GMO Statistics Part 15. GMO Pundit's advice for the statistically challenged: Do different treatments have different effects?

Contingency Tables

Do different treatments cause different effects?

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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.)...



Answers available @ Contingency Tables: www.physics.csbsju.edu