# Minimization of partitions

#### 2021-10-26

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