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Generalized Pairwise Comparisons (GPC) statistical methodology
Generalized pairwise comparisons (GPC) is a non-parametric statistical methodology for comparing treatments or interventions in clinical trials using one
Generalized pairwise comparisons
Generalized_pairwise_comparisons
Estimation procedure for correlated data
In statistics, a generalized estimating equation (GEE) is used to estimate the parameters of a generalized linear model with a possible unmeasured correlation
Generalized estimating equation
Generalized_estimating_equation
Statistical model for pairwise comparisons
The Bradley–Terry model is a probability model for the outcome of pairwise comparisons between items, teams, or objects. Given a pair of items i and j drawn
Bradley–Terry_model
Machine learning technique
through pairwise comparison under the Bradley–Terry–Luce model (or the Plackett–Luce model for K-wise comparisons over more than two comparisons), the maximum
Reinforcement learning from human feedback
Reinforcement_learning_from_human_feedback
Algorithm that arranges lists in order
comparison function (crowdsourced preferences like voting systems), comparisons are very costly (sports), or when it would be impossible to pairwise compare
Sorting_algorithm
Statistical hypothesis test
variable. "a priori comparisons"/ "planned comparisons"- a particular set of comparisons "pairwise comparisons"-all possible comparisons i.e. Fisher's least
F-test
Statement about integration on manifolds
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called
Generalized_Stokes_theorem
Subfield of machine learning
learning typically involves supervised learning using datasets of pairwise preference comparisons, rankings, or other preference information. The main task in
Preference_learning
Inexact statistical measure
estimator. Examples of quasi-likelihood methods include the generalized estimating equations and pairwise likelihood approaches. The term quasi-likelihood function
Quasi-likelihood
Non-parametric statistical test
Friedman's Test (R code)". February 22, 2010. "PMCMRplus: Calculate Pairwise Multiple Comparisons of Mean Rank Sums Extended". 17 August 2022. Daniel, Wayne W
Friedman_test
Computer science metric for string similarity
also denote a larger family of distance metrics. It is closely related to pairwise string alignments. The Levenshtein distance between two strings a , b {\displaystyle
Levenshtein_distance
Set preferred to any other by a majority
JSTOR 1914033. Introduces a version of a generalized Condorcet Criterion that is satisfied when pairwise elections are based on simple majority choice
Smith_set
Economic model of personal preferences
original model assumes that the probabilities of agents' mistakes in pairwise comparisons are independent and identically distributed: all mistakes have the
Random_utility_model
NP-hard problem in combinatorial optimization
also by many travelers) the task to find, for finitely many points whose pairwise distances are known, the shortest route connecting the points. Of course
Travelling_salesman_problem
Non-parametric method for testing whether samples originate from the same distribution
hypothesis testing. pairwise.wilcox.test(airquality$Ozone, airquality$Month, p.adjust.method = "bonferroni") Pairwise comparisons using Wilcoxon rank
Kruskal–Wallis_test
Statistical modeling method
variables according to a study design, the comparisons of interest may literally correspond to comparisons among units whose predictor variables have
Linear_regression
Property of electoral systems
system criterion Green-Armytage, James (2004). "Cardinal-weighted pairwise comparison" (PDF). Voting matters. Retrieved 2024-10-19. Tideman, Nicolaus (2006)
Mutual_majority_criterion
Collection of statistical models
The follow-up tests may be "simple" pairwise comparisons of individual group means or may be "compound" comparisons (e.g., comparing the mean pooling across
Analysis_of_variance
Plot using the dispersal of scattered dots to show the relationship between variables
variables (dimensions) X1, X2, ... , Xk, the scatter plot matrix shows all the pairwise scatter plots of the variables on a single view with multiple scatterplots
Scatter_plot
Generalizability theory Generalized additive model Generalized additive model for location, scale and shape Generalized beta distribution Generalized
List_of_statistics_articles
Comparative politics for electoral systems
the likelihood that a group of elected winners would beat all losers in pairwise races. The Gallagher Index and Loosemore–Hanby index (LH) measure proportionality
Comparison of electoral systems
Comparison_of_electoral_systems
Ambiguous term in statistics
classes count greater than two: Reduction to binary tasks and subsequent pairwise coupling, see Hastie and Tibshirani (1998) Dirichlet calibration, see Gebel
Calibration_(statistics)
Method in computational chemistry
does not yet equal that from the more commonly used generalized Born approximation. The Generalized Born (GB) model is an approximation to the exact (linearized)
Implicit_solvation
Generalization of the one-dimensional normal distribution to higher dimensions
independent. This implies that any two or more of its components that are pairwise independent are independent. But, as pointed out just above, it is not
Multivariate normal distribution
Multivariate_normal_distribution
Statistical method that summarizes and/or integrates data from multiple sources
pair-wise comparisons are required. The alternative methodology uses complex statistical modelling to include the multiple arm trials and comparisons simultaneously
Meta-analysis
17th-century conjecture proved by Andrew Wiles in 1994
there are no solutions to the generalized Fermat equation in positive integers a, b, c, m, n, k with a, b, and c being pairwise coprime and all of m, n, k
Fermat's_Last_Theorem
Curves of genus > 1 over the rationals have only finitely many rational points
n\geq 4} there are at most finitely many primitive integer solutions (pairwise coprime solutions) to a n + b n = c n {\displaystyle a^{n}+b^{n}=c^{n}}
Faltings'_theorem
provided by Wendt, such that the correlation is always positive. Pairwise comparisons Condorcet winner criterion Intransitive dice Wolfe D, Hogg R (1971)
Probability_of_superiority
Metric to compare ordering
rank distance is a metric (distance function) that counts the number of pairwise disagreements between two ranking lists. The larger the distance, the more
Kendall_tau_distance
Algorithm in numerical analysis
mean square errors of pairwise summation actually grow as O ( log n ) {\displaystyle O\left({\sqrt {\log n}}\right)} . Pairwise summation is used by
Kahan_summation_algorithm
Process of reducing the number of random variables under consideration
variance unfolding (MVU). The central idea of MVU is to exactly preserve all pairwise distances between nearest neighbors (in the inner product space) while
Dimensionality_reduction
Minimum Evolution (ME) is a phylogenetic tree building method. It uses a pairwise distance matrix, calculated from a multiple sequence alignment, to generate
Minimum_evolution
Nonparametric test of the null hypothesis
of the U statistic, which corresponds to the number of wins out of all pairwise contests (see the tortoise and hare example under Examples below). For
Mann–Whitney_U_test
Alignment of more than two molecular sequences
Multiple sequence alignments require more sophisticated methodologies than pairwise alignments, as they are more computationally complex. Most multiple sequence
Multiple_sequence_alignment
Process of replacing missing data with substituted values
points in time, while still maintaining complete case comparison for other parameters, pairwise deletion can introduce impossible mathematical situations
Imputation_(statistics)
Method used in statistics, pattern recognition, and other fields
in C classifiers, whose results are combined. Another common method is pairwise classification, where a new classifier is created for each pair of classes
Linear_discriminant_analysis
Inference rule in logic, proof theory, and automated theorem proving
achieve completeness.[citation needed] Traugott's rule is generalized to allow several pairwise distinct subformulas p 1 , … , p m {\displaystyle p_{1}
Resolution_(logic)
Statistical measure of how far values spread from their average
matrix, is the generalized variance det ( C ) {\displaystyle \det(C)} , the determinant of the covariance matrix. The generalized variance can be
Variance
Machine learning problem
calibration with an algorithm as described above and further application of the pairwise coupling algorithm by Hastie and Tibshirani. A method used to assign scores
Probabilistic_classification
Criterion of fair item allocation
{MMS} _{v_{i}}^{1{\text{-out-of-}}n}(C)} . An allocation is a vector of n pairwise-disjoint subsets of C {\displaystyle C} -- one subset per agent. An allocation
Maximin_share
Concept in natural language processing
Other measures calculate the similarity between ontological instances: Pairwise: measure functional similarity between two instances by combining the semantic
Semantic_similarity
Diagnostic plot of binary classifier ability
illustrates the performance of a binary classifier model (although it can be generalized to multiple classes) at varying threshold values. ROC analysis is commonly
Receiver operating characteristic
Receiver_operating_characteristic
Functional programming language created in 2007
(automatically) proven not to enter an infinite loop. Another common example is pairwise addition of two vectors that are parameterized over their length: total
Idris_(programming_language)
Measure of consensus in ratings given by multiple observers
Kendall's τ, or Spearman's ρ {\displaystyle \rho } can be used to measure pairwise correlation among raters using a scale that is ordered. Pearson assumes
Inter-rater_reliability
Clustering algorithm minimizing the sum of distances to k representatives
usage: import kmedoids from sklearn.metrics import pairwise_distances dissimilarity_matrix = pairwise_distances(X, metric="euclidean") fp = kmedoids
K-medoids
Property of electoral systems
competes. The preferences would be: The Kemeny method arranges the pairwise comparison counts in the following tally table: The ranking scores of all possible
Independence of clones criterion
Independence_of_clones_criterion
b. Fisher's inequality is valid for more general classes of designs. A pairwise balanced design (or PBD) is a set X together with a family of non-empty
Fisher's_inequality
Mathematical principles
property. One variant - the pairwise PM - is satisfied exclusively by divisor methods. That is, an apportionment method is pairwise PM if-and-only-if it is
Mathematics_of_apportionment
Fundamental theorem in probability theory and statistics
still, they need not be independent, nor even pairwise independent.[citation needed] By the way, pairwise independence cannot replace independence in the
Central_limit_theorem
the result of a series of coin-flips. This is a stronger condition than pairwise independence. nonparametric regression nonparametric statistics non-sampling
Glossary of probability and statistics
Glossary_of_probability_and_statistics
data mining method especially for studying biological networks based on pairwise correlations between variables. While it can be applied to most high-dimensional
Weighted correlation network analysis
Weighted_correlation_network_analysis
for treating van der Waals forces between bodies which does not assume pairwise additivity of the individual intermolecular forces; that is to say, the
Lifshitz theory of van der Waals force
Lifshitz_theory_of_van_der_Waals_force
Problem in computational complexity theory
S} as a bit vector, computing the set S + S {\displaystyle S+S} of all pairwise sums as a discrete convolution using the fast Fourier transform, and finally
3SUM
Relation between sides of a right triangle
difference in each coordinate between the points. The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not
Pythagorean_theorem
sequence alignment problem is generally based on pairwise sequence alignment and currently, for a pairwise sequence alignment problem, biologists can use
Tree_alignment
2023 edition of the C++ programming language standard
adjacent_view (and std::views::pairwise being equivalent to std::views::adjacent<2>) adjacent_transform_view (and std::views::pairwise_transform being equivalent
C++23
Statistics concept
estimating the covariance matrix is to treat the estimation of each variance or pairwise covariance separately, and to use all the observations for which both variables
Estimation of covariance matrices
Estimation_of_covariance_matrices
transversal of a balanced hypergraph can be expressed as a union of k pairwise-disjoint transversals, and such partition can be obtained in polynomial
Balanced_hypergraph
the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices
Glossary_of_computer_science
Commutative group (mathematics)
rather than an uppercase A. Jacobson (2009), p. 41. Ramík, J., Pairwise Comparisons Method: Theory and Applications in Decision Making (Cham: Springer
Abelian_group
Mathematical model for data types
pop(S); W ← pop(S) } where x, y, and z are any values, and U, V, W are pairwise distinct variables, is equivalent to: { U ← y; V ← z; W ← x } Unlike the
Abstract_data_type
Edges that hit all cycles in a graph
disagrees with as few pairwise outcomes as possible. Alternatively, if these comparisons result in independent probabilities for each pairwise ordering, then
Feedback_arc_set
Method of data analysis
matrix (that is, PX is a random vector with all its distinct components pairwise uncorrelated). A quick computation assuming P {\displaystyle P} were unitary
Principal_component_analysis
Message encoded with more bits than needed
and not any simpler collection of sources. The above pairwise redundancy measure can be generalized to a set of n variables. R e d u n d a n c y = I ( X
Redundancy (information theory)
Redundancy_(information_theory)
Generalizes sine function to polytopes
In geometry, the polar sine generalizes the sine function of angle to the vertex angle of a polytope. It is denoted by psin. Let v1, ..., vn (n ≥ 1) be
Polar_sine
Machine learning technique
Transformers applied attention to unordered sets and relational reasoning, generalizing pairwise interaction models. 2018 Non-local neural networks extended attention
Attention_(machine_learning)
Whether items that propose to measure the same construct produce similar scores
usually measured with Cronbach's alpha, a statistic calculated from the pairwise correlations between items. Internal consistency ranges between negative
Internal_consistency
Impossibility of straightforward game forms
and chooses a random dictator if the coin lands on heads, or chooses the pairwise winner between two random candidates if the coin lands on tails, is strategyproof
Gibbard's_theorem
Statistical framework for analyzing data from dyads
modeling and estimation approaches may be used, such as generalized linear mixed models or generalized estimating equations for dyadic categorical data, to
Actor–partner interdependence model
Actor–partner_interdependence_model
is an ordinal of the form ωα GCH Generalized continuum hypothesis generalized continuum hypothesis The generalized continuum hypothesis states that 2אα
Glossary_of_set_theory
Standardized AI performance test
The full evaluation is by blinded expert pairwise comparison against human expert deliverables, each comparison costing 1 human hour on average. A gold
Language_model_benchmark
Smallest complete lattice containing a partial order
much smaller than the c2 entries in a matrix that specifies all pairwise comparisons between elements. Nourine & Raynaud (1999) show how to compute this
Dedekind–MacNeille_completion
Analysis of sets of categorical sequences
methodological innovations (see Pairwise dissimilarities below) that address limitations of early sequence comparison methods developed in the 20th century
Sequence analysis in social sciences
Sequence_analysis_in_social_sciences
Simultaneous observation and analysis of more than one outcome variable
algorithms to determine a set of synthetic variables that best represent the pairwise distances between records. The original method is principal coordinates
Multivariate_statistics
used in ELECTRE TRI, the assignment of the alternatives is based on pairwise comparisons of the alternatives to predefined category boundaries. Rule-based
Multicriteria_classification
Class of statistics in estimation theory
{\displaystyle f(x_{1},x_{2})=|x_{1}-x_{2}|} , the U-statistic is the mean pairwise deviation f n ( x 1 , … , x n ) = 2 / ( n ( n − 1 ) ) ∑ i > j | x i − x
U-statistic
sets nest cleanly, so the comparison relation is complete and acyclic. This explains why the median rule and the generalized median voter schemes, which
Group_strategyproofness
Causal or moderating relationship between statistical variables
variables, several interaction variables are constructed, with pairwise-products representing pairwise-interactions and higher order products representing higher
Interaction_(statistics)
independently of the others. (Example: How does one feel about Coke?) Pairwise comparison scale – a respondent is presented with two items at a time and asked
Scale_(social_sciences)
Set of methods for supervised statistical learning
kernel trick, representing the data only through a set of pairwise similarity comparisons between the original data points using a kernel function, which
Support_vector_machine
Statement relating differentiable symmetries to conserved quantities
{\displaystyle N} Newtonian particles where the potential only depends pairwise upon the relative displacement. For Q → {\displaystyle {\vec {Q}}} , consider
Noether's_theorem
Algorithm for searching sorted, infinite lists
algorithm based on exponentially increasing the search band solves global pairwise alignment for O ( n s ) {\displaystyle O(ns)} , where n {\displaystyle
Exponential_search
Statistical hypothesis test
{\displaystyle \therefore {\text{failed to reject }}H_{0}} that the median of pairwise differences is different from zero. The p {\displaystyle p} -value for
Wilcoxon_signed-rank_test
Applications of machine learning to quantum physics
and measurements; Engineering of quantum gates from qubit networks with pairwise interactions, using time dependent or independent Hamiltonians. Improving
Machine_learning_in_physics
Algorithms for matrix decomposition
ISSN 2364-415X. Youdong Guo & Timothy E. Holy (2025). "An optimal pairwise merge algorithm improves the quality and consistency of nonnegative matrix
Non-negative matrix factorization
Non-negative_matrix_factorization
Hungarian and American mathematician and physicist (1903–1957)
follows: Any complemented modular lattice L having a "basis" of n ≥ 4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal
John_von_Neumann
Agglomerative hierarchical clustering method
in each cluster) that are closest to each other. The shortest of these pairwise distances that remain at any step causes the two clusters whose elements
Single-linkage_clustering
Type of human genetic study conducted on twins
separated twins agree to be part of twin studies. For a group of twins, pairwise concordance is defined as C/(C+D), where C is the number of concordant
Twin_study
Fewest cliques covering a graph's edges
produce compact letter displays that assist in visualizing multiple pairwise comparisons. These displays are formed by choosing a letter or other visual marker
Intersection number (graph theory)
Intersection_number_(graph_theory)
Variable representing a random phenomenon
random variables Event (probability theory) Multivariate random variable Pairwise independent random variables Observable variable Random compact set Random
Random_variable
Decision rule that selects alternatives that have a majority
defined over a multidimensional option space, then choosing options using pairwise majority rule is unstable. In most cases, there will be no Condorcet winner
Majority_rule
Nonparametric measure of rank correlation
are incorporated). Stata implementation: spearman varlist calculates all pairwise correlation coefficients for all variables in varlist. MATLAB implementation:
Spearman's rank correlation coefficient
Spearman's_rank_correlation_coefficient
Statistical concept
having no missing values include: Listwise deletion/casewise deletion Pairwise deletion Methods which take full account of all information available,
Missing_data
Index of articles associated with the same name
the Condorcet criteria that is the smallest set of candidates that are pairwise unbeaten by every candidate outside of it, will always exist. If voters
Voting_criteria
Least-weight tree connecting graph vertices
computationally-efficient algorithms. In a comparison model, in which the only allowed operations on edge weights are pairwise comparisons, Karger, Klein & Tarjan (1995)
Minimum_spanning_tree
Grouping a set of objects by similarity
(2001-12-14). "Automatic clustering of orthologs and in-paralogs from pairwise species comparisons". Journal of Molecular Biology. 314 (5): 1041–1052. doi:10.1006/jmbi
Cluster_analysis
AI's tendency to abruptly and drastically forget old info after learning new info
each other. Input vectors are said to be orthogonal to each other if the pairwise product of their elements across the two vectors sum to zero. For example
Catastrophic_interference
Long-term high blood pressure in the arteries
(October 2023). "Exercise training and resting blood pressure: a large-scale pairwise and network meta-analysis of randomised controlled trials". British Journal
Hypertension
Non-local formulation of continuum mechanics
is considered broken, and a pairwise force of zero is assigned for all t ≥ t ′ {\displaystyle t\geq t'} . After a comparison between the strain energy density
Peridynamics
Type of fair division of resources
Amodei generalized the notion of super envy-freeness to agents with different entitlements and different comparison signs; they called the generalized notion
Super_envy-freeness
GENERALIZED PAIRWISE-COMPARISONS
GENERALIZED PAIRWISE-COMPARISONS
Girl/Female
Muslim/Islamic
Fountain of paridise
Surname or Lastname
English
English : status name from Middle English squyer ‘esquire’, ‘a man belonging to the feudal rank immediately below that of knight’ (from Old French esquier ‘shield bearer’). At first it denoted a young man of good birth attendant on a knight, or by extension any attendant or servant, but by the 14th century the meaning had been generalized, and referred to social status rather than age. By the 17th century, the term denoted any member of the landed gentry, but this is unlikely to have influenced the development of the surname.
Girl/Female
Arabic
Highest Garden in Paridise; Heaven
GENERALIZED PAIRWISE-COMPARISONS
GENERALIZED PAIRWISE-COMPARISONS
Girl/Female
Arabic, French, Indian, Kannada, Muslim, Sindhi
Ecstasy; Sentiment
Girl/Female
Hindu, Indian
Golden; Diamond
Female
Greek
Variant spelling of Greek Kore, KORI means "maiden."
Boy/Male
Biblical Greek Irish Latin Shakespearean
Praised, praiseworthy.
Surname or Lastname
English
English : habitational name from a lost place in East Sussex, Alchehorne in the parish of Buxted, which was last recorded in 1592.
Girl/Female
Hindu, Indian, Kannada, Tamil, Telugu
Rock Born; Very Hard and Strong
Girl/Female
Biblical
A liar, sliding away.
Boy/Male
Hindu, Indian, Traditional
Lord Ram
Boy/Male
Hindu, Indian, Marathi
Ascending a Tree
Boy/Male
Indian, Punjabi, Sikh
Pure One
GENERALIZED PAIRWISE-COMPARISONS
GENERALIZED PAIRWISE-COMPARISONS
GENERALIZED PAIRWISE-COMPARISONS
GENERALIZED PAIRWISE-COMPARISONS
GENERALIZED PAIRWISE-COMPARISONS
imp. & p. p.
of Mineralize
v. t.
To impregnate with a mineral; as, mineralized water.
v. t.
To apply to other genera or classes; to use with a more extensive application; to extend so as to include all special cases; to make universal in application, as a formula or rule.
a.
Capable of being generalized, or reduced to a general form of statement, or brought under a general rule.
v. t.
To make universal; to generalize.
v. t.
To bring under a genus or under genera; to view in relation to a genus or to genera.
p. pr. & vb. n.
of Generalize
n.
a court of entrance to, or an inclosed space before, a church; hence, a church porch; -- sometimes formerly used as place of meeting, as for lawyers.
n.
Alt. of Parvise
n.
The system by which power is centralized, as in a government.
n.
One who takes general or comprehensive views.
adv.
Horizontally.
n.
A generalized concept of magnitude.
imp. & p. p.
of Federalize
v. i.
To form into a genus; to view objects in their relations to a genus or class; to take general or comprehensive views.
imp. & p. p.
of Generalize
imp. & p. p.
of Centralize
v. t.
To derive or deduce (a general conception, or a general principle) from particulars.
adv.
In the manner of a pale or pales; by perpendicular lines or divisions; as, to divide an escutcheon palewise.
a.
Comprising structural characters which are separated in more specialized forms; synthetic; as, a generalized type.