Fruit trees are subject to a bacteria-caused disease commonly called fire blight (because the resulting dead branches look like they have been burned). One can imagine several different treatments for this disease: treatment A: no action (a control group), treatment B: careful removal of clearly affected branches, and treatment C: frequent spraying of the foliage with an antibiotic in addition to careful removal of clearly affected branches. One can also imagine several different outcomes from the disease: outcome 1: tree dies in same year as the disease was noticed, outcome 2: tree dies 2-4 years after disease was noticed, outcome 3: tree survives beyond 4 years. A group of N trees are assorted into one of the treatments (i.e., every tree falls into exactly one of the following treatment categories [ A | B | C ] ) and over the next few years the outcome is recorded (i.e., every tree falls into exactly one of the following outcome categories [ 1 | 2 | 3 ] ). If we count the number of trees in a particular treatment/outcome pair (e.g., the number of trees that received treatment B and lived beyond 4 years: #B3), we can display the results in a table called a contingency table:
| Treatment | ||||
|---|---|---|---|---|
| Outcome | A | B | C | Row Totals |
| 1 | #A1 | #B1 | #C1 | total 1 |
| 2 | #A2 | #B2 | #C2 | total 2 |
| 3 | #A3 | #B3 | #C3 | total 3 |
| Column Totals | total A | total B | total C | Grand Total |
For example the contingency table
A B C
1 5 3 2 10
2 2 3 4 9
3 0 2 3 5
7 8 9 24
reports that of the 24 trees selected for this experiment, 4 of them
received the full treatment (C) and nevertheless died in 2-4 years (outcome 2).
It looks as if either treatments B or C are better than the
control, and maybe C is a better treatment than B. But how
can we quantify whether treatment actually helps?
The null hypothesis is that the probabilities for each outcome are independent of the treatment. For example, since overall 10/24 of the trees died in the first year, if treatment has no effect one might estimate that 10/24 is the probability for early death; that, in turn, would suggest that in the control group early death should have been expected for 7×10/24=2.92 trees whereas 5 were observed. Following this logic through, under the null hypothesis the expected contingency table would be:
A B C
1 2.92 3.33 3.75 10
2 2.63 3.00 3.38 9
3 1.46 1.67 1.88 5
7 8 9 24
One can now use chi-square (actually X2) to compare the expected contingency table
to the observed contingency table. Please note a problem: we said chi-square
is suspect if expected values are less than 5; all expected values are too
small here! For what its worth, chi-square reports the null hypothesis
is still OK with this data. (In fact, re-binning: putting outcomes 1&2
together into "tree death" and putting treatments B&C together under
"some action" and using Fisher Exact Test on the resulting 2x2, still
shows a viable null hypothesis.)
Possible solutions to the problem of sparsely populated cells is discussed in greater detail here.
The previous example is called a 3x3 contingency table; more generally we have #row x #column contingency tables.
While contingency tables are most commonly analyzed using X2, there is an exact method (Fisher Exact Test) which avoids the concerns of small expected values, but which is more difficult to compute. This server will compute exact p for up to 6x6 contingency tables. Click here for more information.