Measures | Association Analysis | Vantage Analytics Library - Association Rule Measures

Support

Support is a measure of the generality of an entire association rule, its antecedent or consequent, or a single item that it references.

For an entire rule, antecedent, or consequent, Support is the percentage of groups that contain all items referenced by the rule, antecedent, or consequent.

For a single item, Support is the percentage of groups that contain it.

Assume these definitions:

N is the total number of customers.
L is the number of customers who own the set of products in the antecedent.
R is the number of customers who own the set of products in the consequent.
LR is the number of customers who own all products in the association rule (this notation does not mean L*R).

These are the formulas for support:

Support (L) = L/N
Support (R) = R/N
Support (L R) = LR/N

For example, assume there are 10 customers (N=10). Six have a checking account (L=6), five have a savings account (R=5), and four have both (LR=4). Support (L) = 6/10 = 0.6, Support (R) = 5/10 = 0.5, and Support (L R) = 4/10 = 0.4.

Confidence

The Confidence of an association rule is the probability of R occurring in an item group given that L is in the item group:

Confidence (L R) = Support (L R) = ) / Support (L)

Equivalently, Confidence is the percentage of groups containing L that also contain R:

Confidence (L R) = LR/L

For example, the Confidence that checking account ownership implies savings account ownership is 4/6.

Expected Value

The Expected Value of an association rule is the number of customers expected to have both L and R if there is no relationship between L and R. No relationship between L and R means customers who have L are neither more nor less likely to have R than are customers who do not have L.

These formulas for Expected Value (E_Value) are equivalent:

E_Value (L R) = (L*R)/N
E_Value (L R) = (Support (L)) * (Support (R)) * N

The Expected Value of the number of customers with checking and savings is (6*5)/10 = 3.

Expected Confidence

The Expected Confidence of an association rule is the Confidence that results if there is no relationship between L and R:

E_Confidence (L R) = R/N

Because owning L has no effect on owning R, the Expected Confidence of the rule is also the percentage of customers who own R:

E_Confidence (L R) = Support (R)

The Expected Confidence of the rule that having a checking account implies having a savings account is 5/10.

Lift

The Lift of an association rule is how much the probability of R is increased by the presence of L in an item group. For example:

Lift (L R) = 1 means there are exactly as many occurrences of R as expected. The presence of L neither increases nor decreases the probability of R.
Lift (L R) = 5 means there are 5 times as many occurrences of R than expected. The presence of L increases the probability of R by 5.
Lift (L R) = 0.5 means there are half as many occurrences of R as expected. The presence of L decreases the probability of R by half.

This is a formula for Lift:

Lift (L R) = LR / E_Confidence (L R)

Equivalently, lift is the ratio of Confidence to Expected Confidence, and you can calculate it by either of these formulas:

Lift (L R) = (Confidence (L R))/E_Confidence (L R)
Lift (L R) = (Confidence (L R)) * (Support (R)) * N

The Lift of the rule that having a checking account implies having a savings account is 4/3.

The Lift formulas do not account for ordered items. For more information, see Sequence Analysis.

Z-Score

The Z-score of an association rule is the statistical difference between the actual and expected results. For example:

Z-score (L R) = 0 means the actual and expected results are the same. The presence of L neither increases nor decreases the likelihood of owning R.
Z-score (L R) = 1 means the actual result is 1 standard deviation more than the expected result. The presence of L increases the likelihood of owning R.
Z-score (L R) = -3 means the actual result is 3 standard deviations less than the expected result. The presence of L decreases the likelihood of owning R.

A Z-score greater than 3 or less than -3 is statistically significant, which means the difference between the actual and expected result is very unlikely to be due to chance.

A Z-score helps answer the question of how confident you can be about the observed relationship between L and R, but does not directly indicate the magnitude of the relationship.

These formulas for Z-score are equivalent:

Z-score (L R) = (LR - E_Value (L R)) / SQRT (E_Value (L R)(1 - (E_Value (L R) / N)))
Z-score (L R) = ((N * Support (LR) - N) * Support (L) * Support (R)) / SQRT (N * Support (L) * Support (R) * (1 - Support (L) * Support (R)))

The mean value is E_Value (L R). The expected value is 6*5/10, so the mean value is 3.

The actual value is LR, which is 4.

The standard deviation is SQRT (E_Value (L R) * (1 - (E_Value (L R) / N)). The standard deviation is SQRT(3*(1-3/10)) = 1.449.

Therefore, the Z-score is (4 - 3) / 1.449 = .690.

The Z-score formulas do not account for ordered items. For more information, see Sequence Analysis.