Four methods of selecting PNR (probability non-replacement) samples are described and their properties examined. The four are based on a common theme and provide samples whereinPi(the probability thejth element is included in a sample) is proportional to the size of the element. The four methods differ primarily in the degree of control that is exercised overPij(the probability that elementsiandjare both in the sample.) In methods 1 and 2 the sampler has considerable control over the combination of elements that comprise each possible sample. Method 3 is completely objective, the samples being generated by a weighted system of randomizations. Method 4 as presented requires an iterative solution to achieve the desired control over thePij.