THE SAMPLING METHOD 



Sampling with unequal probabilities of selecting a sample unit frequently can 

 result in increased efficiency in estimating population totals. The technique 

 requires that each unit in the population be characterized by a "size" variable that 

 is correlated with the variable of real interest to be measured for that unit. If a 

 list of the size variable can be obtained cheaply for all units in the population, 

 then sampling with probability proportional to size (pps) according to the method 

 developed by Hartley (1966) can be more efficient than pps sampling by other methods. 

 How much more efficient depends on the character of the functional relation between 

 size and the variable of real interest. 



The distinguishing feature of Hartley's method is that the units of the population 

 are sampled at uniform intervals of accumulated size from a random start in the list 

 of units sorted into order by size. Hence, the sample that is drawn is more likely 

 to contain both small and large units than would a pps sample drawn from a randomly 

 ordered list. Hartley has provided an expression for the true variance of estimates 

 based on this method of selecting the sample. However, the sample estimate of this 

 variance that he suggests is based on the assumption that the ratio of the variable 

 of interest to the size variable is a linear function of the size variable. In a 

 subsequent part of this paper, the assumption concerning the linearity of this relation 

 is relaxed to include polynomial functions of the size variable. 



PPSORT--A COMPUTER SUBROUTINE FOR 

 SELECTING THE SAMPLE 



PPSORT is a FORTRAN IV computer subroutine that implements Hartley's procedure 

 for drawing samples from a population list sorted by size. The program is complete 

 with its own composite random number generator (Marsaglia and Bray 1968; Grosenbaugh 

 1969) and sorting algorithm. The execution speed of the version of PPSORT listed in 

 appendix I was markedly improved over earlier versions by adopting a sorting method 

 developed by D. L. Shell, as it was programed by Robert M. Russell of the Pacific 

 Southwest Forest and Range Experiment Station. 



The pps selection method used by PPSORT is without replacement. Accordingly, 

 the number of sample units must be less than the measure of size accumulated over the 

 entire population divided by the size of the largest element in the population. The 

 logic of PPSORT is such that if this condition is not satisfied for the number of 

 sample units requested, then the population is divided into two strata. The largest 

 unit in the population is assigned to one stratum in which every unit will be desig- 

 nated for measurement with absolute certainty. The other stratum is comprised of the 

 remaining units of the population. The number of sample units to be drawn is reduced 

 by one, and the inequality is tested again. If the number of sample units is still 

 too large, the process is repeated until the inequality is satisfied. Fortunately, 

 the expression for estimating the population total can ignore the separation into 



