Covering arrays for words of length (Formula presented.) over a (Formula presented.) -letter alphabet are (Formula presented.) arrays with entries from the alphabet so that for each choice of (Formula presented.) columns, each of the (Formula presented.) (Formula presented.) -letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case known as partitioning hash families, words are equivalent if they induce the same partition of a (Formula presented.) element set. In the second case, words of the same weight are equivalent. In both cases, we produce logarithmic upper bounds on the minimum size (Formula presented.) of a covering array. Definitive results for (Formula presented.), as well as general results, are provided.