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The R Book by Michael J. Crawley

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The Danger of Contingency Tables

We have already dealt with simple contingency tables and their analysis using Fisher's exact test or Pearson's chi-squared (see p. 308). But there is an important further issue to be dealt with. In observational studies we quantify only a limited number of explanatory variables. It is inevitable that we shall fail to measure a number of factors that have an important influence on the behaviour of the system in question. That's life, and given that we make every effort to note the important factors, there is little we can do about it. The problem comes when we ignore factors that have an important influence on the response variable. This difficulty can be particularly acute if we aggregate data over important explanatory variables. An example should make this clear.

Suppose we are carrying out a study of induced defences in trees. A preliminary trial has suggested that early feeding on a leaf by aphids may cause chemical changes in the leaf which reduce the probability of that leaf being attacked later in the season by hole-making insects. To this end we mark a large cohort of leaves, then score whether they were infested by aphids early in the season and whether they were holed by insects later in the year. The work was carried out on two different trees and the results were as follows:

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There are four variables: the response variable, count, with ...

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