Why are all my lift values the same?

Question

I'm trying to measure association in my basket based on transactions. Let's imagine I have 5 products. Then, I calculate the number of client who bought product A and B, A and C, B and C and so on.

Total columns is the number of clients who bought the product.

I calculate penetration rate (or support when we use association rule) = number of client who bought product A and B / number of client (in my example 5)

Then, I calculate kind of confidence = number of client who bought product A and B / number of client who bought product A

Finally, I calculate my lift = confidence / support.

Results are here:

1. Why my lift value are all the same by line?

=> I know why. Because support = p(a) * p(b) and confidence = p(a) * p(b) / p(a) so lift = 1/p(a)

2. I need to take into account the volume effect because the products highlighted in my analysis are always those that people buy the most, so what techniques do you know for this problem?

Answer 1

You want to know how buying B affects buying A, that's the probability that someone has bought A given that they bought B which is written as $P(A|B)$.

$P(A|B)=\frac{P(A \text{ and } B)}{P(B)}$

The probability of someone buying B is computed from your sales data. Suppose you have $N$ customers, then $P(B)=\frac{\text{number buying B}}{N}$

Likewise, the probability of someone buying A and B is $P(A \text{ and } B)=\frac{\text{number buying A and B}}{N}$

Putting those expressions into the first formula gives:

$P(A|B)=\frac{\frac{\text{number buying A and B}}{N}}{\frac{\text{number buying B}}{N}}= \frac{\text{number buying A and B}}{\text{number buying B}}$

Is this the kind of association you were trying to calculate? Perhaps you were looking for a more complicated way to measuring the relationship.