Nonparametric, ModelAssisted Estimation for a TwoStage Sampling Design Mark Delorey, F. Jay Breidt,

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Nonparametric, Model-Assisted Estimation for a
Two-Stage Sampling Design Mark Delorey, F. Jay
Breidt, Colorado State University
Abstract In aquatic resources, a two-stage
sampling design can be employed to make the best
use of what are often limited time and financial
resources. Even with the ability to focus such
resources, it is often the case that the sample
sizes are not sufficiently large to make
model-free inferences. The presence of auxiliary
information for the regions of interest suggests
employing a model in our inferences. Breidt,
Claeskens, and Opsomer (2003) propose
incorporating this auxiliary information through
a class of model-assisted estimators based on
penalized spline regression in single stage
sampling. Zheng and Little (2003) also use
penalized spline regression in a model-based
approach for finite population estimation in a
two-stage sample. In a survey context, weights
computed from a set of auxiliary information are
often applied to many study variables. With this
approach, model-assisted estimators should fare
better than model-based estimators. We compare
the two through a series of simulations.

• Two-Stage Sampling
• The population of elements U 1,, k,, N is
  partitioned into clusters or primary sampling
  units (PSUs), U1,, Ui,, . So,where Ni
  is the number of elements or secondary sampling
  units (SSUs) in Ui.
• First stage A sample of clusters, sI, is
  selected based on a design, pI(?) with inclusion
  probabilities ?Ii and ?Iij.
• ?Ii and ?Iij are the first and second order
  inclusion probabilities, respectively
• Second stage For every i ? sI, a sample si is
  drawn from Ui based on the design pi(? sI)
• Typically require second stage design to be
  invariant and independent of the first stage
• Two-Stage Sampling with Aquatic Resources
• Time and expense constraints may make two-stage
  sampling more efficient
• Auxiliary information may be available on
  different scales

• The Estimators (for population totals)
• Horvitz-Thompson (HT)where
• Model-assistedwhere is the PSU total
  predicted by the model
• Model-basedwhere
  is the ith cluster mean predicted by
  the model

• Notes on the Models and Model Parameters
• 3 different models used
• Linear
• Penalized spline with random effect for PSU
• Penalized spline with no random effect for PSU
• In a survey context, such as those found in
  environmental monitoring, it is often desirable
  to obtain a single set of survey weights that can
  be used to predict any study variable. To
  accommodate this
• Smoothing parameter for spline is selected by
  fixing the degrees of freedom for the smooth
  rather than using a data driven approach
• Variance component for PSU effect is computed for
  the linear model and resulting covariance matrix
  and corresponding survey weights are applied to
  samples from other data sets
• In this kind of survey context, model-assisted
  estimators have good efficiency properties and
  should be superior to model-based estimators
  which rely on correct specification of variance
  components

• Case A Cluster Level Auxiliaries (Our focus)
• The auxiliary information is available for all
  clusters in the population
• Leads to regression modeling of quantities
  associated with the clusters, such as cluster
  totals
• Cluster quantities can be computed for all
  clusters
• Population quantities can be computed from
  cluster estimates
• Example Lake represents a cluster auxiliary
  information is elevation

• Generating Responses
• 500 PSUs the number of SSUs per cluster
  Uniform(50, 400)
• ?PSU m(?I) ?, where m(?) is one of the eight
  functions below and ? N(0, ?2I)
• We use first order inclusion probabilities
  proportional to size (pps)
• Auxiliary data is often proportional to size of
  cluster
• Response of interest yij ?i ?ij. where yij is
  the jth element in the ith cluster and ?ij iid
  N(0, ?2)

• Comments on Simulation Results
• 500 samples from each of the populations were
  drawn
• H-T Horvitz-Thompson estimatorM-A lin
  Model-assisted estimator using a linear
  modelM-B pmmra Model-based estimator using a
  penalized spline and including a random effect
  for PSUM-A pmm Model-assisted estimator using
  a penalized spline with no random effect for PSU
• Point represents MSEEstimatorMSEModel-assisted
  estimator with radom effect for PSU
• Vertical black bars represent approximate 95
  confidence intervals
• Model-assisted estimator with random effect for
  PSU is as efficient or more efficient than
  model-based estimator we do not appear to lose
  efficiency (with respect to MSE) by using
  model-assisted non-parametric methods

• Case B Complete Element Level Auxiliaries
• The auxiliary information is available for all
  elements in the population
• Leads to regression modeling of quantities
  associated with the elements
• Cluster and population quantities can then be
  computed from element estimates and observations
• Example EMAP hexagon is cluster lake is
  element auxiliary information is elevation

• Case C Limited Element Level Auxiliaries
• The auxiliary information is available for all
  elements in selected clusters only
• Leads to regression modeling of quantities
  associated with the elements
• Regression estimators can be used for
  cluster-level quantities only for the clusters
  selected in the first-stage sample
• Example Aerial photography of selected sites
  (clusters) for each point (element) in site, we
  have percent forested, urban, industrial

• Case D Limited Cluster Level Auxiliaries
• The auxiliary information is available for all
  clusters in the first-stage sample
• Not a very interesting case
• Design-based estimator can be used for population
  quantities
• In some cases, good estimators for population
  quantities are not available
• Example Cluster is lake auxiliary information
  is measure of size which is not available until
  site is visited