Hankin D., Mohr M.S., Newman K.B. Sampling Theory: For the Ecological and Natural Resource Sciences

Oxford: Oxford University Press, 2019. — 360 p.Sampling theory considers how methods for selection of a subset of units from a finite population (a sample) affect the accuracy of estimates of descriptive population parameters (mean, total, proportion). Although a sound knowledge of sampling theory principles would seem essential for ecologists and natural resource scientists, the subject tends to be somewhat overlooked in contrast to other core statistical topics such as regression analysis, experimental design, and multivariate statistics. This introductory text aims to redress this imbalance by specifically targeting ecologists and resource scientists, and illustrating how sampling theory can be applied in a wide variety of resource contexts. The emphasis throughout is on design-based sampling from finite populations, but some attention is given to model-based prediction and sampling from infinite populations.The design-based paradigm
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Recommendations for instructors
Sampling theory A brief history
Basic concepts
Terminology
Components of a sampling strategy
Selection methods
Properties of estimators
Sampling distribution of an estimator
Judgment sampling versus random sampling
Equal probability sampling
Without replacement sampling
Estimation of the population mean, proportion,and total
Sampling variance
Estimation of sampling variance
Bernoulli sampling
With replacement sampling
Estimation of the population mean, proportion, and total
Sampling variance and variance estimation
Rao–Blackwell theorem
Relative performance of alternative sampling strategies
Measures of relative performance
An example SRS/mean-per-unit estimation versus SWR
Sample size to achieve desired level of precision
Approximate normality of sampling distributions
Confidence interval construction
Sample size determination
Nonresponse and oversampling
Sampling in R
SRS and SWR
Sample Spaces
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Problems
Systematic sampling
Linear systematic sampling
N /k is integer-valued
Relative efficiency
N/k is not integer-valued
Unbiased estimation
Selection methods that guarantee fixed n
Circular systematic sampling
Fractiona linterval random start
Estimation of sampling variance
Biased estimation
SRS proxy
Estimation in presence of linear trend
Unbiased estimation
m samples selected independently
m samples selected without replacement
Unpredictable trend in sampling variance with n
Warning Pathological settings
Nonresponse and oversampling
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Problems
Stratified sampling
Estimation of the population mean
Expected value
Sampling variance
Numerical examples
Estimation of the population proportion
Estimation of the population total
Estimation of sampling variance
Allocation of the sample across strata
Optimal allocation Graphical analysis
Optimal allocation Analytical analysis
Use of Lagrange multipliers
Comments on optimal allocation
Sample size determination
Relative efficiency
Proportional allocation
Estimation of finite population variance
Effective degrees of freedom
Post-stratification
Unconditional sampling variance
Conditional sampling variance
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Problems
Single-stage cluster sampling Clusters of equal size
Estimation of the population mean
Sampling variance
ANOVA/mean squares approach
Intracluster correlation approach
Estimation of the population total and proportion
Estimation of sampling variance
Estimation of finite population variance
Sample size determination
Relative efficiency
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Problems
Ratio and regression estimation
Estimation of the mean and total
Graphical representation
Sample space illustration
Bias
Sampling variance
Ratio estimator
Regression estimator
Estimation of sampling variance
Sample size determination
Ratio estimation
Regression estimation
Relative efficiency
Ratio estimator
Regression estimator
Ratio estimation of a proportion
Ratio estimation with stratified sampling
Combined estimator
Separate estimator
Choosing between combined and separate estimators
A model-based perspective
Estimation of model parameters
Mean model
Linear models
Prediction of population parameters
Mean model
Ratio model
Regression model
Prediction error
Mean model
Ratio model
Regression model
Prediction variance estimators
Monte Carlo performance evaluation
Mark-recapture estimation
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Problems
Unequal probability sampling
Unbiased ratio estimator
Sampling with replacement
Hansen–Hurwitz estimator
Unbiasedness
Sampling variance and variance estimation
Sampling without replacement
Horvitz–Thompson estimator
Unbiasedness
Sampling variance and variance estimation
Alternative selection methods
Chao’s method
Sunter’s method
Strategy performance comparisons
Survey cost comparisons
Sampling distribution
Systematic sampling
Generality of Horvitz-Thompson estimation
Generalized Horvitz-Thompson estimation
Variance, covariance, and correlation estimators
Mean-per-unit, ratio, and regression estimators
Performance of generalized Horvitz–Thompson estimators
Poisson sampling
Nonresponse and oversampling
Hansen–Hurwitz estimator
Horvitz–Thompson estimator
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Problems
Multi-stage sampling
Two-stage sampling Clusters of equal size
Estimation of the population mean
Expectation
Sampling variance and its estimation
Estimation of first- and second-stage contributions
Optimal allocation
Net relative efficiency
Estimation of finite population variance
Two-stage sampling Clusters of unequal size
Single-stage cluster sampling
Estimation of the population mean and total
Sample space illustration
Two-stage estimation of the population mean and total
Sampling variance and its estimation
General expressions
SRS within clusters
Sample space illustration Horvitz-Thompson estimation
More than two stages
Optimal allocation
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Generality of the multi-stage framework
Taking advantage of ecological understanding
Implications for large-scale natural resource surveys
Problems
Multi-phase sampling
Two-phase estimation of the population mean and total
Estimators
Sampling variance and its estimation
Sample space illustration
Optimal allocation
Two-phase ratio estimation
Two-phase regression estimation
Net relative efficiency
Graphical analysis
Two-phase ratio estimation of a proportion
Two-phase sampling with stratification
Estimation of the population mean and total
Sampling variance and its estimation
Optimal allocation
Net relative efficiency
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Problems
Adaptive sampling
Adaptive cluster sampling
Basic scheme
Definitions
Inclusion probabilities and expected sample size
Estimators and relative efficiency
Adaptive mean-per-unit estimation
Adaptive Horvitz–Thompson estimation
Other adaptive sampling designs
Single-stage strip and systematic designs
Two-stage complete allocation cluster sampling
Murthy’s estimator for Ti
Two-stage estimators
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Problems
Spatially balanced samplingFinite populations
Generalized random tessellation stratified sampling
Procedure
Application coastal salmonid monitoring
Balanced acceptance sampling
Spatial balance and Voronoi polygons
van der Corput sequence
Halton sequence
Procedure
Procedure with Halton frame
Halton boxes
Halton frame
Sample selection
Estimation
Neighborhood variance estimator
Infinite populations
Generalized random tessellation stratified sampling
Balanced acceptance sampling
Estimation
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Sampling through time
Sampling on two occasions
Design Full retention of units across occasions
Design Independent SRS on each occasion
Comparison of full retention and independent SRS designs
Design Partial retention/partial replacement
Sampling variance
Monitoring design
Membership design
Revisit design
Shorthand notation
Estimation of status and trend
Design-based estimation
Estimators for some specific designs
Sample size determination
Dual frame sampling
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APPENDIX A Mathematical foundations
Counting techniques
Permutations
k-Permutations
Combinations
Partitions
Basic principles of probability theory
Random experiment
Sample space
Outcome probability
Event
Event relations
Union and intersection
Conditional probability
Other probability relations
Discrete random variablesDiscrete random variables are central to probability
Definition
Probability distributions
Probability relations
Expectation
Variance and coefficient of variation
Covariance and correlation
Total expectation and variance
Total expectation
Total variance
Indicator variables
Key discrete probability distributions
Uniform
Bernoulli
Multinoulli
Binomial
Multinomial
Hypergeometric
Multivariate hypergeometric
Population variables
Definition
Population parameters
Probability sampling
Sampling experiment
Sampling designs
Inclusion probabilities
Inclusion indicator variables
Estimators
Definition
Sampling distribution
Statistical properties
Confidence intervals
Delta method
Lagrange multipliers