Suppose a grant program allows city administrators to allocate the grant across three mutually exclusive categories 1, 2, 3.
City administrators are surveyed to elicit their preferred allocations. A response by administrator i is a triplet R_i = (p1, p2, p3), where by definition: 0 <= pi <= 100 for all i in {1,2,3} and p3 = 100 - p1 - p2. In words, p1, p2, p3 represent the proportion of grant money he or she would allocate to each of three categories 1, 2, 3.
Suppose 80 administrators are surveyed. Of these 34 selected at random had previously received a prompt meant to shift the distribution of allocations in favor of p1, at the expense of the other two items. The motivation is that the grantor of money has preference ordering 1 > 2 > 3.
I want to test the null that elicited budget allocations across treatment and control are the same, against the one-sided alternative of a location shift towards p1. Due to small sample size, I am mostly interested in non-parametric approaches.
An obvious possibility is to test p1_T = p1_C, for treatment and control, against p1_T > p1_C. But I am not sure this makes the most effective use of the information, as any such difference must also imply a decline in p2_T or p3_T or both relative to control. In addition, I am not sure marginal test of homogeneity (like Chi square test) offer a directed alternative.
For these reasons I am considering a weighted response like Y = p1*1 + p2*2 + p3*3 for each administrator, as the categories are ordered. The alternative would then be Y_T < Y_C.
Is this correct? Any suggestions and rationale?
