Author: Ingo Rohlfing
Date: 2026-05-17

# Minimization of partitions

#### Ingo Rohlfing

#### 2026-05-17

    library(QCAcluster)
    library(knitr) # nicer html tables

Two functions allow empirical researchers to partition clustered data on one or two dimensions and to derive solutions for the pooled data and for each partition.

- `partition_min()` is available for producing conservative or parsimonious models;
- `partition_min_inter()` should be used for intermediate models. For programming purposes, we opted for a separate function for the intermediate solution.

## Panel data: Minimization of cross sections and time series

We first illustrate how one can decompose panel data on two dimensions. In a between-unit perspective, the panel is partitioned into multiple cross sections with the `time` argument that specifies the cross section ID. In a within-unit perspective, the data is decomposed into multiple time series with the `units` argument that specifies the unit (or time series) ID. The arguments of the functions are:

- `n_cut`: Frequency threshold for pooled data
- `incl_cut`: Inclusion threshold (a.k.a. consistency threshold) for pooled data
- `solution` (only for `partition_min()`): Either `C` for conservative solution (a.k.a. complex solution) or `P` for parsimonious solution
- `BE_cons` and `WI_cons`: Inclusion thresholds for cross sections and time series. The length of the numeric vector should equal the number of units and time series.
- `BE_ncut` and `WI_ncut`: Frequency thresholds for the cross sections and time series. The length of the numeric vector should equal the number of units and time series.

### Conservative and parsimonious solution

We first illustrate the parsimonious solution with dataset from Thiem (2011).

    # load data (see data description for details)
    data(Thiem2011)
    # partition data into time series (within-unit) and cross sections (between-unit)
    Thiem_pars <- partition_min(
      dataset = Thiem2011,
      units = "country", time = "year",
      cond = c("fedismfs", "homogtyfs", "powdifffs", "comptvnsfs",
               "pubsupfs", "ecodpcefs"),
      out = "memberfs",
      n_cut = 6, incl_cut = 0.8,
      solution = "P", # parsimonious solution
      BE_cons = c(0.9, 0.8, 0.7, 0.8, 0.85, 0.8, 0.8, 0.8, 0.8, 0.8, 0.8),
      BE_ncut = rep(1, 11),
      WI_cons = c(0.75, 0.8, 0.9, 0.8, 0.85, rep(0.75, 10)),
      WI_ncut = rep(1, 15))
    kable(Thiem_pars)

type | partition | solution | model | consistency | coverage
--- | --- | --- | --- | --- | ---
pooled | - | comptvnsfs+fedismfs * pubsupfs | 1 | 0.8976935 | 0.7113797
pooled | - | comptvnsfs+fedismfs * ecodpcefs | 2 | 0.8949502 | 0.7158019
pooled | - | comptvnsfs+homogtyfs * pubsupfs | 3 | 0.8780259 | 0.7342767
between | 1996 | fedismfs * comptvnsfs | 1 | 0.9030303 | 0.3748428
between | 1996 | comptvnsfs * pubsupfs | 2 | 0.9885057 | 0.4327044
between | 1997 | ~powdifffs | 1 | 0.9064748 | 0.6339623
between | 1997 | comptvnsfs | 2 | 0.8910675 | 0.5144654
between | 1997 | pubsupfs * ~ecodpcefs | 3 | 0.8672769 | 0.4767296
between | 1998 | comptvnsfs | 1 | 0.9288703 | 0.6090535
between | 1999 | ~powdifffs+fedismfs * ecodpcefs | 1 | 0.8876404 | 0.7623643
between | 1999 | ~powdifffs+fedismfs * ~homogtyfs+homogtyfs * pubsupfs * ecodpcefs | 2 | 0.8961039 | 0.7490953
between | 2000 | comptvnsfs+fedismfs * pubsupfs | 1 | 0.9684685 | 0.6508577
between | 2000 | comptvnsfs+fedismfs * ecodpcefs | 2 | 0.9417476 | 0.6851665
between | 2000 | comptvnsfs+fedismfs * ~homogtyfs+homogtyfs * pubsupfs | 3 | 0.9708333 | 0.7053481
between | 2001 | fedismfs+comptvnsfs | 1 | 0.9028436 | 0.7689203
between | 2002 | fedismfs+~powdifffs+pubsupfs | 1 | 0.8149780 | 0.7467205
between | 2002 | fedismfs+comptvnsfs+pubsupfs | 2 | 0.8214665 | 0.7800202
between | 2003 | pubsupfs+~ecodpcefs | 1 | 0.7985213 | 0.8529121
between | 2004 | fedismfs+~ecodpcefs | 1 | 0.9184290 | 0.8260870
between | 2004 | pubsupfs+~ecodpcefs | 2 | 0.9081726 | 0.8958333
between | 2005 | pubsupfs+~ecodpcefs | 1 | 0.9002695 | 0.9076087
between | 2005 | fedismfs+homogtyfs+ecodpcefs | 2 | 0.8868101 | 0.8586957
between | 2006 | comptvnsfs+~pubsupfs | 1 | 0.8982118 | 0.7829736
between | 2006 | ~pubsupfs+fedismfs * ~ecodpcefs | 2 | 0.8335725 | 0.6966427
within | AT | All truth table rows inconsistent | - | NA | NA
within | BE | No variation in all conditions | - | NA | NA
within | DE | All truth table rows consistent | - | NA | NA
within | DK | ~pubsupfs | 1 | 0.8297389 | 0.9798928
within | DK | ~ecodpcefs | 2 | 0.9469154 | 0.8847185
within | ES | All truth table rows consistent | - | NA | NA
within | FI | No variation in all conditions | - | NA | NA
within | FR | All truth table rows consistent | - | NA | NA
within | GR | All truth table rows inconsistent | - | NA | NA
within | IE | All truth table rows inconsistent | - | NA | NA
within | IT | No variation in all conditions | - | NA | NA
within | LU | homogtyfs | 1 | 0.7629630 | 0.8131579
within | NL | All truth table rows consistent | - | NA | NA
within | PT | All truth table rows inconsistent | - | NA | NA
within | SE | All truth table rows inconsistent | - | NA | NA
within | UK | All truth table rows consistent | - | NA | NA

The output of `partition_min()` is a dataframe summarizing the solutions for the pooled data and the partitions and the consistency and coverage values for the solution. The column `model` shows whether model ambiguity is given for the pooled data or individual partitions if one can derive any model from the data in the first place.

There are different reasons why one might not be able to derive a partition-specific solution:

- All rows could be consistent
- All rows could be inconsistent
- There is no variation across cases of a partition and all cases belong to the same truth table row.

When one the reason applies, it is listed in the column `solution`.

### Intermediate solution

The intermediate solution is derived with `partition_min_inter()`. The only command that is new compared to `partition_min()` is `intermediate` that is available for specifying the directional expectations. The data structure for Schwarz 2016 is an unbalanced panel with eight countries, ten years and 74 observations in total. We assume that one is only interested in the between-unit dimension and wants to derive one solution per cross section. For this reason, the argument for the within-unit dimension (`unit`) is not specified.

    # load data (see data description for details)
    data(Schwarz2016)
    # partition data into cross sections
    Schwarz_inter <- partition_min_inter(
      Schwarz2016,
      time = "year",
      cond = c("poltrans", "ecotrans", "reform", "conflict", "attention"),
      out = "enlarge",
      n_cut = 1, incl_cut = 0.8,
      WI_cons = rep(0.8, 8), BE_cons = c(0.75, 0.75, 0.75, 0.75, 0.75,
                                         0.8, 0.8, 0.8, 0.8, 0.8),
      WI_ncut = rep(1, 8), BE_ncut = rep(1, 10),
      intermediate = c("1", "1", "1", "1", "1"))
    kable(Schwarz_inter)

type | partition | solution | model | consistency | coverage
--- | --- | --- | --- | --- | ---
pooled | - | poltrans * ecotrans * reform+poltrans * reform * conflict * attention | 1 | 0.8008497 | 0.7783001
between | 04 | All inconsistent | - | NA | NA
between | 05 | All inconsistent | - | NA | NA
between | 06 | All inconsistent | - | NA | NA
between | 07 | poltrans * ecotrans * reform+poltrans * reform * ~conflict | 1 | 0.7552752 | 0.8692104
between | 08 | poltrans * ecotrans * reform * conflict | 1 | 0.7626173 | 0.8482275
between | 09 | All consistent | - | NA | NA
between | 10 | poltrans * ecotrans * reform * attention+poltrans * reform * conflict * attention | 1 | 0.8760953 | 0.8125806
between | 11 | poltrans * conflict * attention+poltrans * ecotrans * reform * attention | 1 | 0.8195671 | 0.9566749
between | 12 | poltrans * conflict+poltrans * ecotrans * reform | 1 | 0.8411864 | 0.8865839
between | 13 | All consistent | - | NA | NA

## Multilevel data

Clustered data can be partitioned on a single dimension if there is only one dimension as an in multilevel data where lower-level units are nested in higher-level units. The analysis is then similar to the partition of panel data along one dimension. We use the dataset by Grauvogel and von Soest (2014) for illustrating the analysis of multilevel data. The study analyzes the effect of sanctions on authoritarian regimes. The data distinguishes between the source of the sanction (`Sender`) and the target country (`Target`). All sanctions have been imposed by the EU, UN or US, which means that target countries are nested in three different senders. We partition the data on the dimension of senders to see how solutions differ across senders.

    # load data (see data description for details)
    data(Grauvogel2014)
    # partition data by sender country (higher-level unit)
    GS_pars <- partition_min(
      dataset = Grauvogel2014,
      units = "Sender",
      cond = c("Comprehensiveness", "Linkage", "Vulnerability",
               "Repression", "Claims"),
      out = "Persistence",
      n_cut = 1, incl_cut = 0.75,
      solution = "P",
      BE_cons = rep(0.75, 3),
      BE_ncut = rep(1, 3))
    kable(GS_pars)

type | partition | solution | model | consistency | coverage
--- | --- | --- | --- | --- | ---
pooled | - | ~Comprehensiveness * Claims+~Linkage * Claims+~Repression * Claims+~Comprehensiveness * ~Linkage * ~Repression+Comprehensiveness * ~Vulnerability * Repression | 1 | 0.7758164 | 0.7336208
pooled | - | ~Comprehensiveness * Claims+~Linkage * Claims+~Repression * Claims+~Comprehensiveness * ~Linkage * ~Repression+Linkage * ~Vulnerability * Repression | 2 | 0.7776948 | 0.7245792
within | EU | Vulnerability+~Comprehensiveness * ~Repression+Linkage * Claims | 1 | 0.6293355 | 0.9145825
within | EU | Vulnerability+~Linkage * ~Repression+Linkage * Claims | 2 | 0.6327684 | 0.9049634
within | EU | Vulnerability+Linkage * Repression+~Repression * Claims | 3 | 0.6310549 | 0.9022701
within | EU | Vulnerability+Linkage * Claims+~Repression * Claims | 4 | 0.6320277 | 0.9126587
within | EU | Vulnerability+~Comprehensiveness * Linkage+~Comprehensiveness * ~Repression+Comprehensiveness * Claims | 5 | 0.6273610 | 0.8945748
within | EU | Vulnerability+~Comprehensiveness * Linkage+Comprehensiveness * Repression+~Repression * Claims | 6 | 0.6238361 | 0.9022701
within | EU | Vulnerability+~Comprehensiveness * Linkage+Comprehensiveness * Claims+~Linkage * ~Repression | 7 | 0.6253391 | 0.8868796
within | EU | Vulnerability+~Comprehensiveness * Linkage+Comprehensiveness * Claims+~Repression * Claims | 8 | 0.6261682 | 0.9022701
within | EU | Vulnerability+~Comprehensiveness * ~Repression+Comprehensiveness * Claims+Linkage * Repression | 9 | 0.6303763 | 0.9022701
within | UN | Comprehensiveness+Linkage+Claims | 1 | 0.7458176 | 0.9207195
within | US | Comprehensiveness * ~Linkage * ~Vulnerability+Linkage * ~Repression * Claims+~Vulnerability * Repression * Claims | 1 | 0.7864914 | 0.5616704
within | US | Comprehensiveness * ~Vulnerability * Claims+Linkage * ~Repression * Claims+~Vulnerability * Repression * Claims | 2 | 0.7866918 | 0.5396568
within | US | Comprehensiveness * ~Linkage * ~Vulnerability+Linkage * ~Vulnerability * Claims+~Vulnerability * Repression * Claims+Comprehensiveness * Linkage * Vulnerability * ~Repression | 3 | 0.7960289 | 0.5710586
within | US | Comprehensiveness * ~Vulnerability * Claims+Linkage * ~Vulnerability * Claims+~Vulnerability * Repression * Claims+Comprehensiveness * Linkage * Vulnerability * ~Repression | 4 | 0.7975690 | 0.5522823

### Other packages used in this vignette

Yihui Xie (2021): knitr: A General-Purpose Package for Dynamic Report Generation in R. R package version 1.33.

Yihui Xie (2015): Dynamic Documents with R and knitr. 2nd edition. Chapman and Hall/CRC. ISBN 978-1498716963

Yihui Xie (2014): knitr: A Comprehensive Tool for Reproducible Research in R. In Victoria Stodden, Friedrich Leisch and Roger D. Peng, editors, Implementing Reproducible Computational Research. Chapman and Hall/CRC. ISBN 978-1466561595