### Abstract:

A person's preferences for m objects are often expressed as a ranking or permutation of all these m objects. Sometimes just some, say the top q (≤m) objects are listed - this will be called a q-permutation. If people are indifferent to the objects, all permutations have the same probability. This leads to the uniform distribution on permutations. However, people usually have preferences, and they may differ from group to group. Hence, it is necessary to have non-uniform distributions and methods to fit them. Goodness-of-fit tests of such models will be needed. Very few non-uniform distributions have been suggested in the literature. The only test of uniformity deals with full, not partial permutations. This thesis suggests models for q-permutations, goodness-of-fit tests for them including a test for uniformity of q-permutations and a test whether different groups rank similarly. Two new statistical models for q-permutations are introduced. The first model is defined by successive sampling from q vases that may have different proportions of balls labeled l,...,m. A vase is an urn where the number of balls may be infinite. The second is the Log-Linear model. It may be obtained by modifying the sampling scheme on these vases. The sufficient statistics of the Log-Linear models are the numbers of rankers who rank the ith object jth. The likelihood ratio test of the Log-Linear model versus an unrestricted alternative has an asymptotic chi-square distribution whose degrees of freedom are derived here. The Log-Linear model contains the uniform distribution; the likelihood ratio test of the uniform versus the Log-Linear alternative is shown to have (m-l)q degrees of freedom. A hierarchical family of Log-Linear models can be defined that range from the uniform distribution on q-permutations to the full Log-Linear model. This family may be used to test if the object that is ranked ith has a uniform distribution over all m objects. Log-Linear models are fitted to two different datasets. The Log-Linear models seem to fit well and are parsimonious. These data sets have several groups of rankers. Separate Log-Linear models maybe fitted to each group or one model may be fitted to the entire dataset. The formula for asymptotic degrees of freedom for the likelihood ratio test of the separate models versus one common model, are derived. In this way we can test if the rankings of two or more groups are just different samples from the same distribution on q-permutations or are from different distributions. Iterative numerical methods are given for fitting both the vase and Log-Linear models.