Probability proportional to size (PPS) sampling includes a number of sample selection methods in which the probability of selection for a sampling unit is directly proportional to a size measure, Xit which is known for all sampling units and thought to be approximately proportional to the unknown F;. The Xi must all be strictly greater than 0. In single-stage sampling it can be used to reduce the variance of survey estimates. If the observed values, Yit are exactly proportional to Xi, the variance of an estimated total will be exactly 0. When the F; are approximately proportional to the Xi, the variance can still be greatly reduced relative to equal probability sampling schemes. PPS sampling is also used in the early stages of multi-stage samples ...