Alternative Sampling Techniques

Unequal Probabilities
 * Reasons for sampling with unequal probabilities
 * Disproportional stratification(due to one subset of population being larger than the other for instance)
 * Multistage sampling
 * PPS sampling

Multistage
 * One type of sampling method in first stage eg: Cluster Sampling
 * Do another method of sampling in the next stage of sampling
 * So on and so forth as sampling design requires
 * Calculating probability of selection is just multiplication of each probability of selection at each stage together

Probability Proportional to Size(PPS) Sampling
 * The probability of a sample being selected is proportional to the size variable in the population(pop'n 60 has 40 of x so 40/60 = 2/3 chance of x being selected)
 * In practice PPS is done without replacement usually. PPS can also be done with replacement sampling.
 * PPS sampling better in SRS in terms of variance
 * PPS total estimate = t(est) = 1/n summation sample/probability of selection
 * E(tpps(est)) is unbiased. Take the 1/n and summation outside of E because has no i. Summation of sample/probability of selection = t therefore unbiased.
 * For PPS, the t(est) is the same for all.
 * Constant of proportionality k = n/Sigma(M).
 * n is sample size.
 * M is size of the variable in population.
 * Sigma M would be the total size of the variable in the population
 * The probability of selecting a variable is thus kM. Constant of proportionality * Size measure.