Tests for Linear Trends in Proportions and Frequencies

One frequently encounters data consisting of a series of proportions, occurring in groups which fall into some natural order. The question usually asked is then not so much whether the proportions differ significantly, but whether they show a significant trend, upwards or downwards, with the ordering of the groups, In the data shown in Table 1, for, instance, the usual test for a 2 X 3 contingency table yields a x2 equal to 7.89 on 2 degrees of freedom, corresponding to a probability of about 0.02. But this calculation takes no account of the fact that the carrier rate increases with the tonsil size, and it is reasonable to believe that a test specifically designed to detect a trend in the carrier rate as the tonsil size increases would show a much higher degree of significance.