# Does this quantity related to independence have a name?

Obviously events A and B are independent iff Pr$(A\cap B)$ = Pr$(A)$Pr$(B)$. Let's define a related quantity Q:

$Q\equiv\frac{\mathrm{Pr}(A\cap B)}{\mathrm{Pr}(A)\mathrm{Pr}(B)}$

So A and B are independent iff Q = 1 (assuming the denominator is nonzero). Does Q actually have a name though? I feel like it refers to some elementary concept that is escaping me right now and that I will feel quite silly for even asking this.

• I think it depends on the context. Note that $$Q = \frac{\Pr(A|B)}{\Pr(A)} = \frac{\Pr(B|A)}{\Pr(B)}$$ so that $\Pr(A|B) = Q \Pr(A)$ and $\Pr(B|A) = Q \Pr(B)$. This form has more of a Bayesian inference flavor.
– vqv
Jan 7 '11 at 14:23
• This SE could do with some more "quite silly" questions. It's very intimidating, even for someone who enjoyed basic undergrad level statistics. +1 for silliness Feb 8 '12 at 13:49
• This question was asked in Mathematics: About joint probability divided by the product of the probabilities.
– user10525
Oct 22 '12 at 18:55
• Go for "Migdal Probability" ;) Oct 22 '12 at 18:56
• @PiotrMigdal Thanks for the kind offer. I would prefer to see your own answer. Maybe including how you came up with this question and how that quantity can be useful.
– user10525
Oct 22 '12 at 19:16

It's observed to expected ratio (abbreviation: o/e).

Quoting an answer to About joint probability divided by the product of the probabilities at Math.SE (pointed out by Procrastinator):

Then, at least in the environmental, medical and life sciences literature, P(A∩B)/(P(A)P(B)) is called the observed to expected ratio (abbreviation o/e). The idea is that the numerator is the actual probability of A∩B while the denominator is what it would be if A and B were independent.

I think that you are looking for Lift (or improvement). Lift is the ratio of the probability that A and B occur together to the multiple of the two individual probabilities for A and B. It is used to interpret the importance of a rule in association rule mining. Lift is a way to measure how much better a model is over benchmark and it is defined as the confidence divided by the benchmark, where any value that is greater that one suggest that there is some usefulness to the rule. See this page also as another example.

• (+1) Nice answer. The arules vignette also have some good references about lift.
– chl
Jan 7 '11 at 9:45
• Thanks, that's probably where I've seen it before. I think I've seen lift with a slightly different definition in machine learning context before though...I hate that sometimes there is a lack of consensus over a definition while other times there are many terms for the same concept. Jan 7 '11 at 14:59

The correspondence analysis folk call one of these quantities a contingency ratio, in the context of cross-tabulated counts. The distances of multiple such ratios from 1 are what biplots visualize. See e.g. Greenacre (1993) ch.13.

The old-school machine learning feature selection folk call the log of this quantity pointwise mutual information. See e.g. Manning and Schütze (1999) p.66.

• Thanks for pointing out "contingency ratio" and "pointwise mutual information". Nov 3 '12 at 14:09

In Data Mining it seems they call this lift.

Maybe you are asking how this quantity is related to the Odds Ratio, as a quantity for measuring independence.

I think you are searching for "Relation to statistical independence". See http://en.wikipedia.org/wiki/Odds_ratio