After a sample has been selected and the sample items have been reviewed or tested, the sample results are extrapolated to the population. Several methods can be used for extrapolation, each with its own advantages and disadvantages. One factor to consider for each estimator is the bias associated with that estimate. An estimator is biased if it consistently results in an estimate that is different from the true value of the unknown parameter. A second factor to consider when selecting an estimator is precision. The precision of an estimator is defined as the accuracy of the estimator; alternatively, it may be thought of as the consistency of the result if the estimation were performed multiple times.
The following diagram will clarify these competing, but important concepts.
Imagine three different archers aiming for the same target. Archer 1 is unbiased as the average (or center) of his four attempts is the bull's eye. Archer 1, however, is not very precise. That is, he is not consistently hitting the same mark - whether it is the bull's eye or any other mark. Archer 2 is very precise but is biased. That is, he consistently hits the same spot, but his mark is not close to the bull's eye. Archer 3 is precise and unbiased, as he is consistently hitting the bull's eye.
For the sampling context, one can imagine that each archer is an estimation methodology, and the target is the population parameter that must be estimated from a sample. Ideally, one would generally elect to use an unbiased, precise estimator. However, bias and precision are not the only factors when considering an estimation methodology. In some cases, an estimator that is slightly biased may be preferred. In these situations, a statistician must clearly weigh the trade-off between bias and precision.
To illustrate this scenario, we consider the following example. We assume we have a population of 100 health insurance claims that can accrue interest based on the number of days that they have been left unpaid. Interest begins accruing on the eighth day that the claim has been left unpaid at the monthly rate of 10 percent. Therefore, the interest for each claim is calculated as
It is believed that there may be error in the determination of the number of unpaid days per claim. The company is clearly interested in determining the amount of interest that is due for all 100 claims based on the accurate number of days the claim has been unpaid. Determining the correct number of days unpaid is a costly and time-consuming process. Therefore, a sample of 10 is selected from the population. (A sample of 10 is used to simplify the example here. In most cases, a sample of 10 would not be sufficient to draw precise conclusions.)
The following table provides relevant measures for the population and sample:
Through a review of records, the correct number of days that a claim was unpaid is established for each item in the sample. The following table provides the sample results for the 10 sample items:
Using the correct days and interest for the sample, we can estimate the interest due for the entire population of 100 claims. Moreover, because the interest payments must ultimately be made to individuals, it is necessary to obtain an interest amount for each claim in the population, not just for the sample, and not just for the population in the aggregate. Thus, after calculating the interest for the entire population, we will allocate the estimated interest for the population to individual claims.
There are multiple approaches to this problem. We will provide three possible solutions. Several other types of estimators could also be considered here.
- Calculate the average change in the number of days from the sample and add this average to the number of days for each population claim. Then calculate the interest based on the new estimated number of days unpaid.
- Calculate the sample weighted average change in the number of days, weighted by original claim amount. Add the weighted average to the number of days for each population claim, and calculate the interest based on the new estimated number of days unpaid; and
- Calculate a new interest amount for the sample that is based on the analyzed number of days for each claim in the sample. Then, estimate the total interest that is due for the population using the average interest calculation from the sample results.
For each of the three scenarios described above, we will calculate the point estimate and the precision and bias of the estimate.
Scenario One
Calculate the average change in the number of days from the sample and add this average to the number of days for each population claim. Then calculate the interest based on the new estimated number of days unpaid.
The average change in days is calculated as:
The estimated days and estimated interest for each claim in the population is then calculated as:
The bias of the estimate is calculated by comparing the interest amount using the correct, analyzed number of days to the interest amount using the estimated days across all items in the sample. That is, the bias is calculated as:
