ABSTRACT
The exploratory analysis of Data Set 3, the Uruguayan rice –elds, in Chapters 7 and 8 indicated that irrigation effectiveness plays an important role in distinguishing high-yielding from low-yielding –elds. Irrigation effectiveness is represented by the ordinal variable Irrig, which is the expert agronomist’s rating of irrigation effectiveness at each measured site. Can the association between Irrig and Yield be scaled down? That is, is irrigation effectiveness consistently associated with yield at the single –eld scale, or, alternatively, is this an example of the ecological fallacy discussed in Section 11.5.2? Figure 12.1 shows a trellis plot, made with the lattice package (Sarkar, 2008) function xyplot(), of yield versus irrigation effectiveness for each –eld. Yield generally seems to increase with irrigation effectiveness, although there is a lot of variability, and some –elds actually seem to show a decreasing yield with increasing effectiveness. The individual plots generally look as though they would be appropriate for linear regression. The variable Irrig, however, is an ordinal scale variable, and as such the operations of multiplication and addition, which are used to compute the regression coef–cients, are not meaningful when applied to its values. In principle, it is not appropriate to interpret a regression as indicating that yield increases by a certain percent for each unit increase in irrigation effectiveness. The issue of treating an ordinal scale variable as a response variable in a regression model was discussed in Section 8.3.3, and we will not go over that again. For the present application, our interest is not in predicting a value of yield based on irrigation effectiveness but rather on computing the regression coef–cient in order to test the null hypothesis that this coef–cient is equal to zero. We will consider this an acceptable breach of the rules.
