Publications

    Survey Methodology

    Browse by

    4 Applications

    Peter M. Aronow and Cyrus Samii

    Previous | Next

    Proposition 1 shows that the bias of the Horvitz-Thompson variance estimator under non-measurability is

    A= ∑ k∈U ∑ l∈{U\k: π kl =0} y k y l . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qacaWGbbGaeyypa0Zaaabua8aabaWdbmaaqafapaqaa8qacaWG 5bWdamaaBaaaleaapeGaam4AaaWdaeqaaOWdbiaadMhapaWaaSbaaS qaa8qacaWGSbaapaqabaaabaWdbiaadYgacqGHiiIZcaGG7bGaamyv aiaacYfacaWGRbGaaiOoaiabec8aW9aadaWgaaadbaWdbiaadUgaca WGSbaapaqabaWcpeGaeyypa0JaaGimaiaac2haaeqaniabggHiLdaa l8aabaWdbiaadUgacqGHiiIZcaWGvbaabeqdcqGHris5aOGaaiOlaa aa@5642@

    This expression, along with the fact that A * ≥A, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qacaWGbbWdamaaCaaaleqabaWdbiaacQcaaaGccqGHLjYScaWG bbGaaiilaaaa@3E6E@  makes it evident that the degree of bias in Var ^ ( t ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qadaqiaaWdaeaapeGaaeOvaiaabggacaqGYbaacaGLcmaacaGG OaGabmiDayaajaGaaiykaaaa@3F5D@  and Var ^ C ( t ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qadaqiaaWdaeaapeGaaeOvaiaabggacaqGYbaacaGLcmaapaWa aSbaaSqaa8qacaWGdbaapaqabaGcpeGaaiikaiqadshagaqcaiaacM caaaa@4099@  depends a great deal on the number of pairs with zero pairwise inclusion probabilities. For designs where this number is small, Var ^ ( t ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qadaqiaaWdaeaapeGaaeOvaiaabggacaqGYbaacaGLcmaacaGG OaGabmiDayaajaGaaiykaaaa@3F5D@  may provide a reasonable and conservative estimator for cases where y k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qacaWG5bWdamaaBaaaleaapeGaam4AaaWdaeqaaaaa@3BB0@  takes the same sign for all k, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qacaWGRbGaaiilaaaa@3B08@  and Var ^ C ( t ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qadaqiaaWdaeaapeGaaeOvaiaabggacaqGYbaacaGLcmaapaWa aSbaaSqaa8qacaWGdbaapaqabaGcpeGaaiikaiqadshagaqcaiaacM caaaa@4099@  may provide a reasonable and conservative estimator for cases where y k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qacaWG5bWdamaaBaaaleaapeGaam4AaaWdaeqaaaaa@3BB0@  may take different signs for some k. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aai aac6caaaa@3AEA@  An example that arises frequently is stratified sampling where for a relatively small proportion of cases, we have small strata from which we draw only one unit.

    For designs that result in many pairs having zero inclusion probabilities, Var ^ ( t ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qadaqiaaWdaeaapeGaaeOvaiaabggacaqGYbaacaGLcmaacaGG OaGabmiDayaajaGaaiykaaaa@3F5D@  and Var ^ C ( t ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qadaqiaaWdaeaapeGaaeOvaiaabggacaqGYbaacaGLcmaapaWa aSbaaSqaa8qacaWGdbaapaqabaGcpeGaaiikaiqadshagaqcaiaacM caaaa@4099@  could be wildly over-conservative and other estimators may be preferred in terms of criteria such as mean square error. A prominent example is systematic sampling. Indeed, Särndal et al. (1992, page 76) propose that under systematic sampling, the Horvitz-Thompson variance estimator, Var ^ ( t ^ ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qadaqiaaWdaeaapeGaaeOvaiaabggacaqGYbaacaGLcmaacaGG OaGabmiDayaajaGaaiykaiaacYcaaaa@400D@  can give a "non-sensical result.� The expression for A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyqaa aa@3A0E@  makes it clear why this would be the case. Wolter (2007, Chapter 8) shows that simpler biased estimators, such as the with-replacement (Hansen-Hurwitz) variance estimator, can be reliable, if slightly conservative, in a broad range of data scenarios under equal probability and probability proportional to size (PPS) systematic sampling. Nonetheless, the with-replacement estimator fails to account adequately for sampling variance when outcome variance within systematic sample clusters is smaller than the between cluster variance. In such cases, Var ^ ( t ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qadaqiaaWdaeaapeGaaeOvaiaabggacaqGYbaacaGLcmaacaGG OaGabmiDayaajaGaaiykaaaa@3F5D@  would bound this variance in expectation when outcomes are all of the same sign, and Var ^ C ( t ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qadaqiaaWdaeaapeGaaeOvaiaabggacaqGYbaacaGLcmaapaWa aSbaaSqaa8qacaWGdbaapaqabaGcpeGaaiikaiqadshagaqcaiaacM caaaa@4099@  would always bound this variance in expectation. Of course, it may still be the case that the bias is too large to be of much use, and so we would not suggest that Var ^ ( t ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qadaqiaaWdaeaapeGaaeOvaiaabggacaqGYbaacaGLcmaacaGG OaGabmiDayaajaGaaiykaaaa@3F5D@  and Var ^ C ( t ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qadaqiaaWdaeaapeGaaeOvaiaabggacaqGYbaacaGLcmaapaWa aSbaaSqaa8qacaWGdbaapaqabaGcpeGaaiikaiqadshagaqcaiaacM caaaa@4099@  provides a full solution to the variance estimation problem for systematic sampling under high intra-cluster correlation.

    Results from simulation studies are available in a supplement (at https://files.nyu.edu/cds2083/public/docs/smj_suppl.pdf). They illustrate how Var ^ ( t ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qadaqiaaWdaeaapeGaaeOvaiaabggacaqGYbaacaGLcmaacaGG OaGabmiDayaajaGaaiykaaaa@3F5D@  and Var ^ C ( t ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdHiVc=bYP0xb9sq=fFjea0RXxb9qr0dd9q8qi0lf9 Fve9Fve9vapdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaaeaaaaaa aaa8qadaqiaaWdaeaapeGaaeOvaiaabggacaqGYbaacaGLcmaapaWa aSbaaSqaa8qacaWGdbaapaqabaGcpeGaaiikaiqadshagaqcaiaacM caaaa@4099@  perform relative to commonly-used alternatives in applied scenarios. The simulations demonstrate situations when these estimators are preferable to the alternatives. For one-unit-per-stratum sampling, we show that these estimators are less biased than the "collapsed stratum� estimator in a range of scenarios. For PPS systematic sampling, these estimators perform favorably when the population exhibits substantial periodicity, a case when the commonly-used with-replacement estimator may be grossly negatively biased.

    Previous | Next

    Date modified: