# Weight of partitions

#### 2021-10-26

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

### Conservative and parsimonious solution

We use the data from Thiem (2011) for illustrating how the function wop() calculates the weights of partitions. The weight of a partition is defined on the level of individual models and can be calculated for the consistency and coverage value of a model that has been derived from the pooled data. The weight of a partition for the consistency value of the pooled solution is calculated by applying the consistency formula only to the cases that belong to a partition. The weight of partition is calculated in absolute terms by calculating separately its contribution to the numerator and denominator of the formula. When one divides the partition-specific absolute contribution to the numerator by the contribution to the denominator, then one receives the partition-specific consistency or coverage score (depending on the type of formula).

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: Either C for conservative solution (a.k.a. complex solution) or P for parsimonious solution
• amb_selector: Numerical value for selecting a single model in the presence of model ambiguity. Models are numbered according to their order produced by minimize by the QCA package.
# load data (see data description for details)
data("Thiem2011")
# calculate weight of partitions
wop_pars <- wop(
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",
amb_selector = 1)
kable(wop_pars)
type partition denom_cons num_cons denom_cov num_cov
pooled - 80.64 72.39 101.76 72.39
between 1996 7.46 5.49 7.95 5.49
between 1997 6.77 5.25 7.95 5.25
between 1998 6.79 5.34 7.29 5.34
between 1999 7.31 6.31 8.29 6.31
between 2000 6.66 6.45 9.91 6.45
between 2001 6.80 6.57 9.91 6.57
between 2002 6.85 6.59 9.91 6.59
between 2003 7.24 7.02 10.13 7.02
between 2004 7.73 7.68 11.04 7.68
between 2005 8.22 8.09 11.04 8.09
between 2006 8.81 7.60 8.34 7.60
within AT 4.32 1.89 2.28 1.89
within BE 9.99 7.71 7.86 7.71
within DE 9.65 9.65 10.73 9.65
within DK 1.73 1.59 7.46 1.59
within ES 8.41 7.95 9.83 7.95
within FI 0.60 0.60 5.26 0.60
within FR 9.88 9.86 10.87 9.86
within GR 1.46 1.29 3.80 1.29
within IE 0.73 0.21 0.21 0.21
within IT 9.21 9.21 10.77 9.21
within LU 0.33 0.33 3.80 0.33
within NL 6.26 5.73 7.06 5.73
within PT 0.81 0.81 3.80 0.81
within SE 6.94 5.27 7.16 5.27
within UK 10.32 10.29 10.87 10.29

When one aggregates the partition-specific absolute weights for the between-dimension or within-dimension, one gets the absolute value for the pooled solution. We illustrate this with the following chunk

# sum over all cross-sections for consistency denominator
sum(wop_pars[wop_pars$type == "between", ]$denom_cons)
#> [1] 80.64
# sum over all time series for coverage  numerator
sum(wop_pars[wop_pars$type == "within", ]$num_cov)
#> [1] 72.39

On the basis of the absolute weights, one can calculate the relative weight of a partition by dividing its absolute contribution by the corresponding value for the pooled solution.

# relative contribution of cross sections to denominator for consistency
wop_between  <- wop_pars[wop_pars$type == "between", ] wop_between$rel_denom_cons <- round(wop_between$denom_cons / sum(wop_between$denom_cons), digits = 2)
kable(wop_between)
type partition denom_cons num_cons denom_cov num_cov rel_denom_cons
2 between 1996 7.46 5.49 7.95 5.49 0.09
3 between 1997 6.77 5.25 7.95 5.25 0.08
4 between 1998 6.79 5.34 7.29 5.34 0.08
5 between 1999 7.31 6.31 8.29 6.31 0.09
6 between 2000 6.66 6.45 9.91 6.45 0.08
7 between 2001 6.80 6.57 9.91 6.57 0.08
8 between 2002 6.85 6.59 9.91 6.59 0.08
9 between 2003 7.24 7.02 10.13 7.02 0.09
10 between 2004 7.73 7.68 11.04 7.68 0.10
11 between 2005 8.22 8.09 11.04 8.09 0.10
12 between 2006 8.81 7.60 8.34 7.60 0.11

### Intermediate solution

The weight of partitions for intermediate solutions is produced with wop_inter(). We use data from Schwarz 2016 to illustrate the function.

# load data (see data description for details)
data("Schwarz2016")
# calculating weight of partitions
Schwarz_wop_inter <- partition_min_inter(
Schwarz2016,
units = "country", time = "year",
cond = c("poltrans", "ecotrans", "reform", "conflict", "attention"),
out = "enlarge",
n_cut = 1, incl_cut = 0.8,
intermediate = c("1", "1", "1", "1", "1"))
kable(Schwarz_wop_inter)

### 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