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1. Building discrete distributional
data
Survey data are usually stored as microdata: one row per
respondent, with some categorical variables (gender, faculty, age class,
…) and a set of items measured on an ordinal scale (e.g. a 1-7 Likert
scale). When the interest is in groups of respondents
rather than in single individuals, each group can be described, for each
item, by the frequency distribution of the scores given
by its members. The resulting table, with groups in the rows, items in
the columns and a discrete distribution in each cell, is a discrete
distributional data table: the object of the
DDAWASS package.
This vignette shows how to build such a table (a MatD
object) from standard data.
The data: evaluation of the orientation services
We use the univer data of the CUB
package: a survey on the evaluation of the orientation services of the
University of Naples Federico II (2002), with 2179 students who rated,
on a 1-7 scale (1 = extremely unsatisfied, 7 = extremely satisfied), the
information received (informat), the willingness of the
staff (willingn), the office hours (officeho),
the competence of the staff (compete) and the global
satisfaction (global).
data(univer, package = "CUB")
str(univer)
#> 'data.frame': 2179 obs. of 12 variables:
#> $ faculty : int 1 1 1 1 1 1 1 1 1 1 ...
#> $ freqserv: int 0 0 1 1 0 0 1 0 0 0 ...
#> $ age : int 20 21 22 23 21 22 21 22 20 21 ...
#> $ gender : int 0 0 0 0 0 1 0 0 1 0 ...
#> $ diploma : int 2 2 5 6 6 2 6 1 2 2 ...
#> $ residenc: int 3 2 3 3 1 2 1 2 2 2 ...
#> $ changefa: int 0 0 0 0 1 0 0 0 0 1 ...
#> $ informat: int 5 6 6 7 5 1 1 5 6 5 ...
#> $ willingn: int 6 7 6 7 5 7 2 5 7 6 ...
#> $ officeho: int 7 5 6 6 4 2 2 3 6 5 ...
#> $ compete : int 5 6 6 7 5 5 2 4 6 5 ...
#> $ global : int 7 7 7 7 6 7 1 4 6 6 ...
The covariates are coded as integers: we recode the ones we use as
factors (the faculties are anonymous codes in the original data).
univer$Faculty <- factor(sprintf("F%02d", univer$faculty))
univer$Gender <- factor(univer$gender, levels = 0:1, labels = c("M", "W"))
univer$User <- factor(univer$freqserv, levels = 0:1, labels = c("occasional", "regular"))
items <- c("informat", "willingn", "officeho", "compete", "global")
From microdata to a MatD
data2MatD() groups the rows by one or more categorical
variables and computes, for each group and each item, the distribution
of the scores. With support = 1:7 all the distributions
share the seven categories of the scale (categories never chosen get a
zero frequency), which makes plots and tables easier to read.
by_faculty <- data2MatD(univer, unit = "Faculty", vars = items, support = 1:7)
by_faculty
#> a matrix of discrete distributions
#> 13 rows x 5 columns
#> informat willingn officeho
#> F01 [m=5.28 s=1.73 k=7] [m=6.28 s=1.06 k=7] [m=5.64 s=1.48 k=7]
#> F02 [m=6.54 s=0.948 k=7] [m=6.57 s=0.871 k=7] [m=6.26 s=1.27 k=7]
#> F03 [m=5.39 s=1.42 k=7] [m=6.04 s=1.11 k=7] [m=5.27 s=1.35 k=7]
#> F04 [m=5.82 s=1.12 k=7] [m=6.11 s=1.2 k=7] [m=5.4 s=1.21 k=7]
#> F05 [m=6.06 s=0.91 k=7] [m=6.38 s=0.801 k=7] [m=6.21 s=0.992 k=7]
#> F06 [m=4.43 s=1.7 k=7] [m=4.99 s=1.69 k=7] [m=4.31 s=1.7 k=7]
#> F07 [m=6.05 s=1.07 k=7] [m=6.37 s=0.931 k=7] [m=4.61 s=1.8 k=7]
#> F08 [m=5.93 s=1.18 k=7] [m=6.43 s=0.917 k=7] [m=5.78 s=1.25 k=7]
#> F09 [m=5.26 s=1.4 k=7] [m=5.86 s=1.37 k=7] [m=5.04 s=1.4 k=7]
#> F10 [m=5.73 s=1.37 k=7] [m=5.92 s=1.44 k=7] [m=4.92 s=1.82 k=7]
#> compete global
#> F01 [m=5.76 s=1.5 k=7] [m=6.06 s=1.32 k=7]
#> F02 [m=6.56 s=0.879 k=7] [m=6.55 s=0.917 k=7]
#> F03 [m=5.36 s=1.5 k=7] [m=5.57 s=1.17 k=7]
#> F04 [m=5.83 s=1.19 k=7] [m=5.87 s=1.12 k=7]
#> F05 [m=6.2 s=0.88 k=7] [m=6.26 s=0.837 k=7]
#> F06 [m=4.6 s=1.72 k=7] [m=4.78 s=1.6 k=7]
#> F07 [m=6.16 s=1.03 k=7] [m=6.05 s=0.92 k=7]
#> F08 [m=6.03 s=0.958 k=7] [m=6.19 s=0.833 k=7]
#> F09 [m=5.6 s=1.23 k=7] [m=5.46 s=1.15 k=7]
#> F10 [m=5.23 s=1.15 k=7] [m=5.38 s=0.923 k=7]
#> ... (only the first rows/columns shown)
#> k = number of support values
Groups can be defined by crossing several variables. Very small
groups give unreliable distributions: min.n drops the
groups with fewer observations than a threshold.
M <- data2MatD(univer, unit = c("Faculty", "Gender"), vars = items,
support = 1:7, min.n = 10)
dim(M@M)
#> [1] 26 5
get.MatD.stats(M, "n")$mat[1:6, "global"] # group sizes
#> F01_M F01_W F02_M F02_W F03_M F03_W
#> 31 19 84 91 127 128
The 26 rows are the faculty-by-gender groups. This table is used in
all the vignettes.
Inspecting the data
Each cell is a distributionD object, with the support
x, the probabilities p, the cumulative
distribution cdf, the number of observations
n, the mean m and the standard deviation
s.
d <- get.cell.MatD(M, "F02_W", "global")
d
#> Discrete distribution (7 support values, n = 91)
#> x p cdf
#> 1 0.00000 0.00000
#> 2 0.01099 0.01099
#> 3 0.02198 0.03297
#> 4 0.03297 0.06593
#> 5 0.10989 0.17582
#> 6 0.13187 0.30769
#> 7 0.69231 1.00000
#> mean = 6.40659 std = 1.06882
compQ(d, c(0.25, 0.5, 0.75)) # quartiles
#> [1] 6 7 7
compP(d, 5) # proportion of scores <= 5
#> [1] 0.1758242
A grid of bar charts shows a part of the table:
plot(M[c("F02_M", "F02_W", "F06_M", "F06_W"), ])
get.MatD.stats() returns a matrix of summary statistics
(means, standard deviations, quantiles, skewness, …) of all the
cells:
round(get.MatD.stats(M, "mean")$mat[1:6, ], 2)
#> informat willingn officeho compete global
#> F01_M 5.35 6.19 5.77 5.45 5.87
#> F01_W 5.16 6.42 5.42 6.26 6.37
#> F02_M 6.62 6.65 6.30 6.64 6.71
#> F02_W 6.47 6.49 6.23 6.48 6.41
#> F03_M 5.27 5.97 5.14 5.35 5.54
#> F03_W 5.52 6.12 5.40 5.37 5.59
round(get.MatD.stats(M, "std")$mat[1:6, ], 2)
#> informat willingn officeho compete global
#> F01_M 1.73 1.23 1.41 1.66 1.50
#> F01_W 1.72 0.67 1.57 1.02 0.87
#> F02_M 0.79 0.75 1.19 0.77 0.68
#> F02_W 1.07 0.97 1.34 0.96 1.07
#> F03_M 1.44 1.11 1.40 1.39 1.18
#> F03_W 1.39 1.10 1.28 1.59 1.16
Other ways of building the data
From a frequency table. Aggregated data often come
as counts by group, item and score. long2MatD() reads this
long format.
tab <- as.data.frame(table(unit = univer$Faculty, value = univer$global))
tab$variable <- "global"
head(tab)
#> unit value Freq variable
#> 1 F01 1 1 global
#> 2 F02 1 0 global
#> 3 F03 1 1 global
#> 4 F04 1 2 global
#> 5 F05 1 0 global
#> 6 F06 1 20 global
G <- long2MatD(tab, unit = "unit", variable = "variable", value = "value", freq = "Freq")
G[1:3, ]
#> a matrix of discrete distributions
#> 3 rows x 1 columns
#> global
#> F01 [m=6.06 s=1.32 k=6]
#> F02 [m=6.55 s=0.917 k=6]
#> F03 [m=5.57 s=1.17 k=7]
#> k = number of support values
Single distributions can be created from
probabilities, counts or raw observations:
distributionD(x = 1:5, p = c(0.1, 0.1, 0.2, 0.3, 0.3))
#> Discrete distribution (5 support values)
#> x p cdf
#> 1 0.1 0.1
#> 2 0.1 0.2
#> 3 0.2 0.4
#> 4 0.3 0.7
#> 5 0.3 1.0
#> mean = 3.6 std = 1.28062
distributionD(x = 1:5, counts = c(2, 2, 4, 6, 6))
#> Discrete distribution (5 support values, n = 20)
#> x p cdf
#> 1 0.1 0.1
#> 2 0.1 0.2
#> 3 0.2 0.4
#> 4 0.3 0.7
#> 5 0.3 1.0
#> mean = 3.6 std = 1.28062
distributionD(c(4, 5, 5, 3, 5, 4, 2))
#> Discrete distribution (4 support values, n = 7)
#> x p cdf
#> 2 0.14286 0.14286
#> 3 0.14286 0.28571
#> 4 0.28571 0.57143
#> 5 0.42857 1.00000
#> mean = 4 std = 1.06904
and arranged in a matrix with MatD(). From
histograms: a MatH of the HistDAWass package can
be discretized with MatH2MatD(). Finally,
MatD2long() exports a MatD as a long data
frame, e.g. for plotting with ggplot2 or for saving to
a file.
head(MatD2long(M))
#> row variable x p cdf
#> 1 F01_M informat 1 0.09677419 0.09677419
#> 2 F01_M informat 2 0.03225806 0.12903226
#> 3 F01_M informat 3 0.00000000 0.12903226
#> 4 F01_M informat 4 0.00000000 0.12903226
#> 5 F01_M informat 5 0.25806452 0.38709677
#> 6 F01_M informat 6 0.38709677 0.77419355
Next steps
The vignette 2. Basic statistics of discrete distributional
data shows how to describe the table (summary statistics of the
cells, Wasserstein barycenters, variance, covariance and correlation).
The other vignettes cover clustering, principal component analysis and
regression.
References
- Crocetta C., Irpino A., Antonucci L., Marin C. (2025). A performance
indicator and its decomposition according to the impacts of different
aspects based on distributional data. Quality & Quantity,
59(3), 2091-2110. doi:10.1007/s11135-024-01956-0
- Iannario M., Piccolo D., Simone R. (2024). CUB: A Class of Mixture
Models for Ordinal Data. R package.
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