full package tutorial is available in the ThSQCA documentation, and the methodology is described in Toyoda (2026), Quality & Quantity.

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

The four sweeps at a glance

Setup: the same practice data as part 1

library(ThSQCA)

make_demo_data <- function(n = 400, seed = 2026) {
  set.seed(seed)
  clip <- function(x) pmin(10, pmax(0, round(x)))
  QUA <- clip(rnorm(n, 6, 2))
  SER <- clip(rnorm(n, 6, 2))
  ENV <- clip(rnorm(n, 6, 2))
  core <- pmax(pmin(QUA, SER), pmin(QUA, ENV) - 1)
  LOY  <- clip(core + rnorm(n, 0, 0.7))
  data.frame(LOY, QUA, SER, ENV)
}

demo <- make_demo_data()

Remember the story behind the data: loyalty (LOY) is high when quality (QUA) and service (SER) are both high, or, a little less strongly, when quality and the store environment (ENV) are both high.

Sweep 1: one condition with ctSweepS()

Suppose you are unsure where to draw the line for “high service delivery”. Is 6 high enough? 7? 9? Fix the outcome at LOY >= 7, fix the other conditions at 7, and sweep the threshold of SER from 5 to 9.

res_cts <- ctSweepS(
  dat          = demo,
  outcome      = "LOY",
  conditions   = c("QUA", "SER", "ENV"),
  sweep_var    = "SER",
  sweep_range  = 5:9,
  thrY         = 7,
  thrX_default = 7
)

summary(res_cts)

CTS (single) Summary
==================== 

Analysis Parameters:
  Outcome: LOY 
  Conditions: QUA, SER, ENV 
  Consistency cutoff: 0.8 
  Frequency cutoff: 1 

Results by Threshold:

 threshold   expression inclS  covS n_solutions
         5  No solution    NA    NA           0
         6  No solution    NA    NA           0
         7      QUA*SER 0.894 0.656           1
         8      QUA*SER 0.909 0.333           1
         9 QUA*SER*~ENV 0.917 0.122           1

The new arguments are sweep_var (the condition whose threshold is swept), thrY (the fixed outcome threshold), and thrX_default (the fixed threshold for all other conditions).

Reading the table from top to bottom: when “high service” is defined as 5 or 6, there is no solution, because the definition is so loose that the group of “highly served” customers is no longer reliably loyal. At 7 and 8 the solution is QUA*SER. At 9 the solution gains a third element, QUA*SER*~ENV, and coverage falls sharply, from 0.656 at 7 to 0.122 at 9. You can see why by counting how many customers are behind the strictest definition:

sum(demo$QUA >= 7 & demo$SER >= 9)                 # customers high on both quality and service

[1] 16

sum(demo$QUA >= 7 & demo$SER >= 9 & demo$ENV < 7)  # customers satisfying the solution QUA*SER*~ENV

[1] 12

Of the 400 customers, 16 are high on both quality and service, and only 12 of them also have a store-environment score below 7, which is what the solution QUA*SER*~ENV requires. The result there rests on very few cases. This is a useful habit: when a solution appears only at the edge of the grid, check how many cases are behind it.

Sweep 2: several conditions with ctSweepM()

Now let the thresholds of all three conditions vary at once: each of QUA, SER, and ENV is tried at 6, 7, and 8. That is 3 x 3 x 3 = 27 combinations. The outcome stays at LOY >= 7.

res_mcts <- ctSweepM(
  dat        = demo,
  outcome    = "LOY",
  conditions = c("QUA", "SER", "ENV"),
  sweep_list = list(QUA = 6:8, SER = 6:8, ENV = 6:8),
  thrY       = 7
)

The output has 27 rows. To keep it readable, show only the combinations that produced a solution:

tab <- res_mcts$summary
tab[tab$n_solutions > 0, ]

   combo_id           threshold                  expression     inclS      covS
5         5 QUA=7, SER=7, ENV=6                     QUA*SER 0.8939394 0.6555556
6         6 QUA=8, SER=7, ENV=6                     QUA*SER 0.9189189 0.3777778
8         8 QUA=7, SER=8, ENV=6                     QUA*SER 0.9090909 0.3333333
9         9 QUA=8, SER=8, ENV=6                     QUA*SER 0.9047619 0.2111111
12       12 QUA=8, SER=6, ENV=7                QUA*SER*~ENV 0.8000000 0.3111111
14       14 QUA=7, SER=7, ENV=7                     QUA*SER 0.8939394 0.6555556
15       15 QUA=8, SER=7, ENV=7                QUA*SER*~ENV 1.0000000 0.3000000
17       17 QUA=7, SER=8, ENV=7                     QUA*SER 0.9090909 0.3333333
18       18 QUA=8, SER=8, ENV=7                QUA*SER*~ENV 1.0000000 0.1888889
21       21 QUA=8, SER=6, ENV=8                     QUA*ENV 0.8888889 0.1777778
23       23 QUA=7, SER=7, ENV=8                     QUA*SER 0.8939394 0.6555556
24       24 QUA=8, SER=7, ENV=8           QUA*SER + QUA*ENV 0.9200000 0.5111111
26       26 QUA=7, SER=8, ENV=8                     QUA*SER 0.9090909 0.3333333
27       27 QUA=8, SER=8, ENV=8 QUA*SER*~ENV + QUA*~SER*ENV 0.9393939 0.3444444
   n_solutions
5            1
6            1
8            1
9            1
12           1
14           1
15           1
17           1
18           1
21           1
23           1
24           1
26           1
27           1

Some patterns to look for:

• Fourteen of the 27 combinations have a solution. In every combination where “high quality” is defined as 6, there is no solution at all.
• With QUA at 7 and SER at 7 or 8, the solution is QUA*SER no matter which threshold is used for ENV. This is a stable region.
• With QUA at 8, the solution depends on ENV (and on SER). With ENV at 6 it is QUA*SER (when SER is 7 or 8). With ENV at 7 it is QUA*SER*~ENV. With ENV at 8 it varies, from QUA*ENV alone to two-path solutions such as QUA*SER + QUA*ENV. When “high quality” is defined strictly, the definition of the store environment starts to matter.

The grid shows both where a conclusion is robust and where it is fragile, which a single analysis cannot.

Sweep 3: the outcome and the conditions together with dtSweep()

The dual threshold sweep combines the two ideas: it varies the target level of the outcome and the conditions’ definitions at the same time. Here we use 3 outcome thresholds (6, 7, 8) and 8 condition combinations (QUA, SER, ENV each at 6 or 7), so 24 cells.

res_dts <- dtSweep(
  dat           = demo,
  outcome       = "LOY",
  conditions    = c("QUA", "SER", "ENV"),
  sweep_list_X  = list(QUA = 6:7, SER = 6:7, ENV = 6:7),
  sweep_range_Y = 6:8
)

res_dts$summary

   combo_id thrY                thrX        expression     inclS      covS
1         1    6 QUA=6, SER=6, ENV=6           QUA*SER 0.8630137 0.7118644
2         1    7 QUA=6, SER=6, ENV=6       No solution        NA        NA
3         1    8 QUA=6, SER=6, ENV=6       No solution        NA        NA
4         2    6 QUA=7, SER=6, ENV=6           QUA*SER 0.9215686 0.5310734
5         2    7 QUA=7, SER=6, ENV=6       No solution        NA        NA
6         2    8 QUA=7, SER=6, ENV=6       No solution        NA        NA
7         3    6 QUA=6, SER=7, ENV=6           QUA*SER 0.8932039 0.5197740
8         3    7 QUA=6, SER=7, ENV=6       No solution        NA        NA
9         3    8 QUA=6, SER=7, ENV=6       No solution        NA        NA
10        4    6 QUA=7, SER=7, ENV=6           QUA*SER 0.9848485 0.3672316
11        4    7 QUA=7, SER=7, ENV=6           QUA*SER 0.8939394 0.6555556
12        4    8 QUA=7, SER=7, ENV=6       No solution        NA        NA
13        5    6 QUA=6, SER=6, ENV=7           QUA*SER 0.8630137 0.7118644
14        5    7 QUA=6, SER=6, ENV=7       No solution        NA        NA
15        5    8 QUA=6, SER=6, ENV=7       No solution        NA        NA
16        6    6 QUA=7, SER=6, ENV=7 QUA*SER + QUA*ENV 0.9076923 0.6666667
17        6    7 QUA=7, SER=6, ENV=7       No solution        NA        NA
18        6    8 QUA=7, SER=6, ENV=7       No solution        NA        NA
19        7    6 QUA=6, SER=7, ENV=7           QUA*SER 0.8932039 0.5197740
20        7    7 QUA=6, SER=7, ENV=7       No solution        NA        NA
21        7    8 QUA=6, SER=7, ENV=7       No solution        NA        NA
22        8    6 QUA=7, SER=7, ENV=7 QUA*SER + QUA*ENV 0.9259259 0.5649718
23        8    7 QUA=7, SER=7, ENV=7           QUA*SER 0.8939394 0.6555556
24        8    8 QUA=7, SER=7, ENV=7       No solution        NA        NA
   n_solutions
1            1
2            0
3            0
4            1
5            0
6            0
7            1
8            0
9            0
10           1
11           1
12           0
13           1
14           0
15           0
16           1
17           0
18           0
19           1
20           0
21           0
22           1
23           1
24           0

Reading it: at LOY >= 6, every one of the 8 condition definitions gives a solution, usually QUA*SER, and sometimes QUA*SER + QUA*ENV. At LOY >= 7, only two definitions still give a solution, the ones where QUA and SER are both at 7. At LOY >= 8, none do. In this data set, raising the target level is the strongest driver of change.

A picture makes the same point faster. The next chunk draws the solution coverage of all 24 cells as a heat map. Each row is one definition of the three conditions, each column is one outcome threshold, and “none” marks a cell with no solution.

s <- res_dts$summary
m <- tapply(s$covS, list(s$combo_id, s$thrY), function(x) x[1])
lab <- as.character(s$thrX[match(rownames(m), s$combo_id)])

op <- par(mar = c(4.5, 11, 3, 1))
image(x = seq_len(ncol(m)), y = seq_len(nrow(m)), z = t(m),
      col = colorRampPalette(c("#FFFFFF", "#233A9A"))(50), zlim = c(0, 1),
      xaxt = "n", yaxt = "n", xlab = "Outcome threshold (thrY)", ylab = "",
      main = "Solution coverage in the 24 cells")
axis(1, at = seq_len(ncol(m)), labels = colnames(m))
axis(2, at = seq_len(nrow(m)), labels = lab, las = 1, cex.axis = 0.8)
for (i in seq_len(nrow(m))) for (j in seq_len(ncol(m))) {
  if (is.na(m[i, j])) text(j, i, "none", col = "grey50", cex = 0.8)
  else text(j, i, sprintf("%.2f", m[i, j]), cex = 0.8)
}
box()

par(op)

The column for LOY >= 6 is filled in every row, the column for LOY >= 7 in only two, and the column for LOY >= 8 in none.

How big is a sweep?

Every cell is a full QCA analysis, so the number of cells matters for run time and for how readable the results are.

Choosing a range you can defend

A sweep is easy to run, which is exactly why it needs discipline. Trying many thresholds and reporting only the convenient ones would add researcher degrees of freedom instead of removing them. Some rules of thumb:

1. Decide the range before you look at the results. Base it on what the scale means, on theory, and on practical targets (for example “acceptable”, “good”, “very good”).
2. Include your baseline. The threshold you would have used in an ordinary analysis should be inside the grid, so readers can compare.
3. Avoid the neutral zone. On a 0 to 10 scale, scores around the midpoint do not clearly mean “high” for anyone, so ranges usually start above it.
4. Keep the grid small enough to read. A few meaningful values per variable are better than every possible value.
5. Check the case counts at the edges. Cells at the extremes of the grid often rest on very few cases.
6. Report the whole grid. Show the stable and the unstable regions, not only the cells that support your story.

What comes next

In part 3 we look at the three solution types (complex, parsimonious, and intermediate) and at what to do when a cell has more than one minimal solution.

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

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.

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)
  }
}

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.

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:

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

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.

A report that records the whole sweep

A sweep produces many results, and a reader needs to know what range was explored, under which settings, and what came out at each threshold. generate_report() writes this into a Markdown file. It comes in two formats.

• "simple" is a short report for the main text or an appendix. It has a summary table, the solution at each threshold, and code for checking the result with the QCA package.
• "full" is for supplementary material. It adds an overview of the analysis (the sweep range and the cutoffs), descriptive statistics, and, for every threshold, the truth table, the solution fit, per-term metrics, a configuration chart, the QCA package output, and the settings needed to reproduce it. A comparison across thresholds follows.

report_file <- tempfile(fileext = ".md")
generate_report(res_ots, report_file, dat = demo, format = "simple")

Here are the first lines of the file it wrote. On your own computer, give the file a real name, such as "ots_report_simple.md", and it will appear in your working folder.

cat(head(readLines(report_file), 25), sep = "\n")

# QCA Analysis Report

*(Auto-generated by ThSQCA package)*

**Generated**: 2026-10-08 23:37

---

## Summary

| thrY | expression | inclS | covS | n_solutions |
|---|---|---|---|---|
| 5 | QUA\*SER + QUA\*ENV | 0.981 | 0.403 | 1 |
| 6 | QUA\*SER + QUA\*ENV | 0.926 | 0.565 | 1 |
| 7 | QUA\*SER | 0.894 | 0.656 | 1 |
| 8 | No solution | NA | NA | 0 |


---

## Solutions Overview

### LOY >= 5

**Solution**: QUA\*SER + QUA\*ENV -> LOY

A few options are worth knowing:

generate_report(res_ots, "r.md", dat = demo, format = "full")              # more detail
generate_report(res_ots, "r.md", dat = demo, include_chart = FALSE)        # no charts
generate_report(res_ots, "r.md", dat = demo, chart_symbol_set = "latex")   # LaTeX symbols
generate_report(res_ots, "r.md", dat = demo, include_raw_output = FALSE)   # omit QCA output

Configuration charts

QCA papers often show solutions as a chart: one column per path, one row per condition, a filled circle for “present”, a crossed circle for “absent”, and a blank where the path does not restrict the condition (neither presence nor absence is required). You can draw one from a list of paths with config_chart_from_paths().

paths <- c("A*B*~C", "A*D", "B*E")
cat(config_chart_from_paths(paths))

● = presence, ⊗ = absence, blank = don’t care

If you prefer plain characters, for example for a journal that cannot print the circle symbols, use the ASCII version:

cat(config_chart_from_paths(paths, symbol_set = "ascii"))

O = presence, X = absence, blank = don’t care

How many cases stand behind each threshold?

A sweep table gives solutions and fit values, but not the number of cases behind them. A reader should be able to see that, because a result that rests on a handful of cases deserves less trust. For the outcome sweep of part 1, we can count the customers directly from the data, without ThSQCA.

thr <- 5:8
n_out  <- sapply(thr, function(t) sum(demo$LOY >= t))
n_both <- sapply(thr, function(t) sum(demo$LOY >= t & demo$QUA >= 7 & demo$SER >= 7))
cases <- rbind("High loyalty (LOY >= thrY)" = n_out,
               "... and high on QUA and SER" = n_both)
colnames(cases) <- thr

bp <- barplot(cases, beside = TRUE, col = c("#233A9A", "#F28018"),
              ylim = c(0, max(cases) * 1.15),
              xlab = "Outcome threshold (thrY)", ylab = "Number of customers",
              main = "Cases behind each threshold",
              legend.text = rownames(cases),
              args.legend = list(x = "topright", bty = "n", cex = 0.8))
text(bp, cases, labels = cases, pos = 3, cex = 0.8)

As the target rises, the high-loyalty group shrinks from 263 customers to 29, so the solutions at the strict end of a sweep are built on far fewer cases. Reporting these counts next to the sweep table is a simple way to show how much each row can bear.

Check a cell against the QCA package

ThSQCA does not invent new results. Every cell is an ordinary QCA analysis, so you can reproduce any cell by hand. This is a good way to build trust in the tool, and to answer a reviewer who asks. Here we reproduce the cell at an outcome threshold of 7.

bin <- function(x, t) as.integer(x >= t)

d7 <- data.frame(
  LOY = bin(demo$LOY, 7), QUA = bin(demo$QUA, 7),
  SER = bin(demo$SER, 7), ENV = bin(demo$ENV, 7)
)

tt  <- truthTable(d7, outcome = "LOY", conditions = c("QUA", "SER", "ENV"),
                  incl.cut = 0.8, n.cut = 1, show.cases = FALSE)
sol <- minimize(tt)
sol

M1: QUA*SER -> LOY

The solution printed by minimize() matches the row for thrY = 7 in the sweep table from part 1 (QUA*SER). In both cases the data were binarized in the same way (a score of 7 or above is a member), and in both cases the same QCA functions did the minimization.

Record your software versions

Results can depend on the versions of R and the packages. Print them and keep them with your report:

c(R      = R.version.string,
  QCA    = as.character(packageVersion("QCA")),
  ThSQCA = as.character(packageVersion("ThSQCA")))

A short checklist for reporting a sweep

Because a sweep gives you freedom to explore, it also asks you to be transparent. Before you report one, make sure that you can answer yes to each of these questions.

1. Did I decide the threshold range, and the step between values, before looking at the results?
2. Do I state every setting that was held fixed (consistency cutoff, frequency cutoff, solution type, directional expectations)?
3. Did I report every cell of the grid, including the cells with no solution?
4. Did I check how many cases stand behind the cells at the edges of the grid?
5. If a cell had several minimal solutions, did I say so?
6. Did I avoid presenting the sweep as a search for the best threshold?
7. Did I record the versions of R, QCA, and ThSQCA?

Where to go from here

You now know the whole workflow: choose a sweep, set a defensible range, run it, read it, and report it. If you want more, the package documentation has the full tutorial with additional examples, and the methodology is explained in the paper.

• ThSQCA documentation and tutorial
• ThSQCA on CRAN
• Toyoda, Y. (2026). Threshold-sweep QCA: threshold dependence as an analytical dimension. Quality & Quantity. https://doi.org/10.1007/s11135-026-03092-3
• Browser tools: QCA Quickstart and Sweep Builder

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

The cutoff is something to vary

Threshold-sweep QCA (ThSQCA) does not fix the cutoff in advance. It tries several. You run the same analysis at several cutoffs, with everything else held constant, and line the answers up in one table. You see where the answer holds steady, where it changes shape, and where it disappears.

If you have used top-box versus top-2-box scoring, you already know the problem. A result that looks solid at top-2-box can look very different at top-box. The sweep makes that comparison systematic.

A small example (simulated data)

To show the idea without using real customer data, we simulated 400 customers of a café brand. Each customer rated the brand from 0 to 10 on quality, service, and store environment, and also gave a loyalty score from 0 to 10. To keep things simple, we treat each rating as a yes/no split at 7 (crisp-set QCA). We then asked: which combinations of “high” ratings (7 or above) are enough for customers to reach each loyalty target?

Read down the table as the bar rises.

• A modest goal has options. There are two routes, so a manager can choose the cheaper one: improve quality and service, or improve quality and the store.
• A demanding goal has fewer options. At 7 or above, only the quality and service route remains. The store environment stops being a way in.
• A very demanding goal can outrun the data. At 8 or above, none of the three strengths, alone or in combination, was enough. That is not proof that no route exists; it tells you that these three strengths do not explain the very top. It is a cue to revisit additional conditions, measurement, the number of cases, and the analysis settings.

A single analysis at “7 or above” would have told the first manager something misleading: that quality and service is the only route, when for a broader goal there is a second one.

Questions where this helps

• Satisfaction, recommendation intent (such as the NPS question), or purchase intent: does the recipe for “satisfied” equal the recipe for “delighted”?
• Segment definitions: if “heavy user” means 10 purchases a year versus 20, do the drivers change?
• Campaign targets: what does it take to move customers from “consider” to “prefer”?
• Reporting: instead of defending one cutoff, show the range and let readers see how much depends on it.

Two honest cautions

1. It shows combinations that go together with the outcome. It does not prove cause. This is true of any analysis of observational data. Treat the results as a map for where to look and what to test.
2. Do not hunt for the cutoff that gives the nicest answer. Decide the range before you look, say why, and report all of it. The value of the sweep is in being transparent about how much the answer depends on the definition.

Try it on your own survey

You do not need to write code from scratch.

1. QCA Quickstart creates a first analysis script from the names of your columns.
2. Sweep Builder turns that setup into a threshold-sweep script.
3. Paste the script into R. The package is free: install.packages("ThSQCA").

If you would like a gentler walk through the same ideas, the beginner’s series explains every line with a practice data set. The method itself is described in Toyoda (2026), Threshold-sweep QCA: threshold dependence as an analytical dimension, Quality & Quantity, and the package is on CRAN.

Why QCA is attractive for decisions

How it differs from regression (roughly)

Neither is better than the other. You choose according to the question.

What QCA cannot do (and what is also true of other methods)

• QCA is not the only method that cannot prove causation. Analyses of observational data, regression included, cannot fully establish causation. A sufficiency solution from QCA does not, by itself, guarantee the causal effect of carrying out an intervention. A causal reading needs support from research design, temporal order, theory, and case knowledge. The results are best used as material for thinking about candidate interventions and hypotheses to test.
• It needs theory and knowledge of the cases. The analyst decides which conditions to include and where “high” begins.

The pitfall is the threshold

That last point matters more than it first appears. In QCA, the analyst decides the threshold that separates “high” from “low”. But a decision target is exactly such a threshold: do you aim for “reasonably satisfied” or for “loved”? When the threshold changes, the answer can change too.

Threshold-sweep QCA (ThSQCA) addresses this problem. The next post continues from here.

Next: Is “good enough” the same goal as “loved”?

References

• Ragin, C. C. (2008). Redesigning Social Inquiry: Fuzzy Sets and Beyond. University of Chicago Press.
• Fiss, P. C. (2011). Building better causal theories: A fuzzy set approach to typologies in organization research. Academy of Management Journal, 54(2), 393-420.
• Woodside, A. G. (2013). Moving beyond multiple regression analysis to algorithms: Calling for adoption of a paradigm shift from symmetric to asymmetric thinking in data analysis and crafting theory. Journal of Business Research, 66(4), 463-472.

The idea

Changing a threshold changes which cases belong to a set. That can change the truth table, the fit measures, and the Boolean minimization. The relationship between a threshold t and the sufficiency solution S is therefore better described as a mapping, t → S(t), than as a single solution.

This is not the same question as a robustness check, which asks whether a favored solution survives alternative settings. It is also different from a search for the best threshold. Threshold-sweep QCA asks a descriptive question: how are sufficiency structures distributed across threshold definitions, and where do they persist, appear, or disappear? The methodology is developed in the paper below.

What the package does

ThSQCA does not implement its own minimization algorithm. Truth table construction and minimization are delegated to the QCA package, so the computational core is the same as in a standard analysis. ThSQCA adds the workflow around it:

• the same consistency threshold, frequency threshold, solution type, and directional expectations are applied at every step of the sweep;
• product terms, solution consistency, and coverage are collected into one common format;
• cases with no solution or with multiple minimal solutions are recorded rather than lost;
• results can be shown as tables and configuration charts, and written out as a Markdown report; the summary table can also be plotted with ordinary R graphics.

Four sweep functions

Getting started

install.packages("ThSQCA")
library(ThSQCA)

The package includes a small example data set, sample_data, with an outcome Y and three conditions X1, X2, and X3, all scored from 0 to 10. The following is a minimal outcome-threshold sweep that runs as it is:

library(ThSQCA)

head(sample_data)

res <- otSweep(
  dat         = sample_data,
  outcome     = "Y",
  conditions  = c("X1", "X2", "X3"),
  sweep_range = 6:9,
  thrX        = c(X1 = 7, X2 = 7, X3 = 7)
)

summary(res)

This sweep asks how the sufficiency solution changes when “high outcome” is defined as Y >= 6, Y >= 7, Y >= 8, and Y >= 9. Each condition counts as “high” when its score is 7 or above. If you want a slower, step-by-step explanation of every row and of how to read the result table, see part 1 of the beginner’s series.

Running it prints a table like this (only the result part is shown):

 thrY                  expression inclS  covS n_solutions
    6 ~X1*X3 + ~X2*X3 + X1*X2*~X3 0.906 0.853           1
    7 ~X1*X3 + ~X2*X3 + X1*X2*~X3 0.906 0.879           1
    8                 No solution    NA    NA           0
    9                 No solution    NA    NA           0

Each row is one outcome threshold. In the expression column, * means AND, + means OR, and ~ means NOT. At Y >= 6 and Y >= 7, the same three sufficient paths appear. At Y >= 8 and Y >= 9, no solution meets the consistency requirement. A single-threshold analysis cannot show where an answer stops holding; a sweep can.

The package vignette covers all four functions with worked examples.

A note on use

ThSQCA is not a device for finding the threshold that gives the most appealing result. The range of thresholds should be justified in advance by the meaning of the scale, theory, and practical goals, and the range explored should be reported in full. The generate_report() function is designed to keep that record.

Links

• Paper: Toyoda, Y. (2026). Threshold-sweep QCA: threshold dependence as an analytical dimension. Quality & Quantity. https://doi.org/10.1007/s11135-026-03092-3
• Package on CRAN: ThSQCA
• Source code: GitHub
• Documentation: im-research-yt.github.io/ThSQCA
• Browser tools: QCA Quickstart and Sweep Builder