Support, Confidence, and Lift

A rule with 90% confidence can be worthless. Lift tells you which of your discovered rules carry any information at all.

Ch 7 7.6

Misconception

Unsupervised Learning & Text

18 mindifficulty 3/5

Controls

Transactionsn4000

Popularity of milk0.600

Pushes milk toward appearing in nearly every basket.

Planted diapers → beer lift2.500

The one genuine dependence in the data. Below 1 it becomes a repulsion.

Minimum support2.0%

Minimum confidence50%

Rank rules by

Random seed

Every result on this page is a deterministic function of the seed and the controls.

Confidence against lift

Everything above the horizontal line at lift = 1 carries information. Everything below it is a warning.

planted diapers → beerrule concluding “milk”passes both thresholdsfiltered out

Rules passing filters

of 42 candidates

Passing but lift < 1

A is evidence against B

Planted rule confidence

21.9%

lift 1.90

Top rule by your ranking

diapers → milk

lift 1.02

Top rules by confidence

Sort by confidence and by lift, and compare the two lists.

rule	support	confidence	lift	leverage	conviction
diapers → milk	0.062	0.734	1.018	0.0011	1.05
bread → milk	0.249	0.730	1.013	0.0032	1.04
beer → milk	0.084	0.727	1.009	7.82e-4	1.02
coffee → milk	0.139	0.715	0.992	-0.0011	0.98
eggs → milk	0.177	0.709	0.983	-0.0030	0.96
batteries → milk	0.037	0.692	0.960	-0.0015	0.91

Confidence rewards popularity, not association

“diapers → milk” has confidence 73.4% and lift 1.02. Milk is simply in most baskets, so almost any antecedent predicts it. Lift divides by P(milk) and the apparent finding collapses.

Predict before you look

Milk appears in 90% of baskets. You mine a rule “bread → milk” with confidence 0.91. How strong is the association?

Scenarios

Teacher-ready starting points.

Documentation

What this shows

A simulated basket dataset over seven items in which exactly one dependence is planted (diapers → beer, with a lift you set) and one item (milk) can be made almost ubiquitous. Every single-antecedent rule is mined, scored on five interest measures, and plotted in the confidence–lift plane so the difference between popularity and association is visible in one figure.

Five measures of one 2×2 table

Every rule A → B reduces to a 2×2 contingency table of counts. Support is the joint probability P(A ∩ B): it measures how often the rule applies at all, and thresholding it is what makes Apriori tractable, since any superset of an infrequent itemset is infrequent.

Confidence is the conditional probability P(B | A). Its flaw is that it never looks at P(B): if B appears in 90% of baskets then every rule concluding B has confidence near 0.9, whatever A is. Lift divides confidence by P(B), so lift = 1 means exactly independence, greater than 1 means positive association, and less than 1 means the antecedent is evidence against the consequent.

Leverage measures the same departure additively, P(A∩B) − P(A)P(B), which keeps rare-item deviations in proportion. Conviction, (1 − P(B))/(1 − confidence), asks how much more often the rule would be violated under independence, and diverges as confidence approaches 1. Lift and leverage are symmetric in A and B — which is the clearest statement that these are measures of association, not of direction, and certainly not of cause.

support(A → B) = P(A ∩ B)

confidence(A → B) = P(A ∩ B) / P(A) = P(B | A)

lift(A → B) = confidence / P(B) = P(A ∩ B) / (P(A)P(B))

leverage = P(A ∩ B) − P(A)P(B)

conviction = (1 − P(B)) / (1 − confidence)

Try this

Set the milk popularity to 0.9 and rank by confidence. What do the top rules have in common?
Switch the ranking to lift. Where did the milk rules go, and what took their place?
Set the planted lift to 1.0. Can any measure still tell that diapers and beer are independent?
Set the planted lift to 0.6 and find the rules with lift below 1. What would acting on them do?
With n = 500 and minimum support 0.002, press New draw repeatedly and watch the top-lift rule change.

Common misconception

“This rule has 92% confidence, so buying A strongly predicts buying B.”

Compare it with P(B). If B is in 90% of baskets, a 92% confidence rule is barely distinguishable from independence and its lift is 1.02 — the rule tells you almost nothing you did not know from B's marginal frequency. High confidence is necessary but not sufficient; report lift or leverage alongside it, and remember all three are symmetric associations rather than causal claims.

DSPA3 Chapter 7 — Association rule learning