6 Weights and Horvitz-Thompson

Jeremy Strief and Glen Meeden

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The usual definition of the weight assigned to a unit in the sample is the inverse of its inclusion probability. One is encouraged to think of a unit's weight as being the number of units in the population which it represents. The resulting estimator of the population total is the Horvitz-Thompson (HT) estimator and is design unbiased. As we have already noted the unbiased estimate of its variance depends on the joint selection probabilities of the all the pairs of units appearing in the sample. Since in practice this can be impossible to compute the approximation in equation (5.4) is often used.

The HT estimator works best when $y_i$ is approximately proportional to its selection probability. To compare its behavior to the WDP method we conducted a small simulation experiment. We constructed the variable $x$ by drawing a random sample of 2,000 from a gamma distribution with shape parameter 5 and scale parameter 1 and adding 20 to each value. To generate $y$ we let the conditional distribution of $y_i$ given $x_i$ be a normal distribution with mean $5x_i$ and standard deviation 20. The correlation of the resulting population was 0.49. We denoted this population by A. We created a second population, B, by using the same vector of $x$ values but adding 400 to each $y_i$ value. Our sampling plan used $x$ to do sampling proportional to size, i.e., $\text{pps}(x).$We used the R package sampling so that the inclusion probabilities were exact. Under this design we expect that the HT estimator would work well for population A but perform less well for population B. We also considered a third estimator, NHT, which is just the weights of the HT estimator rescaled so that they sum to the population size. We generated 500 samples of size 50. The results are giving in Table 6.1.

Table 6.1
Results for populations A and B based on 500 samples of size 50. The NHT estimator is the HT estimator renormalized so that the weights sum to the population size, N = 2,000. The nominal coverage for each method is 0.95.
Table summary
This table displays the results of results for populations a and b based on 500 samples of size 50. the nht estimator is the ht estimator renormalized so that the weights sum to the population size. The information is grouped by population (appearing as row headers), method, ave. abs err, ave. len and freq of coverage (appearing as column headers).

Population	Method	Ave. abs err	Ave. len	Freq of coverage
A	HT	4.628	21.898	0.940
B	HT	8.965	43.914	0.960
A & B	WDP	4.706	24.381	0.960
A	NHT	5.051	21.897	0.896
B	NHT	5.051	43.919	0.998

Although not shown in the table both the HT and WDP estimators are unbiased for both populations. As expected the HT estimator is the best for population A although its performance falls off dramatically for population B. On the other hand the WDP performance for both populations is exactly the same. As a point estimator the NHT does much better than the HT estimator for population B but not as well for population A. Overall the WDP is clearly performs the best. What is an explanation for these differences?

In population A, $y_i\propto x_i$ and calculations show that ${\gamma }_{dw}$ is almost always negative and its absolute value is small compared to ${\sigma }_{dw}$. In other words, when the HT estimator is appropriate it is essentially using the variance of the constructed population based on its weights to get its estimate of variance.

The only difference between populations A and B is that a constant has been added to the $y$value of each unit. Now if the sample weights allow us to make a good guess for the population in the first case what goes wrong in the in the second case to cause the HT estimator to preform so poorly? To see the problem consider the following.

In the HT estimate the sum of the weights in the sample almost never equal $N,$the population size. Given a sample in population B the HT estimate is

${\displaystyle \sum _{i=1}^{50}{w}_i{y}_i={\displaystyle \sum _{i=1}^{50}{w}_i{y^{\prime }}_i}+400{\displaystyle \sum _{i=1}^{50}{w}_i}}$

where ${y^{\prime }}_i$ denotes the unit's corresponding value in population A and $y_i$ its value in population B. Note the second term in the above equation is adding additional variablity to the HT estimator. In population B calculations show that the term ${\gamma }_{dw}$ in equation (5.5) is positive and can be quite large. It is accounting for the extra variablity in the HT estimator in population B which results from that fact that here $y_i\propto x_i+400$ and not $x_i.$

We note that Zheng and Little (2003) argued that when estimating a finite population total and when using a probability-proportional to size sampling design that a penalized spline, nonparametric, model based estimator generally outperformed the Horvitz-Thompson estimator. Zheng and Little (2005) developed methods to estimate the variance of their estimator. Some related work can be found in Zheng and Little (2004).

The WDP weights only use the constraint that simulated complete copies of the population should have the correct population mean for $x.$ This is a more robust assumption than the one which underlies the HT estimator. But to be fair to the HT estimator it should be remembered (as was pointed out by a referee) that it was developed with the limited goal of obtaining linear unbiased estimators of the population total. Today however its simplicity no longer seems so important when more complicated and efficient estimators are much easier to compute. The superior performance of the stepwise Bayes method here suggests that if one believes that they have a set of weights for the sampled units which sums to the population size and which yields a good guess for the population, then they should use the variance of their good guess for the population to construct an estimate of the variance of their estimate of the population mean rather than equation (5.4). This is particularly true for large surveys containing several $y$characteristics of interest. It would be very surprising if all of them satisfied the assumptions necessary to make equation (5.4) a good estimate of variance of a sample mean. Analogous to the observation in Royall and Cumberland (1981) and Royall and Cumberland (1985) that good balanced samples (the sample mean is close to the population mean) can lead to improved performance one should base their inference on simulated complete copies of the population which incorporate the available prior information contained in the auxiliary variables.

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