The second-order stochastic dominance criterion for inequality analysis introduced by Atkinson (1970) covers nearly all well-known inequality indices. The same cannot be said, in respect of poverty indices, for the second-order stochastic dominance criterion for poverty analysis introduced by Atkinson (1987). Indeed, two of the best known poverty indices, the head count ratio and the Sen indix are excluded by it. This paper introduces a more general 'mixed' dominance criterion which provides a more comprehensive coverage of poverty indice. By establishing the relationship between welfare and poverty functions, it also generalizes the proofs given by Atkinson (1987) to include non-separable as well as separable functions.