Inverse Sampling

Applications

Applications for inverse sampling can have broad appeal. One such application is the ability to determine the better of two binomial populations (or the one with the highest probability of success). For example, in a drug trial, where the outcome is success or failure, inverse sampling can be used to determine the better of the two treatment options and has been shown to be as efficient, and potentially less costly, than a fixed sample size design. Milton Sobel and George Weiss present two inverse sampling techniques to conduct such an analysis: (1) vector-at-a-time (VT) sampling and (2) play-the-winner (PW) sampling. VT inverse sampling involves two observations, one from each population, that are drawn simultaneously. Sampling continues until r successful observations are drawn from one of the populations. PW inverse sampling occurs when one of the populations is randomly selected and an observation is randomly selected from that population. Observations continue to be selected from that population until a failure occurs, at which point sampling is conducted from the other population. Sampling continues to switch back and forth between populations until r successful observations are selected in one of the populations. Under both VT and PW, the population from which r successes are first observed is determined the better population. In clinical trials PW is an advantageous design because it has the same probability requirements as VT, but the expected number of trials on the poorer population is always smaller. Sobel and Weiss have also extended this methodology for k > 2 populations.

Inverse sampling is also used to estimate the number of events that occur in an area of interest based on a Poisson distribution. In these situations, one can use inverse sampling to estimate the total number of events or the number of events during a certain period by selecting a sampling unit and counting the number of events that occur in that unit. A series of independent units are sequentially selected until the total number of events across all of the selected units meets or exceeds a pre-assigned number of events. The number of trials needed to reach the pre-assigned number of events is then used to estimate the mean number of events that will occur. This design assumes a Poisson distribution but not a Poisson process. Therefore, not every sampling unit selected has to have a Poisson distribution, but all the sampling units combined do have a Poisson distribution. An example of this design would be to estimate the number of accidents on the road. Because the number of accidents depends on the day of the week, a week would be the smallest sampling unit that one could assume had a Poisson distribution. If one day were the sampling unit, then a Poisson distribution might not always hold.

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