---
title: A Probabilistic Comparison of Commonly Used Interest Measures for Association
  Rules
author: "Michael Hahsler"
abstract: "This document contains a comprehensive collection of commonly used measures of
  significance and interestingness (sometimes also called strength) 
  for association rules and itemsets. 
  Interest measures are usually defined in terms of itemset support and counts. 
  Here, we also present their relationship with estimating probabilities and 
  conditional probabilities."
output:
  pdf_document:
    toc: yes
    toc_depth: '2'
  html_document:
    toc: yes
    toc_depth: 2
    toc_float:
      collapsed: no
      smooth_scroll: no
bibliography: association_rules.bib
link-citations: yes
editor_options: 
  markdown: 
    wrap: 72
---

# About this Document

**Latest update:** January 10, 2024

Please
cite this document as **Michael Hahsler, A Probabilistic Comparison of
Commonly Used Interest Measures for Association Rules, 2015, URL:
<https://mhahsler.github.io/arules/docs/measures>**

A PDF version of this document is available at
<https://mhahsler.github.io/arules/docs/measures.pdf>. An annotated
bibliography of association rules can be found at
<https://mhahsler.github.io/arules/docs/association_rules.html>.

<img src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png" alt="CC BY-SA 4.0" style="float:left; margin:4px;"/>
This work is licensed under the
<a href="http://creativecommons.org/licenses/by-sa/4.0/">Creative
Commons Attribution Share Alike 4.0 International License.</a> 



## Code and Implementation

All measures discussed on this page are implemented in function
[interestMeasure()](https://search.r-project.org/CRAN/refmans/arules/html/interestMeasure.html)
in the freely available software:

* R package: [arules](https://github.com/mhahsler/arules)
* Python module: [arulespy](https://pypi.org/project/arulespy/)

## Corrections and Feedback

For corrections and missing measures on this page or in the
implementation in the package arules, please [open an issue on
GitHub](https://github.com/mhahsler/arules/issues) or contact
[me](http://michael.hahsler.net) directly.

# Definitions

@arules:Agrawal:1993 define [association rule
mining](https://en.wikipedia.org/wiki/Association_rule_learning) in the
following way:

Let $I=\{i_1, i_2,\ldots,i_m\}$ be a set of $m$ binary attributes called
**items.** Let $D = \{t_1, t_2, \ldots, t_n\}$ be a set of transactions
called the **database**. Each transaction $t \in D$ has a unique
transaction ID and contains a subset of the items in $I$, i. e.,
$t \subseteq I$. A **rule** is defined as an implication of the form
$X \Rightarrow Y$ where $X, Y \subseteq I$ and $X \cap Y = \emptyset$.
The sets of items (for short **itemsets**) $X$ and $Y$ are called
antecedent (left-hand side or LHS) and consequent (right-hand side or
RHS) of the rule, respectively. Measures of importance (interest) can be
defined for itemsets and rules. The support-confidence framework defines
the measures [support](#support) and [confidence](#confidence). Rules
that satisfy a user-specified minimum thresholds on support and
confidence are called **association rules.**

Interest measures are usually defined in terms of itemset
[support](#support), here we also present them using probabilities and,
where appropriate, counts. The probability $P(E_X)$ of the event that
all items in itemset $X$ are contained in an arbitrarily chosen
transaction can be estimated from a database $D$ using maximum
likelihood estimation (MLE) by

$$\hat{P}(E_X) = \frac{|\{t \in D; X \subseteq t\}|}{n}$$

where $n_X = |\{t \in D; X \subseteq t\}|$ is the count of the number of
transactions that contain the itemset $X$ and $n = |D|$ is the size
(number of transactions) of the database. For conciseness of notation,
we will drop the hat and the $E$ from the notation for probabilities. We
will use in the following $P(X)$ to mean $\hat{P}(E_X)$ and
$P(X \cap Y)$ to mean $\hat{P}(E_X \cap E_Y) = \hat{P}(E_{X \cup Y})$,
the probability of the intersection of the events $E_X$ and $E_Y$
representing the probability of the event that a transaction contains
all items in the union of the itemsets $X$ and $Y$. The event notation
should not be confused with the set notation used in measures like
support, where $supp(X \cup Y)$ means the support of the union of the
itemsets $X$ and $Y$.

**Note on probability estimation:** The used probability estimates will
be very poor for itemsets with low observed frequencies. This needs to
be always taken into account since it affects most measured discussed
below.

**Note on null-transactions:** Transaction datasets typically contain a
large number of transactions that do not contain either $X$ or $Y$.
These transactions are called null-transactions, and it is desirable
that measures of rule strength are not influenced by a change in the
number of null-transactions. However, most measures are affected by the
number of null-transactions since the total number of transactions is
used for probability estimation. Measures that are not influenced by a
change in the number of null-transactions are called null-invariant
[@arules:Tan:2004; @arules:Wu:2010].

Good overview articles about different association rule measures are

-   @arules:Tan:2004 Selecting the right objective measure for
    association analysis. *Information Systems,* 29(4):293-313, 2004

-   @arules:Geng:2006 Interestingness measures for data mining: A
    survey. *ACM Computing Surveys,* 38(3):9, 2006.

-   @arules:Lenca:2007 Association Rule Interestingness Measures:
    Experimental and Theoretical Studies. *Studies in Computational
    Intelligence (SCI)* 43, 51--76, 2007.

# Measures Defined on Itemsets

## Support {#support}

**Reference:** @arules:Agrawal:1993

$$
supp(X) 
= \frac{n_X}{n}
= P(X)
$$

Support is defined on itemsets and gives the proportion of transactions
that contain $X$. It is used as a measure of significance (importance)
of an itemset. Since it uses the count of transactions, it is often
called a **frequency constraint.** An itemset with support greater than
a set minimum support threshold, $supp(X) > \sigma$, is called a
**frequent or large itemset.**

For rules the support defined as the support of all items in the rule,
i.e., $supp(X \Rightarrow Y) = supp(X \cup Y) = P(X \cap Y)$.

Support's main feature is that it possesses the **downward closure
property (anti-monotonicity),** which means that all subsets of a
frequent set are also frequent. This property (actually, the fact that
no superset of an infrequent set can be frequent) is used to prune the
search space (usually thought of as a lattice or tree of itemsets with
increasing size) in level-wise algorithms (e.g., the Apriori algorithm).

The disadvantage of support is the **rare item problem.** Items that
occur very infrequently in the data set are pruned, although they would
still produce interesting and potentially valuable rules. The rare item
problem is important for transaction data which usually have a very
uneven distribution of support for the individual items (typical is a
power-law distribution where few items are used all the time and most
items are rarely used).

**Range:** $[0, 1]$

## Support Count {#count}

**Alias:** Absolute Support Count

**Range:** $[0, n]$ where $n$ is the number of transactions.

## All-Confidence {#allconfidence}

**Reference:** @arules:Omiecinski:2003

All-confidence is defined on itemsets (not rules) as

$$\textrm{all-confidence}(X) 
= \frac{supp(X)}{max_{x \in X}(supp(x))}
= \frac{P(X)}{max_{x \in X}(P(x))}
= min\{P(X|Y), P(Y|X)\}$$

where $max_{x \in X}(supp(x \in X))$ is the support of the item with the
highest support in $X$. All-confidence means that all rules which can be
generated from itemset $X$ have at least a confidence of
$\textrm{all-confidence}(X)$. All-confidence possesses the
downward-closed closure property and thus can be effectively used inside
mining algorithms. All-confidence is null-invariant.

**Range:** $[0, 1]$

## Cross-Support Ratio {#crosssupportratio}

**Reference:** @arules:Xiong:2003

Defined on itemsets as the ratio of the support of the least frequent
item to the support of the most frequent item, i.e.,

$$\textrm{cross-support}(X) = \frac{min_{x \in X}(supp(x))}{max_{x \in X}(supp(x))}$$

a ratio smaller than a set threshold. Normally many found patterns are
cross-support patterns which contain frequent as well as rare items.
Such patterns often tend to be spurious.

**Range:** $[0, 1]$

# Measures Defined on Rules

## Contingency Table {#table}

A $2 \times 2$ [contingency
table](https://en.wikipedia.org/wiki/Contingency_table) with counts for
rule $X \Rightarrow Y$ in the transaction dataset. The counts are:

|                |         $Y$         |         $\overline{Y}$         |
|:--------------:|:-------------------:|:------------------------------:|
|      $X$       |      $n_{XY}$       |      $n_{X\overline{Y}}$       |
| $\overline{X}$ | $n_{\overline{X}Y}$ | $n_{\overline{X}\overline{Y}}$ |

$n_{XY}$ is the number of transactions that contain all items in $X$ and
$Y$. All other measures for rules can be calculated using these counts.

## Confidence {#confidence}

**Alias:** Strength

**Reference:** @arules:Agrawal:1993

$$conf(X \Rightarrow Y) = \frac{supp(X \Rightarrow Y)}{supp(X)} = 
\frac{supp(X \cup Y)}{supp(X)} =
\frac{n_{XY}}{n_X} =
\frac{P(X \cap Y)}{P(X)} = P(Y | X)$$

Confidence is defined as the proportion of transactions that contain $Y$
in the set of transactions that contain $X$. This proportion is an
estimate for the probability of seeing the rule's consequent under the
condition that the transactions also contain the antecedent.

Confidence is directed and gives different values for the rules
$X \Rightarrow Y$ and $Y \Rightarrow X$. Association rules have to
satisfy a minimum confidence constraint,
$conf(X \Rightarrow Y) \ge \gamma$.

Confidence is not downward closed and was developed together with
support by Agrawal et al. (the so-called support-confidence framework).
Support is first used to find frequent (significant) itemsets exploiting
its downward closure property to prune the search space. Then confidence
is used in a second step to produce rules from the frequent itemsets
that exceed a min. confidence threshold.

A problem with confidence is that it is sensitive to the frequency of
the consequent $Y$ in the database. Caused by the way confidence is
calculated, consequents with higher support will automatically produce
higher confidence values even if there exists no association between the
items.

**Range:** $[0, 1]$

## Added Value {#addedvalue}

**Alias:** AV, Pavillon Index, Centered Confidence

**Reference:** @arules:Tan:2004

Quantifies how much the probability of $Y$ increases when conditioning
on the transactions that contain $X$ Defined as

$$AV(X \Rightarrow Y)) = conf(X \Rightarrow Y) - supp(Y) = P(Y | X) - P(Y)$$

**Range:** $[-.5, 1]$

## Casual Confidence {#casualconfidence}

**Reference:** @arules:Kodratoff:2001

Confidence reinforced by negatives given by

$$\textrm{casual-conf} = 
\frac{1}{2} [conf(X \Rightarrow Y) + conf(\overline{X} \Rightarrow \overline{Y})] =
\frac{1}{2} [P(Y|X) + P(\overline{Y}|\overline{X})]$$

**Range:** $[0, 1]$

## Casual Support {#casualsupport}

**Reference:** @arules:Kodratoff:2001

Support improved by negatives given by

$$\textrm{casual-supp} = 
supp(X \cup Y) + supp(\overline{X} \cup \overline{Y}) =
P(X \cap Y) + P(\overline{X} \cap \overline{Y})$$

**Range:** $[0, 2]$

## Centered Confidence {#centeredconfidence}

**Alias:** relative accuracy, gain

**Reference**: @arules:Nada:1999

$$CC(X \Rightarrow Y) = conf(X \Rightarrow Y) - supp(Y)$$

**Range:** $[-1, 1 - 1/n]$

## Certainty Factor {#certainty}

**Alias:** CF, Loevinger

**Reference:** @arules:Galiano:2002

The certainty factor is a measure of the variation of the probability
that $Y$ is in a transaction when only considering transactions with
$X$. An increasing CF means a decrease in the probability that $Y$ is
not in a transaction that $X$ is in. Negative CFs have a similar
interpretation.

$$CF(X \Rightarrow Y) = \frac{conf(X \Rightarrow Y)-supp(Y)}{supp(\overline{Y})}
= \frac{P(Y|X)-P(Y)}{1-P(Y)}$$

**Range:** $[-1, 1]$ (0 indicates independence)

## Chi-Squared {#chisquared}

**Reference:** @arules:Brin:1997b

For the analysis of $2 \times 2$ contingency tables, the [chi-squared
test statistic](https://en.wikipedia.org/wiki/Chi-squared_test) is a
measure of the relationship between two binary variables ($X$ and $Y$).
The chi-squared test statistic is used as a test for independence
between $X$ and $Y$. The chi-squared test statistic is:

$$
\begin{aligned}
\textrm{chi-squared}(X \Rightarrow Y) 
& = \sum_i \frac{(O_i - E_i)^2}{E_i} \\
& = \frac{\left( n_{XY} - \frac{n_X n_Y}{n} \right)^2}{\frac{n_X n_Y}{n}}
+ \frac{\left( n_{\overline{X}Y} - \frac{n_{\overline{X}} n_Y}{n} \right)^2}{\frac{n_{\overline{X}} n_Y}{n}}
+ \frac{\left( n_{X\overline{Y}} - \frac{n_X n_{\overline{Y}}}{n} \right)^2}{\frac{n_X n_{\overline{Y}}}{n}}
+ \frac{\left( n_{\overline{X}\overline{Y}} - \frac{n_{\overline{X}} n_{\overline{Y}}}{n} \right)^2}{\frac{n_{\overline{X}} n_{\overline{Y}}}{n}} \\
& = n \frac{P(X \cap Y)P(\overline{X} \cap \overline{Y})  - P(X \cap \overline{Y})P(\overline{X} \cap Y)}{\sqrt{P(X)P(Y)P(\overline{X})P(\overline{Y})}}
\end{aligned}
$$

$O_i$ is the observed count of contingency table cell $i$ and $E_i$ is
the expected count given the marginals.\
The statistic has approximately a $\chi^2$ distribution with 1 degree of
freedom (for a 2x2 contingency table). The critical value for
$\alpha=0.05$ is $3.84$; higher chi-squared values indicate that the
null-hypothesis of independence between LHS and the RHS should be
rejected (i.e., the rule is not spurious). Larger chi-squared values
indicate stronger evidence that the rule represents a strong
relationship. The statistic can be converted into a p-value using the
$\chi^2$ distribution.

**Notes:** The contingency tables for some rules may contain cells with
low expected values (less then 5) and thus [Fisher's exact
test](#fishers-exact-test) might be more appropriate. Each rule
represents a statistical test, and
<a href="https://en.wikipedia.org/wiki/Multiple_comparisons_problem">
correction for multiple comparisons</a> may be necessary.

**Range:** $[0, \infty]$

## Collective Strength {#collectivestrength}

**Reference:** @arules:Aggarwal:1998

$$S(X) = \frac{1-v(X)}{1-E[v(X)]} \frac{E[v(X)]}{v(X)}
= 
\frac{P(X \cap Y)+P(\overline{Y}|\overline{X})}
{P(X)P(Y)+P(\overline{X})P(\overline{Y})}
$$

where $v(X)$ is the violation rate and $E[v(X)]$ is the expected
violation rate for independent items. The violation rate is defined as
the fraction of transactions that contain some of the items in an
itemset but not all. Collective strength gives 0 for perfectly negative
correlated items, infinity for perfectly positive correlated items, and
1 if the items co-occur as expected under independence.

Problematic is that for items with medium to low probabilities, the
observations of the expected values of the violation rate is dominated
by the proportion of transactions that do not contain any of the items
in $X$. For such itemsets, collective strength produces values close to
one, even if the itemset appears several times more often than expected
together.

**Range:** $[0, \infty]$

## Confidence Boost {#boost}

**Reference:** @arules:Balcazar:2013

Confidence boost is the ratio of the confidence of a rule to the
confidence of any more general rule (i.e., a rule with the same
consequent but one or more items removed in the LHS).

$$\textrm{confidence-boost}(X \Rightarrow Y) =  
\frac{conf(X \Rightarrow Y)}{max_{X' \subset X}(conf(X' \Rightarrow Y))} =
\frac{conf(X \Rightarrow Y)}{conf(X \Rightarrow Y) - improvement(X \Rightarrow Y)} $$

Values larger than 1 mean the new rule boosts the confidence compared to
the best, more general rule. The measure is related to the [improvement
measure](#improvement).

**Range:** $[0, \infty]$ ($>1$ indicates a rule with confidence boost)

## Conviction {#conviction}

**Reference:** @arules:Brin:1997

$$\mathrm{conviction}(X \Rightarrow Y) =\frac{1-supp(Y)}{1-conf(X \Rightarrow Y)} 
= \frac{P(X)P(\overline{Y})}{P(X \cap \overline{Y})}$$

where $\overline{Y} = E_{\neg Y}$ is the event that $Y$ does not appear
in a transaction. Conviction was developed as an alternative to
confidence which was found to not capture the direction of associations
adequately. Conviction compares the probability that $X$ appears without
$Y$ if they were dependent on the actual frequency of the appearance of
$X$ without $Y$. In that respect, it is similar to lift (see the section
about lift on this page). However, in contrast to lift, it is a directed
measure since it also uses the information of the absence of the
consequent. An interesting fact is that conviction is monotone in
confidence and lift.

**Range:** $[0, \infty]$ (1 indicates independence; rules that always
hold have $\infty$)

## Cosine {#cosine}

**Reference:** @arules:Tan:2004

Cosine is a null-invariant measure of correlation between the items in
$X$ and $Y$ defined as

$$\mathrm{cosine}(X \Rightarrow Y) 
= \frac{supp(X \cup Y)}{\sqrt{(supp(X)supp(Y))}} 
= \frac{P(X \cap Y)}{\sqrt{P(X)P(Y)}}
= \sqrt{P(X | Y) P(Y | X)}$$

**Range:** $[0, 1]$ ($0.5$ means no correlation)

## Coverage {#coverage}

**Alias:** LHS Support

It measures the probability that a rule $X \Rightarrow Y$ applies to a
randomly selected transaction. It is estimated by the proportion of
transactions that contain the antecedent of the rule $X \Rightarrow Y$.
Therefore, coverage is sometimes called antecedent support or LHS
support.

$$\mathrm{cover}(X \Rightarrow Y) = supp(X) = P(X)$$

**Range:** $[0, 1]$

## Descriptive Confirmed Confidence {#confirmedconfidence}

**Reference:** @arules:Tan:2004

Confidence confirmed by the confidence of the negative rule.

$$\textrm{confirmed-conf} = conf(X \Rightarrow Y) - conf(X \Rightarrow \overline{Y}) 
= P(Y|X) - P(\overline{Y}|X)$$

**Range:** $[-1, 1]$

## Difference of Confidence {#doc}

**Alias:** DOC, Difference of Proportions

**Reference:** @arules:Hofmann:2001

The difference of confidence is the difference of the proportion of
transactions containing $Y$ in the two groups of transactions that do
and do not contain $X$. For the analysis of $2 \times 2$ contingency
tables, this measure of the relationship between two binary variables is
typically called the difference of proportion. It is defined as $$
\mathrm{doc}(X \Rightarrow Y) 
= conf(X \Rightarrow Y) - conf(\overline{X} \Rightarrow Y) 
= P(Y|X) - P(Y|\overline{X})
= n_{XY} / n_X - n_{\overline{X}Y} / n_{\overline{X}}
$$

**Range:** $[-1, 1]$ (0 means statistical independence)

## Example and Counter-Example Rate {#counterexample}

Example rate reduced by the counter-example rate.

Defined as $$\mathrm{ecr}(X \Rightarrow Y) = 
\frac{n_{XY} - n_{X\overline{Y}}}{n_{XY}} =
\frac{P(X \cap Y) - P(X \cap \overline{Y})}{P(X \cap Y)} =
1 - \frac{1}{sebag(X \Rightarrow Y)}
$$

The measure is related to the [Sebag-Schoenauer
Measure](#sebag-schoenauer).

**Range:** $[0, 1]$

## Fisher's Exact Test {#fishersexacttest}

**Reference:** @arules:Hahsler:2007

If $X$ and $Y$ are independent, then the $n_{XY}$ is a realization of
the random variable $C_{XY}$ which has a hypergeometric distribution
with $n_Y$ draws from a population with $n_X$ successes and
$n_{\overline{X}}$ failures. The p-value for [Fisher's one-sided exact
test](https://en.wikipedia.org/wiki/Fisher%27s_exact_test) giving the
probability of observing a contingency table with a count of at least
$n_{XY}$ given the observed marginal counts is

$$
\textrm{p-value} = P(C_{XY} \ge n_{XY}) 
$$

The p-value is related to [hyper-confidence](#hyper-confidence).
Compared to the [Chi-squared test](#chi-squared), Fisher's exact test
also applies when cells have low expected counts. Note that each rule
represents a statistical test, and [correction for multiple
comparisons](https://en.wikipedia.org/wiki/Multiple_comparisons_problem)
may be necessary.

**Range:** $[0, 1]$ (p-value scale)

## Generalized Improvement {#generalizedImprovement}

**Reference:** @arules:Hahsler:2023

Generalizes the [improvement](#improvement) measure to arbitrary interest measures.

$$
\mathrm{generalizedImprovement}(X \Rightarrow Y) 
= min_{X' \subset X}(M(X \Rightarrow Y) - M(X' \Rightarrow Y))
$$

where $M$ can be any measure that increases with interestingness. 
The original definition of improvement uses the measure confidence.

**Range:** $[-\infty, +\infty]$ (the actual range depends on the used measure)

## Generalized Increase Ratio {#ginc}

**Reference:** @arules:Hahsler:2023

Generalizes [lift increase](#lic) to arbitrary interest measures.

$$
\mathrm{INC}(X \Rightarrow Y) 
= min_{X' \subset X} \left[ \frac{M(X \Rightarrow Y)}{M(X' \Rightarrow Y)} \right]
$$

where $M$ can be any measure of interestingness. The original definition of lift increase 
uses the measure lift.

**Range:** $[0, +\infty]$ ($> 1$ means an increase)

## Gini Index {#gini}

**Reference:** @arules:Tan:2004

The [Gini index](https://en.wikipedia.org/wiki/Gini_coefficient)
measures quadratic entropy as

$$\mathrm{gini}(X \Rightarrow Y) =
P(X)    [P(Y|X)^2+P(\overline{Y}|X)^2] +
P(\overline{X}) [P(B|\overline{X})^2+P(\overline{Y}|\overline{X})^2] -
P(Y)^2 - P(\overline{Y})^2
$$

**Range:** $[0, 1]$ (0 means that the rule does not provide any
information for the dataset)

## Hyper-Confidence {#hyperconfidence}

**Reference:** @arules:Hahsler:2007

The confidence level for observation of too high/low counts for rules
$X \Rightarrow Y$ using the hypergeometric model. Since the counts are
drawn from a hypergeometric distribution (represented by the random
variable $C_{XY}$ with known parameters given by the counts $n_X$ and
$n_Y$, we can calculate a confidence interval for the observed counts
$n_{XY}$ stemming from the distribution. Hyper-confidence reports the
confidence level as

$$ 
\textrm{hyper-conf}(X \Rightarrow Y) 
= 1 - P[C_{XY} \ge n_{XY} | n_X, n_Y]
$$

A confidence level of, e.g., $> 0.95$ indicates that there is only a 5%
chance that the high count for the rule has occurred randomly.
Hyper-confidence is equivalent to the statistic used to calculate the
p-value in [Fisher's exact test](#fishers-exact-test). Note that each
rule represents a statistical test and [correction for multiple
comparisons](https://en.wikipedia.org/wiki/Multiple_comparisons_problem)
may be necessary.

Hyper-Confidence can also be used to evaluate that $X$ and $Y$ are
complementary (i.e., the count is too low to have occurred randomly).

$$ 
\textrm{hyper-conf}_\textrm{complement}(X \Rightarrow Y) 
= 1 - P[C_{XY} < n_{XY} | n_X, n_Y]
$$

**Range:** $[0, 1]$

## Hyper-Lift {#hyperlift}

**Reference:** @arules:Hahsler:2007

Adaptation of the lift measure where instead of dividing by the expected
count under independence ($E[C_{XY}] = n_X / n \times n_Y / n$) a higher
quantile of the hypergeometric count distribution is used. This is more
robust for low counts and results in fewer false positives when
hyper-lift is used for rule filtering. Hyper-lift is defined as:

$$
\textrm{hyper-lift}_\delta(X \Rightarrow Y) = 
\frac{n_{XY}}{Q_{\delta}[C_{XY}]}
$$

where $n_{XY}$ is the number of transactions containing $X$ and $Y$ and
$Q_{\delta}[C_{XY}]$ is the $\delta$-quantile of the hypergeometric
distribution with parameters $n_X$ and $n_Y$.\
$\delta$ is typically chosen to use the 99 or 95% quantile.

**Range:** $[0, \infty]$ (1 indicates independence)

## Imbalance Ratio {#imbalance}

**Alias:** IR

**Reference:** @arules:Wu:2010

Measures the degree of imbalance between two events that the LHS and the
RHS are contained in a transaction. The ratio is close to 0 if the
conditional probabilities are similar (i.e., very balanced) and close to
1 if they are very different. It is defined as

$$
\mathrm{IB}(X \Rightarrow Y) 
= \frac{|P(X|Y) - P(Y|X)|}{P(X|Y) + P(Y|X) - P(X|Y)P(Y|X))}
= \frac{|supp(X) - supp(Y)|}{supp(X) + supp(Y) - supp(X \cup Y)}
$$

**Range:** $[0, 1]$ (0 indicates a balanced, typically uninteresting
rule)

## Implication Index {#implicationindex}

**Reference:** @arules:Gras:1996

A variation of the [Lerman similarity](#lerman-similarity) defined as

$$
\mathrm{gras}(X \Rightarrow Y) 
= \sqrt{N} \frac{supp(X \cup \overline{Y}) - supp(X)supp(\overline{Y})}{\sqrt{supp(X)supp(\overline{Y})}}
$$

**Range:** $[0, 1]$

## Importance {#importance}

**Reference:**
<a href="https://docs.microsoft.com/en-us/sql/analysis-services/data-mining/microsoft-association-algorithm-technical-reference">MS
Analysis Services: Microsoft Association Algorithm Technical
Reference.</a>

In the Microsoft Association Algorithm Technical Reference, confidence
is called "probability," and a measure called importance is defined as
the log-likelihood of the right-hand side of the rule, given the
left-hand side of the rule:

$$
\mathrm{importance}(X \Rightarrow Y) 
= log_{10}(L(X \Rightarrow Y) / L(X \Rightarrow \overline{Y}))
$$

where $L$ is the [Laplace corrected
confidence](#laplace-corrected-confidence).

**Range:** $[-\infty, \infty]$

## Improvement {#improvement}

**Reference:** @arules:Bayardo:2000

The improvement of a rule is the minimum difference between its
confidence and the confidence of any proper sub-rule with the same
consequent. A large positive value indicate that the more specific rule
(with an additional item in the LHS) improves the confidence and should
be kept. Improvement is often used to filter redundant rules.

$$
\mathrm{improvement}(X \Rightarrow Y) 
= min_{X' \subset X}(conf(X \Rightarrow Y) - conf(X' \Rightarrow Y))
$$

**Range:** $[0, 1]$

## Jaccard Coefficient {#jaccard}

**Reference:** @arules:Tan:2004

A null-invariant measure for dependence using the [Jaccard similarity
index](https://en.wikipedia.org/wiki/Jaccard_index) between the two sets
of transactions that contain the items in $X$ and $Y$, respectively.
Defined as

$$
\mathrm{jaccard}(X \Rightarrow Y) 
= \frac{supp(X \cup Y)}{supp(X) + supp(Y) - supp(X \cup Y)}
= \frac{P(X \cap Y)}{P(X)+P(Y)-P(X \cap Y)}
$$

**Range:** $[0, 1]$

## J-Measure {#jmeasure}

\<a href= @arules:Smyth:1991

The J-measure is a scaled version of cross entropy to measure the
information content of a rule.

$$
J(X \Rightarrow Y) 
= P(X \cap Y) log\left(\frac{P(Y|X)}{P(Y)}\right) +
P(X \cap \overline{Y})log\left(\frac{P(\overline{Y}|X)}{P(\overline{Y})}\right)
$$

**Range:** $[0, 1]$ (0 means that $X$ does not provide information for
$Y$)

## Kappa {#kappa}

**Alias:** Cohen's $\kappa$

**Reference:** @arules:Tan:2004

[Cohen's kappa
coefficient](https://en.wikipedia.org/wiki/Cohen%27s_kappa) of the rule
(seen as a classifier) given as the rules observed rule accuracy (i.e.,
confidence) corrected by the expected accuracy (of a random classifier).
Kappa is defined as

$$
\kappa(X \Rightarrow Y) 
= \frac{P(X \cap Y) + P(\overline{X} \cap \overline{Y}) - 
P(X)P(Y) - P(\overline{X})P(\overline{Y})}{1- P(X)P(Y) - 
P(\overline{X})P(\overline{Y})}
$$

**Range:** $[-1,1]$ (0 means the rule is not better than a random
classifier)

## Klosgen {#klosgen}

**Reference:** @arules:Tan:2004

Defined as a scaled version of the [added value measure](#added-value).

$$
\begin{aligned}
\mathrm{klosgen}(X \Rightarrow Y) 
& = \sqrt{supp(X \cup Y)}\,(conf(X \Rightarrow Y) - supp(Y)) \\ 
& = \sqrt{P(X \cap Y)}\, (P(Y|X) - P(Y)) \\
& = \sqrt{P(X \cap Y)}\, AV(X \Rightarrow Y)
\end{aligned}
$$

**Range:** $[-1, 1]$ (0 for independence)

## Kulczynski {#kulczynski}

**Reference:** @arules:Wu:2010

Calculate the null-invariant Kulczynski measure with a preference for
skewed patterns.

$$
\begin{aligned}
\mathrm{kulc}(X \Rightarrow Y)
& = \frac{1}{2} 
\left(conf(X \Rightarrow Y) + conf(Y \Rightarrow X) \right)
= \frac{1}{2} 
\left(\frac{supp(X \cup Y)}{supp(X)} + \frac{supp(X \cup Y)}{supp(Y)} \right) \\
& = \frac{1}{2} 
\left(P(X | Y) + P(Y | X) \right)
\end{aligned}
$$

**Range:** $[0, 1]$ (0.5 means neutral and typically uninteresting)

## Lambda {#lambda}

**Alias:** Goodman-Kruskal's $\lambda$, Predictive Association

**Reference:** @arules:Tan:2004

Goodman and Kruskal's lambda assesses the association between the LHS
and RHS of the rule.

$$
\lambda(X \Rightarrow Y) 
= \frac{\Sigma_{x \in X} max_{y \in Y} P(x \cap y) - max_{y \in Y} P(y)}
{n - max_{y \in Y} P(y)}
$$

**Range:** $[0, 1]$

## Laplace Corrected Confidence {#laplace}

**Alias:** Laplace Accuracy, L

**Reference:** @arules:Tan:2004

$$
L(X \Rightarrow Y) = \frac{n_{XY}+1}{n_X+k},
$$

where $k$ is the number of classes in the domain. For association rule
$k$ is often set to 2. It is an approximate measure of the expected rule
accuracy representing 1 - the Laplace expected error estimate of the
rule. The Laplace corrected accuracy estimate decreases with lower
support to account for estimation uncertainty with low counts.

**Range:** $[0, 1]$

## Least Contradiction {#leastcontradiction}

**Reference:** @arules:Aze:2002

$$
\textrm{least-contradiction}(X \Rightarrow Y)
= \frac{supp(X \cup Y) - supp(X \cup \overline{Y})}{supp(Y)}
= \frac{P(X \cap Y) - P(X \cap \overline{Y})}{P(Y)}
$$

**Range:** $[-\infty, 1]$

## Lerman Similarity {#lerman}

**Reference:** Lerman, I.C. (1981). Classification et analyse ordinale
des donnees. Paris.

Defined as

$$
\mathrm{lerman}(X \Rightarrow Y) 
= \frac{n_{XY} - \frac{n_X n_Y}{n}}{\sqrt{\frac{n_X n_Y}{n}}}
= \sqrt{n} \frac{supp(X \cup Y) - supp(X)supp(Y)}{\sqrt{supp(X)supp(Y)}} 
$$

**Range:** $[0, 1]$

## Leverage {#leverage}

**Alias:** Piatetsky-Shapiro, PS

**Reference:** @arules:Piatetsky-Shapiro:1991 $$
\mathrm{PS}(X \Rightarrow Y) = leverage(X \Rightarrow Y) 
= supp(X \Rightarrow Y) - supp(X)supp(Y) 
= P(X \cap Y) - P(X)P(Y)
$$

Leverage measures the difference of $X$ and $Y$ appearing together in
the data set and what would be expected if $X$ and $Y$ were
statistically dependent. The rationale in a sales setting is to find out
how many more units (items $X$ and $Y$ together) are sold than expected
from the independent sells.

Using minimum leverage thresholds incorporates at the same time an
implicit frequency constraint. E.g., for setting a min. leverage
thresholds to 0.01% (corresponds to 10 occurrences in a data set with
100,000 transactions) one first can use an algorithm to find all
itemsets with min. support of 0.01% and then filter the found item sets
using the leverage constraint. Because of this property, leverage also
can suffer from the rare item problem.

Leverage is a unnormalized version of the [phi correlation
coefficient](#phi-correlation-coefficient).

**Range:** $[-1, 1]$ (0 indicates independence)

## Lift {#lift}

**Alias:** Interest, interest factor

**Reference:** @arules:Brin:1997

Lift was originally called interest by Brin et al. Later, lift, the name
of an equivalent measure popular in advertising and predictive modeling
became more common. Lift is defined as

$$
\textrm{lift}(X \Rightarrow Y) 
= \textrm{lift}(Y \Rightarrow X) 
= \frac{conf(X \Rightarrow Y)}{supp(Y)} 
= \frac{P(Y | X)}{P(Y)} 
= \frac{P(X \cap Y)}{P(X)P(Y)}
= n \frac{n_{XY}}{n_X n_Y}
$$

Lift measures how many times more often $X$ and $Y$ occur together than
expected if they were statistically independent. A lift value of 1
indicates independence between $X$ and $Y$. For statistical tests, see
the [Chi-squared test statistic](#chi-squared), [Fisher's exact
test](#fishers-exact-test), and [hyper-confidence](#hyper-confidence).

Lift is not downward closed and does not suffer from the rare item
problem. However, lift is susceptible to noise in small databases. Rare
itemsets with low counts (low probability), which by chance occur a few
times (or only once) together, can produce enormous lift values.

**Range:** $[0, \infty]$ (1 means independence)

## Lift Increase {#lic}

**Reference:** @arules:Lopez:2014

Related to the [improvement](#improvement) measure but uses [lift](#lift). 
It divides the lift of a rules by the largest lift of any proper sub-rule 
with the same consequent.

$$
\mathrm{LIC}(X \Rightarrow Y) 
= min_{X' \subset X} \left[ \frac{lift(X \Rightarrow Y)}{lift(X' \Rightarrow Y)} \right]
$$

@arules:Lopez:2014 suggests that rules need to satisfy $LIC > 1.05$ to justify 
adding an item to the antecedent.

**Range:** $[0, \infty]$ ($>1$ means an increase)


## MaxConfidence {#maxconfidence}

**Reference:** @arules:Tan:2004

Symmetric, null-invariant version of confidence defined as

$$
\textrm{maxConf}(X \Rightarrow Y) 
= max\{ conf(X \Rightarrow Y),\ conf(Y \Rightarrow X) \}
= max\{ P(Y | X),\ P(X | Y) \}
$$

**Range:** $[0, 1]$

## Mutual Information {#mutualinformation}

**Alias:** Uncertainty

**Reference:** @arules:Tan:2004

[Mutual information](https://en.wikipedia.org/wiki/Mutual_information)
measures the information obtained about Y by observing X.

$$
\begin{aligned}
M(X \Rightarrow Y)
& = \frac{\sum_{i \in \{X, \overline{X}\}} \sum_{j \in \{Y, \overline{Y}\}} \frac{n_{ij}}{n} log \frac{n_{ij}}{n_i n_j}}{min(-\sum_{i \in \{X, \overline{X}\}} \frac{n_i}{n} log \frac{n_i}{n}, -\sum_{j \in \{Y, \overline{Y}\}} \frac{n_j}{n} log \frac{n_j}{n})} \\
& = \frac{\sum_{i \in \{X, \overline{X}\}} \sum_{j \in \{Y, \overline{Y}\}} P(i \cap j) log \frac{P(i \cap j)}{P(i) P(j)}}{min(-\sum_{i \in \{X, \overline{X}\}} P(i) log P(i), -\sum_{j \in \{Y, \overline{Y}\}} P(j) log P(j))}
\end{aligned}
$$

**Range:** $[0, 1]$ (0 means that X does not provide information for Y)

## Odds Ratio {#oddsratio}

**Reference:** @arules:Tan:2004

For the analysis of $2 \times 2$ contingency tables, the [odds
ratio](https://en.wikipedia.org/wiki/Odds_ratio) is a measure of the
relationship between two binary variables. It is defined as the ratio of
the odds of a transaction containing Y in the groups of transactions
that do and do not contain $X$.

$$
\mathrm{OR}(X \Rightarrow Y) 
= \frac{\frac{P(Y | X)}{1 - P(Y | X)}}{\frac{P(Y | \overline{X})}{1 - P(Y | \overline{X})}} 
= \frac{\frac{conf(X \Rightarrow Y)}{1 - conf(X \Rightarrow Y)}}{\frac{conf(\overline{X} \Rightarrow Y)}{1 - conf(\overline{X} \Rightarrow Y)}} 
= \frac{n_{XY} n_{\overline{X}\overline{Y}}}
{n_{X\overline{Y}} n_{\overline{X}Y}}
$$

A confidence interval around the odds ratio can be calculated
[@arules:Li:2014] using a normal approximation. $$
\omega = z_{\alpha/2} \sqrt{\frac{1}{n_{XY}} + \frac{1}{n_{X\overline{Y}}} + \frac{1}{n_{\overline{X}Y}} + \frac{1}{n_{\overline{X}\overline{Y}}}}
$$

$$
\mathrm{CI}(X \Rightarrow Y) = [OR(X \Rightarrow Y) \exp(-\omega), OR(X \Rightarrow Y) \exp(\omega)] 
$$

where $\alpha/2$ is the critical value for a confidence level of
$1-\alpha$.

**Range:** $[0, \infty]$ (1 indicates that Y is not associated with X)

## Phi Correlation Coefficient {#phi}

**Reference:** @arules:Tan:2004

The [Phi correlation
coefficient](https://en.wikipedia.org/wiki/Phi_coefficient) between the
transactions containing X and Y represented as two binary vectors. Phi
correlation is equivalent to Pearson's Product Moment Correlation
Coefficient $\rho$ with 0-1 values and related to the [chi-squared test
statistics](#chi-squared) for $2 \times 2$ contingency tables.

$$
\phi(X \Rightarrow Y) 
= \frac{n n_{XY} - n_Xn_Y}{\sqrt{n_X n_Y n_{\overline{X}} n_{\overline{Y}}}}
= \frac{P(X \cap Y) - P(X)P(Y)}{\sqrt{P(X) (1 - P(X)) P(Y) (1 - P(Y))}}
$$

In machine learning, Phi correlation is also known as the Matthews
correlation coefficient (MCC). The magnitude of the correlation is also
related to the chi-squared statistic:

$$
|\phi(X \Rightarrow Y)| = \sqrt{\frac{\chi^2}{n}}
$$

**Range:** $[-1, 1]$ (0 when X and Y are independent)}

## Ralambondrainy {#ralambondrainy}

**Reference:** @arules:Diatta:2007

Defined as the support of the counter examples.

$$ 
\mathrm{ralambondrainy}(X \Rightarrow Y) 
= \frac{n_{X\overline{Y}}}{n}
= supp(X \Rightarrow Y)
= P(X \cap \overline{Y})
$$

**Range:** $[0, 1]$ (smaller is better)

## Relative Linkage Disequilibrium {#rld}

**Reference:** @arules:Kennett:2008

RLD is an association measure motivated by indices used in population
genetics. It evaluates the deviation of the support of the whole rule
from the support expected under independence given the supports of X and
Y.

$$D = \frac{n_{XY} n_{\overline{X}\overline{Y}} - n_{X\overline{Y}} n_{\overline{X}Y}}{n}$$

$$\mathrm{RLD} = \begin{cases} 
 D / (D + min(n_{X\overline{Y}}, n_{\overline{X}Y})) \quad\textrm{if}\quad D>0 \\
 D / (D - min(n_{XY}, n_{\overline{X}\overline{Y}}))  \quad\textrm{otherwise.}\\
\end{cases}$$

**Range:** $[0, 1]$

## Relative Risk {#relativerisk}

**Reference:** @arules:Sistrom:2004

For the analysis of $2 \times 2$ contingency tables, relative risk is a
measure of the relationship between two binary variables. It is defined
as the ratio of the proportion of transactions containing $Y$ in the two
groups of transactions the do and do not contain $X$. In epidemiology,
this corresponds to the ratio of the risk of having disease $Y$ in the
exposed ($X$) and unexposed ($\overline{X}$) groups.

$$
\mathrm{RR}(X \Rightarrow Y)
= \frac{n_{XY} / n_X}{n_{\overline{X}Y} / n_{\overline{X}}} 
= \frac{P(Y | X)}{P(Y | \overline{X})} 
= \frac{conf(X \Rightarrow Y)}{conf(\overline{X} \Rightarrow Y)} 
$$

**Range:** $[0, \infty]$ ($RR = 1$ means $X$ and $Y$ are unrelated)

## Rule Power Factor {#rulepowerfactor}

**Reference:** @arules:Ochin:2008

Weights the confidence of a rule by its support. This measure favors
rules with high confidence and high support at the same time.

Defined as $$
\mathrm{rpf}(X \Rightarrow Y)
= supp(X \Rightarrow Y)\ conf(X \Rightarrow Y)
= \frac{P(X \cap Y)^2}{P(X)}
$$

**Range:** $[0, 1]$

## Right-Hand-Side Support {#rhssupport}

**Alias:** RHS support, consequent support

Support of the right-hand-side of the rule.

$$
\mathrm{RHSsupp}(X \Rightarrow Y) 
= supp(Y)
= P(Y)
$$

**Range:** $[0, 1]$

## Sebag-Schoenauer {#sebag}

**Reference:** @arules:Sebag:1988

Defined as $$
\mathrm{sebag}(X \Rightarrow Y)
= \frac{conf(X \Rightarrow Y)}{conf(X \Rightarrow \overline{Y})}
= \frac{P(Y | X)}{P(\overline{Y} | X)}
= \frac{supp(X \cup Y)}{supp(X \cup \overline{Y})}
= \frac{P(X \cap Y)}{P(X \cap \overline{Y})}
$$ i **Range:** $[0, 1]$

## Standardized Lift {#stdlift}

**Reference:** @arules:McNicholas:2008

Standardized lift uses the minimum and maximum lift that can reach for
each rule to standardize lift between 0 and 1. The possible range of
lift is given by the minimum

$$
\lambda = \frac{\mathrm{max}\{P(X) + P(Y) - 1, 1/n\}}{P(X)P(Y)}.
$$

and the maximum

$$
\upsilon = \frac{1}{\mathrm{max}\{P(X), P(Y)\}}
$$

The standardized lift is defined as

$$
\mathrm{stdLift}(X \Rightarrow Y) 
= \frac{\mathrm{lift}(X \Rightarrow Y) - \lambda}{ \upsilon - \lambda}.
$$

The standardized lift measure can be corrected for minimum support and
minimum confidence used in rule mining by replacing the minimum bound
$\lambda$ with

$$
\lambda^* 
= \mathrm{max}\left\{\lambda, \frac{4s}{(1+s)^2}, \frac{s}{P(X)P(Y)}, \frac{c}{P(Y)}\right\}.
$$

**Range:** $[0, 1]$

## Varying Rates Liaison {#varyingliaison}

**Reference:** @arules:Bernard:1996

Defined as the [lift](#lift) of a rule minus 1 (0 represents
independence).

$$
\mathrm{VRL}(X \Rightarrow Y) = lift(X \Rightarrow Y) -1
$$

**Range:** $[-1, \infty]$ (0 for independence)

## Yule's Q {#yuleq}

**Reference:** @arules:Tan:2004

Yule's Q, also called Yule coefficient of association is a special case
of the [Goodman and Kruskal's
gamma](https://en.wikipedia.org/wiki/Goodman_and_Kruskal%27s_gamma). It
is defined as $$ Q(X \Rightarrow Y) = \frac{\alpha-1}{\alpha+1} $$

where $\alpha = OR(X \Rightarrow Y)$ is the [odds ratio](#odds-ratio) of
the rule.

**Range:** $[-1, 1]$

## Yule's Y {#yuley}

**Reference:** @arules:Tan:2004

Yule's Y is also know as the [coefficient of
colligation](https://en.wikipedia.org/wiki/Coefficient_of_colligation)
to measure the association between two binary variables. It is defined
as $$ Y(X \Rightarrow Y) = \frac{\sqrt{\alpha}-1}{\sqrt{\alpha}+1} $$

where $\alpha = OR(X \Rightarrow Y)$ is the [odds ratio](#odds-ratio) of
the rule.

**Range:** $[-1, 1]$

# References
