Predictive strength of Jonckheere's test for trend: an application to genotypic scores in HIV infection

A problem arising in studies on the human immunodeficiency virus (HIV) infection relate to one-sided tests with ordered alternatives as opposed to the more classical two-sided tests. Patients not having a resistance mutation may have a better virologic response to treatment than patients with a single mutation. In turn, those with a single mutation may have a better response to treatment than those patients having two mutations, and so on. In the presence of a continuous outcome, Jonckheere's test for ordered alternatives is well adapted to this situation. Such an analysis does not provide any measure of prediction or explained variation which can complement these results. A measure of strength of effect would be helpful in quantifying the degree of association between the genotypic score (number of mutations) and some continuous virological response. We suggest a simple measure of 'goodness of split' for Jonckheere's test for trend. Interestingly, the measure can be related to the non-parametric measure of association known as gamma. The variance formula for the measure studied here can be seen to differ from the known variance estimate of the gamma measure, and simulations show it to be more accurate. Expectation and variance under H(0) of the measure are provided and a large simulation study is presented. Methods are applied to a recent clinical data set involving HIV-1 infected patients where the number of resistance mutations are investigated as potential predictors of the amount of HIV-1 RNA reduction at week 4.