# Association Rule Lift

An Association Rule Lift is an association rule performance measure that is the ratio between confidence measure [math]conf(X,\Rightarrow Y)[/math] and support measure [math]supp(X)[/math].

**AKA:**Lift Measure, Interest Measure, Lift (Association Rule Measure).**Context:**- It can be calculated as:
[math]lift(X \Rightarrow Y ) = \dfrac{conf(X \Rightarrow Y )}{ supp(Y)}= \dfrac{P (Y = y|X = x)}{P (Y = y)} [/math].

- It can be calculated as:
**Example(s)****Counter-Example(s):**

**See:** Association Rule Learning Task, Predictive Efficacy, Mean Absolute Error, Mean Squared Error, ROC Analysis, Negative Predictive Value, Positive Predictive Value, Accuracy, Precision, Recall, Sensitivity, Specificity.

## References

### 2018a

- (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Association_rule_learning#Lift Retrieved:2018-10-7.
- The
*lift*of a rule is defined as: [math] \mathrm{lift}(X\Rightarrow Y) = \frac{ \mathrm{supp}(X \cup Y)}{ \mathrm{supp}(X) \times \mathrm{supp}(Y) } [/math] or the ratio of the observed support to that expected if X and Y were independent. For example, the rule [math] \{\mathrm{milk, bread}\} \Rightarrow \{\mathrm{butter}\} [/math] has a lift of [math] \frac{0.2}{0.4 \times 0.4} = 1.25 [/math] .If the rule had a lift of 1, it would imply that the probability of occurrence of the antecedent and that of the consequent are independent of each other. When two events are independent of each other, no rule can be drawn involving those two events.

If the lift is > 1, that lets us know the degree to which those two occurrences are dependent on one another, and makes those rules potentially useful for predicting the consequent in future data sets.

If the lift is < 1, that lets us know the items are substitute to each other. This means that presence of one item has negative effect on presence of other item and vice versa.

The value of lift is that it considers both the confidence of the rule and the overall data set.

- The

### 2018b

- (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Lift_(data_mining) Retrieved:2018-10-7.
- In data mining and association rule learning,
**lift**is a measure of the performance of a targeting model (association rule) at predicting or classifying cases as having an enhanced response (with respect to the population as a whole), measured against a random choice targeting model. A targeting model is doing a good job if the response within the target is much better than the average for the population as a whole. Lift is simply the ratio of these values: target response divided by average response.For example, suppose a population has an average response rate of 5%, but a certain model (or rule) has identified a segment with a response rate of 20%. Then that segment would have a lift of 4.0 (20%/5%).

Typically, the modeller seeks to divide the population into quantiles, and rank the quantiles by lift. Organizations can then consider each quantile, and by weighing the predicted response rate (and associated financial benefit) against the cost, they can decide whether to market to that quantile or not.

Lift is analogous to information retrieval's average precision metric, if one treats the precision (fraction of the positives that are true positives) as the target response probability.

The lift curve can also be considered a variation on the receiver operating characteristic (ROC) curve, and is also known in econometrics as the Lorenz or power curve [1].

- In data mining and association rule learning,

### 2018c

- (Hahsler, 2018) ⇒ Michael Hahsler (2018). "Lift". In: "A Probabilistic Comparison of Commonly Used Interest Measures for Association Rules" Retrieved: 2018-10-05
- QUOTE: Introduced by S. Brin, R. Motwani, J. D. Ullman, and S. Tsur. Dynamic itemset counting and implication rules for market basket data. In Proc. of the ACM SIGMOD Int'l Conf. on Management of Data (ACM SIGMOD '97), pages 265-276, 1997. Lift was originally called interest. It is defined as[math]lift(X\Rightarrow Y)=lift(Y\Rightarrow X)=\dfrac{conf(X\Rightarrow Y)}{supp(Y)}=\dfrac{conf(Y\Rightarrow X)}{supp(X)}=\dfrac{P(X\cap Y)}{P(X)P(Y)}[/math]
Lift measures how many times more often [math]X[/math] and [math]Y[/math] occur together than expected if they were statistically independent. A lift value of 1 indicates independence between [math]X[/math] and [math]Y[/math].

Lift is not downward closed and does not suffer from the rare item problem. Also, 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: [math][0,\infty][/math] (1 means independence).

- QUOTE: Introduced by S. Brin, R. Motwani, J. D. Ullman, and S. Tsur. Dynamic itemset counting and implication rules for market basket data. In Proc. of the ACM SIGMOD Int'l Conf. on Management of Data (ACM SIGMOD '97), pages 265-276, 1997. Lift was originally called interest. It is defined as

### 2017

- (Sammut & Webb, 2017) ⇒ (2017) "Lift". In: (Sammut & Webb, 2017) DOI:10.1007/978-1-4899-7687-1_474
- QUOTE: Lift is a measure of the relative utility of a classification rule. It is calculated by dividing the probability of the consequent of the rule, given its antecedent by the prior probability of the consequent:
[math]lift(x \to y)= P (Y = y|X = x)/P (Y = y) [/math]

In practice, the probabilities are usually estimated from either training data or test data. In this case,

[math]lift(x \to y) = F (Y = y|X = x)/F(Y=y)[/math]

where [math]F(Y = y|X = x)[/math] is the frequency with which the consequent occurs in the data in the context of the antecedent and [math]F(Y = y)[/math] is the frequency of the consequent in the data.

- QUOTE: Lift is a measure of the relative utility of a classification rule. It is calculated by dividing the probability of the consequent of the rule, given its antecedent by the prior probability of the consequent:

### 2011a

- (Han, Pei & Kamber, 2011) ⇒ Jiawei Han, Jian Pei, and Micheline Kamber (2011). "Data mining: concepts and techniques" (PDF). Elsevier. pp. 266 ISBN 978-0-12-381479-1
- QUOTE: Lift is a simple correlation measure that is given as follows. The occurrence of itemset
*A*is**independent**of the occurrence of itemset*B*if [math]P(A \cup B) = P(A)P(B)[/math]; otherwise, itemsets*A*and*B*are**dependent**and**correlated**as events. This definition can easily be extended to more than two itemsets. The lift between the occurrence of*A*and*B*can be measured by computing[math]lift (A, B) =\dfrac{P(A \cup B)} {P(A)P(B)}\quad\quad \quad[/math](6.8)If the resulting value of Eq. (6.8) is less than 1, then the occurrence of

*A*is negatively correlated with the occurrence of*B*, meaning that the occurrence of one likely leads to the absence of the other one. If the resulting value is greater than 1, then*A*and*B*are positively correlated, meaning that the occurrence of one implies the occurrence of the other. If the resulting value is equal to 1, then*A*and*B*are independent and there is no correlation between them.Equation (6.8) is equivalent to [math]P(B|A)/P(B)[/math], or [math]conf (A \Rightarrow B)/sup(B)[/math], which is also referred to as the lift of the association (or correlation) rule [math]A \Rightarrow B[/math]. In other words, it assesses the degree to which the occurrence of one “lifts” the occurrence of the other. For example, if

*A*corresponds to the sale of computer games and*B*corresponds to the sale of videos, then given the current market conditions, the sale of games is said to increase or “lift” the likelihood of the sale of videos by a factor of the value returned by Eq. (6.8).

- QUOTE: Lift is a simple correlation measure that is given as follows. The occurrence of itemset

### 2011b

- (Tuffery, 2011) ⇒ Stéphane Tufféry (2011);
*Data Mining and Statistics for Decision Making*, Chichester, GB: John Wiley & Sons, translated from the French*Data Mining et statistique décisionnelle*(Éditions Technip, 2008).

### 2008

- (Hahsler & Hornik, 2007) ⇒ Michael Hahsler, and Kurt Hornik (2007). "New probabilistic interest measures for association rules". Intelligent Data Analysis, 11(5), 437-455. arXiv:0803.0966
- QUOTE: Typically, rules mined using minimum support (and confidence) are filtered or ordered using their lift value. The measure lift (also called interest [9] is defined on rules of the form [math]X \Rightarrow Y[/math] as [math]lift(X \Rightarrow Y ) = \dfrac{conf(X \Rightarrow Y )}{ supp(Y)}\quad\quad\quad[/math] (6)
A lift value of 1 indicates that the items are co-occurring in the database as expected under independence. Values greater than one indicate that the items are associated. For marketing applications it is generally argued that lift > 1 indicates complementary products and lift < 1 indicates substitutes [6, 2].

- QUOTE: Typically, rules mined using minimum support (and confidence) are filtered or ordered using their lift value. The measure lift (also called interest [9] is defined on rules of the form [math]X \Rightarrow Y[/math] as

### 1998

- (Aggarwal & Yu,1998) ⇒ Charu C. Aggarwal, and Philip S. Yu (1998, May). "A new framework for itemset generation". In Proceedings of the seventeenth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems (pp. 18-24). ACM.

### 1997

- (Brin et al.,1997) ⇒ Sergey Brin, Rajeev Motwani, Jeffrey D. Ullman, and Shalom Tsur (1997). "Dynamic itemset counting and implication rules for market basket data". Acm Sigmod Record, 26(2), 255-264.

### 1990

- (Betancourt & Gautschi,1990) ⇒ Roger Betancourt, and David Gautschi (1990). "Demand complementarities, household production, and retail assortments". Marketing Science, 9(2), 146-161.