Checking independence in a table

What you will be able to do

• check independence in a table by comparing conditional probabilities;
• work out the counts a table would have if two answers were independent;
• explain why small differences in a real sample may be just chance.

Does knowing one answer change the other?

In the choir table, $P(\text{instrument} \mid \text{choir}) = 0.6$, but $P(\text{instrument} \mid \text{not choir}) = \dfrac{36}{160} = 0.225$. Knowing a student's choir answer changes the chance of the instrument answer by a lot. So the two answers are not independent. We say they are associated.

Here is a different made-up survey of $300$ adults: are you left-handed, and do you drink coffee every day?

$P(\text{coffee} \mid \text{left}) = \dfrac{12}{30} = 0.4$ and $P(\text{coffee} \mid \text{right}) = \dfrac{108}{270} = 0.4$. Both are equal to $P(\text{coffee}) = \dfrac{120}{300} = 0.4$. Knowing the hand tells you nothing about coffee. These two answers are independent.

Three tests that agree

For two answers $A$ and $B$ in a table, each of these says the same thing:

• $P(A \mid B) = P(A \mid \text{not } B)$: the share of $A$ is the same in both groups;
• $P(A \mid B) = P(A)$: the share of $A$ in the group equals the share overall;
• $P(A \cap B) = P(A) \times P(B)$: the product rule of the course on independence.

For the coffee table, $P(\text{left and coffee}) = \dfrac{12}{300} = 0.04$, and $0.1 \times 0.4 = 0.04$. The product rule holds.

The expected counts

If two answers were independent, what would each count be? The left-handed adults are $30$ of the $300$. If they drank coffee at the same rate as everyone, $0.4$ of them would drink coffee, which is $12$ adults. In general,

For the choir table, the expected count of instrument and choir is $\dfrac{60 \times 40}{200} = 12$. The survey found $24$, twice as many. That is a sign of association.

Try it in Python

The program works out the expected counts for the choir table and compares them with the counts that were observed.

Real samples are never exact

Real data almost never give exactly equal shares, even when two answers are truly independent. A sample is a little lucky or unlucky, as in the coin simulations of the first course. A share of $0.42$ in one group and $0.39$ in the other may be just chance. A share of $0.6$ against $0.225$ in groups this size is very unlikely to be chance. The last chapter of this book shows how to decide.

One more warning. Association is not cause. Choir and instrument go together, but neither has to cause the other. The chapter on study design looks at this.

Your turn

The usual mistakes

• Comparing counts instead of shares. Groups of different sizes need shares: $12$ of $30$ and $108$ of $270$ are the same share.
• Calling a tiny difference proof. In a sample, small differences can come from chance alone.
• Mixing up independent and mutually exclusive. "Left-handed" and "coffee" overlap a lot and are still independent.
• Reading association as cause. A link in a table does not say which answer, if either, causes the other.

Remember these

• Independent in a table: the share of one answer is the same in every group of the other.
• Equivalent tests: $P(A \mid B) = P(A)$, or $P(A \cap B) = P(A)\,P(B)$.
• Expected count if independent: row total times column total, divided by the grand total.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. In the coffee table, find P(left-handed given coffee) and P(left-handed). Are they equal?

2. A made-up table of $400$ people has $100$ who walk to work and $160$ who own a dog. If these were independent, how many dog owners would walk to work?

3. In a made-up club, $30\%$ of the members swim, $30\%$ of the adults swim and $30\%$ of the children swim. Is swimming independent of being an adult?

4. For two answers, $P(A) = 0.5$, $P(B) = 0.2$ and $P(A \cap B) = 0.1$. Are they independent?

5. A made-up sample finds that $52\%$ of one group and $49\%$ of another say yes. Does this prove that the answer depends on the group?

References

• OpenStax, Introductory Statistics, Chapter 3, the sections on independent events and on contingency tables.