cardy 0.4.0

Card Sorting Utilities

Cardy

Low-level card sorting utilities to compare card sorts — including calculating edit distances, d-neighbourhoods, d-cliques, and orthogonality of card sorts.

It is recommended to read Deibel et al. (2005)[^1] and Fossum & Haller (2005)[^2] to familiarize yourself with the metrics covered in this library. In fact, that entire special issue of Expert Systems is excellent reading for anyone interested in analysing card sorting data.

Installation

pip install cardy

For the JavaScript version of this library, see Cardy on JSR.

Usage

Card sorts are represented as collections of sets of cards: Colection[Set[T]] where each set represents a group.

Edit Distance

The edit distance between two sorts can be computed with the distance function:

from cardy import distance

sort1 = ({1, 2, 3}, {4, 5, 6}, {7, 8, 9})
sort2 = ({1, 2}, {3, 4}, {5, 6, 7}, {8, 9})

dist = distance(sort1, sort2)
print("Distance:", dist)  # Distance: 3

When comparing sorts for equality, assert an edit distance of zero:

if distance(sort1, sort2) == 0:
    ...

Maximum and Normalised Edit Distances

Normalized edit distances can be computed with the norm_distance function:

from cardy import norm_distance

sort1 = (
    {"a1", "a2", "a3"},
    {"b1", "b2", "b3", "b4", "b5"},
    {"c1", "c2", "c3", "c4", "c5"},
    {"d1", "d2", "d3", "d4"},
)
sort2 = (
    {"a1", "b1", "b5", "c1", "c5", "d1"},
    {"a2", "b2", "c2", "d2"},
    {"a3", "b3", "c3", "d3"},
    {"b4", "c4", "d4"},
)

print(norm_distance(sort1, sort2))  # 0.92...
print(norm_distance(sort1, sort2, num_groups=4))  # 1.0

The num_groups option specifies the normalized distance should be computed under the assumption the maximum number of groups in either card sort will not exceed num_groups. If this option is not given, distances are normalized with no limit on the number of groups.

The maximum edit distance any other card sort can be from a given card sort can be computed with the max_distance function.

from cardy import max_distance

# Using sort1 from the previous example
print(max_distance(sort1))  # 13
print(max_distance(sort1, num_groups=4))  # 12

As before, the num_groups option places a restriction on the maximum number of groups another card sort may have.

Cliques and Neighbourhoods

Cliques and neighbourhoods can be calculated using the clique and neighbourhood functions. Given a mapping of sort IDs to card sorts: Mapping[K, Collection[Set[T]]], a neighbourhood or clique is represented as a set of IDs: Set[K] of card sorts

Neighbourhoods

Neighbourhoods are always deterministic:

from cardy import neighbourhood

probe = ({1, 2, 3, 4, 5},)
sorts = {
    0: ({1, 2, 3}, {4, 5}),
    1: ({1, 2, 3}, {4, 5}, set()),
    2: ({1, 2}, {3}, {4, 5}),
    3: ({1, 2}, {3, 4}, {5}),
    4: ({1, 2, 4}, {3, 5}),
}

two_neighbourhood = neighbourhood(2, probe, sorts)
print(f"2-neighbourhood around `{probe}`: {two_neighbourhood}")
# 2-neighbourhood around `({1, 2, 3, 4, 5},)`: {0, 1, 4}

Neighbourhoods can be calculated using normalized edit distances by passing a custom edit distance function as a named argument:

dist = lambda probe, sort: norm_distance(probe, sort, num_groups=3)
upper_quart_neighbourhood = neighbourhood(0.75, probe, sorts, distance=dist)
print(f"Sorts within 75% of `{probe}` are {upper_quart_neighbourhood}")
# Sorts within 75% of `({1, 2, 3, 4, 5},)` are {0, 1, 4}

Cliques

Cliques can be non-deterministic — even when using a greedy strategy (default):

from cardy import clique

probe = ({1, 2}, {3})
sorts = {
    0: ({1}, {2}, {3}),
    1: ({2, 3}, {1}),
    2: ({1, 2, 3},),
}
one_clique = clique(1, probe, sorts)
print(f"1-clique around `{probe}`: {one_clique}")
# 1-clique around `({1, 2}, {3})`: {0, 1}
# OR
# 1-clique around `({1, 2}, {3})`: {1, 2}

The clique function allows for various heuristic strategies for selecting candidate card sorts (via ID). Heuristic functions are of the form: (int, Mapping[K, Collection[Set[T]]]) -> K — that is, a function that takes a the maximum clique diameter and a key to card sort mapping of viable candidates, and returns a key of a viable candidate based on some heuristic.

Two heuristic functions have been provided: random_strategy and greedy_strategy. random_strategy will select a candidate at random. greedy_strategy will select a candidate that reduces the size of the candidate pool by the smallest amount. In the case two or more candidates reduce the pool by the same amount, one is selected at random.

This behaviour can be changed by providing a deterministic heuristic function, or a deterministic Selector which provides a select method that picks a candidate in the case of ambiguity:

from cardy import clique
from cardy.clique import Selector, greedy_strategy


class MinSelector(Selector):
    def select(self, collection):
        # selects the candidate with the smallest key in case of ties
        # for greedy strategy
        return min(collection)


probe = ({1, 2}, {3})
sorts = {
    0: ({1}, {2}, {3}),
    1: ({2, 3}, {1}),
    2: ({1, 2, 3},),
}
one_clique = clique(
    1,
    probe,
    sorts,
    strategy=lambda d, c: greedy_strategy(d, c, MinSelector())
)
print(f"1-clique around `{probe}`: {one_clique}")
# 1-clique around `({1, 2}, {3})`: {0, 1}

Alternatively, a seed can be passed to the base Selector constructor.

Orthogonality

The orthogonality of a collection of sorts can be calculated with the orthogonality function:

from cardy import orthogonality

p1 = (
    ({1, 3, 4, 5, 6, 7, 13, 14, 15, 22, 23},
     {2, 8, 9, 10, 11, 12, 16, 17, 18, 19, 20, 21, 24, 25, 26}),
    ({1, 3, 4, 6, 7, 10, 13, 14, 15, 18, 23, 26},
     {2, 5, 8, 9, 11, 12, 16, 17, 19, 20, 21, 22, 24, 25}),
    ({1, 2, 5, 8, 9, 11, 12, 16, 17, 18, 19, 20, 21, 22, 24, 25},
     {3, 4, 6, 7, 10, 13, 14, 15, 23, 26}),
)
p1_orthogonality = orthogonality(p1)
print(f"P1 orthogonality: {p1_orthogonality:.2f}")  # P1 orthogonality: 2.33

Similarity and Dissimilarity Matrices

Card similarity/dissimilarity matrices for a set of card sorts can be computed with the similarity_matrix and dissimilarity_matrix functions. These return an n + 1 by n + 1 similarity matrix where the first row and column are the card labels. For example, given the following four sorts:

sorts = (
    ({"a1", "a2"}, {"a3", "b1", "b2"}),
    ({"a1", "a2"}, {"a3", "b1", "b2"}),
    ({"a1", "a3", "b1"}, {"a2", "b2"}),
    ({"a1", "a3"}, {"a2", "b1", "b2"}),
)

The co-occurrence or co-nonoccurrence of cards can be counted:

from pprint import pprint
from cardy import similarity_matrix, dissimilarity_matrix

pprint(similarity_matrix(sorts))
# [[None, 'a1', 'a2', 'b2', 'a3', 'b1'],
#  ['a1', 4, 2, 0, 2, 1],
#  ['a2', 2, 4, 2, 0, 1],
#  ['b2', 0, 2, 4, 2, 3],
#  ['a3', 2, 0, 2, 4, 3],
#  ['b1', 1, 1, 3, 3, 4]]
pprint(dissimilarity_matrix(sorts))
# [[None, 'a1', 'a2', 'b2', 'a3', 'b1'],
#  ['a1', 0, 2, 4, 2, 3],
#  ['a2', 2, 0, 2, 4, 3],
#  ['b2', 4, 2, 0, 2, 1],
#  ['a3', 2, 4, 2, 0, 1],
#  ['b1', 3, 3, 1, 1, 0]]

Or, scaled to be a proportion of times the cards co-occur/nonoccur by providing the scale flag:

from pprint import pprint
from cardy import similarity_matrix, dissimilarity_matrix

pprint(similarity_matrix(sorts, scale=True))
# [[None, 'a1', 'a2', 'b2', 'a3', 'b1'],
#  ['a1', 1.0, 0.5, 0.0, 0.5, 0.25],
#  ['a2', 0.5, 1.0, 0.5, 0.0, 0.25],
#  ['b2', 0.0, 0.5, 1.0, 0.5, 0.75],
#  ['a3', 0.5, 0.0, 0.5, 1.0, 0.75],
#  ['b1', 0.25, 0.25, 0.75, 0.75, 1.0]]
pprint(dissimilarity_matrix(sorts, scale=True))
# [[None, 'a1', 'a2', 'b2', 'a3', 'b1'],
#  ['a1', 0.0, 0.5, 1.0, 0.5, 0.75],
#  ['a2', 0.5, 0.0, 0.5, 1.0, 0.75],
#  ['b2', 1.0, 0.5, 0.0, 0.5, 0.25],
#  ['a3', 0.5, 1.0, 0.5, 0.0, 0.25],
#  ['b1', 0.75, 0.75, 0.25, 0.25, 0.0]]

Note that the columns are not guaranteed to be in any particular order.

For programmatic use, it may be more convenient to use the similarity_mapping and dissimilarity_mapping functions. These behave in the same way as their matrix counterparts, but return the results as a nested map:

from pprint import pprint
from cardy import similarity_mapping

pprint(similarity_mapping(sorts))
# {'a1': {'a1': 4, 'a2': 2, 'a3': 2, 'b1': 1, 'b2': 0},
#  'a2': {'a1': 2, 'a2': 4, 'a3': 0, 'b1': 1, 'b2': 2},
#  'a3': {'a1': 2, 'a2': 0, 'a3': 4, 'b1': 3, 'b2': 2},
#  'b1': {'a1': 1, 'a2': 1, 'a3': 3, 'b1': 4, 'b2': 3},
#  'b2': {'a1': 0, 'a2': 2, 'a3': 2, 'b1': 3, 'b2': 4}}

As before, a scale flag can be given to scale results to a proportion of sorts.

Counting Sorts

The n_sorts function returns the number of possible configurations in which c cards can be sorted. This is, in effect, the cth Bell Number^3:

from cardy import n_sorts

print(n_sorts(5))  # 52
print(n_sorts(10))  # 115975
print(n_sorts(15))  # 1382958545
print(n_sorts(20))  # 51724158235372

Providing a second parameter to n_sorts will count the number of possible configurations of sorting c cards into at most g groups:

from cardy import n_sorts

print(n_sorts(5, 3))  # 41
print(n_sorts(10, 3))  # 9842
print(n_sorts(15, 3))  # 2391485
print(n_sorts(20, 3))  # 581130734

To compute the number of card sorts of c cards into exactly g groups, subtract n_sorts(c, g - 1) from n_sorts(c, g):

from cardy import n_sorts

print(n_sorts(5, 3) - n_sorts(5, 2))  # 25
print(n_sorts(10, 3) - n_sorts(10, 2))  # 9330
print(n_sorts(15, 3) - n_sorts(15, 2))  # 2375101
print(n_sorts(20, 3) - n_sorts(20, 2))  # 580606446

[^1]: Deibel, K., Anderson, R. and Anderson, R. (2005), Using edit distance to analyze card sorts. Expert Systems, 22: 129-138. https://doi.org/10.1111/j.1468-0394.2005.00304.x

[^2]: Fossum, T. and Haller, S. (2005), Measuring card sort orthogonality. Expert Systems, 22: 139-146. https://doi.org/10.1111/j.1468-0394.2005.00305.x