Sampling Design  M. Burgman & J. Carey 2002

Types of Samples Point samples (including neighbour distance samples) Transects line intercept sampling line intersect sampling belt transects Plots circular, square, rectangular plots quadrats nested quadrats Permanent or temporary sites

Arrangement of Samples Subjective (Haphazard, Judgement) Systematic Sampling Search Sampling Probability Sampling –Random: Simple Stratified (restricted) –Multistage –Cluster –Multiphase:Double Variable Probability Sampling PPS/PPP

Samples are selected systematically according to a pre-determined plan. e.g. grid samples evaluation of spatial patterns simplicity of site location (cost) guaranteed coverage of an area representation of management units facilitation of mapping Systematic Sampling

If the ordering of units in a population is random, any predesignated positions will be a simple random sample. Bias may be introduced if there is a spatial pattern in the population. Formulae for random samples may not be applicable.

Assumptions of Systematic Sampling Assumptions no spatial or temporal trends in the variable no natural strata no correlations among individual samples Given these assumptions, a systematic sample will, on average, estimate the true mean with the same precision as a simple random sample or a stratified random sample of the same size.

s 2 =  (x i - x) 2 Simple Random Sampling sample mean (unbiased estimate of  ) 1 n n i=1 sample variance (unbiased estimate of  2 ) 1 n n-1 i=1 x =  x i

Stratified Random Sampling A population is classified into a number of strata. Each stratum is sampled independently. Simple random sampling is employed within strata. fewer samples are required to obtain a given level of precision independent sampling of strata is useful for management, administration, and mapping.

Stratified Random Sampling mean m i=1 where m = number of strata, and p i = proportion of the total made up by the i th stratum. e.g. p i = A i / A x all =  p i x i

Stratified Random Sampling standard error of overall mean s x =  p i 2 = where A i is the area of a stratum, A is the total area, s x is the standard error of the mean within the i th stratum, and n i is the number of sampling units in the i th stratum. m i=1 s 2  A i 2 s x 2 n i A 2   i all i

Stratified Random Sampling confidence limits for the mean CL mean = x all ± s x t [ , n-1] confidence limits for the whole population CL pop = A (x all ± s x t [ , n-1] ) where A = total number of units over all strata (e.g. total area in m 2, when x all has been calculated per m 2 ) all

Allocation of Samples proportional to area: n i = p i N = N where p i = proportion of total area in stratum i, N = total number of samples, and n i = number of samples allocated to stratum i. to minimize variance: A i s i.  A i s i where s i = standard deviation in stratum i AiAAiA n i = N [ ]

Random Sampling within Blocks Combination of systematic and random sampling. Gives coverage of an area, together with some protection from bias.

Cluster Sampling Clusters of individuals are chosen at random, and all units within the chosen clusters are measured. Useful when population units cluster together, either naturally, or because of sampling methods.

Cluster Sampling Examples: schools of fish clumps of plants leaves on eucalypt trees pollen grains in soil core samples vertebrates in quadrat samples Two-stage cluster sampling: clusters are selected, and a sample is taken from each cluster (i.e. each cluster is subsampled)

The division of a population into primary sampling units, only some of which are sampled. Each of those selected is further subdivided into secondary sampling units, providing a hierarchical subdivision of sampling units. Motivations include access, stratification, and efficiency. Multistage Sampling

Procedure for Multistage Sampling A study area (or a population) is partitioned into N large units (termed first-stage or primary units) A first-stage sample of n of these is selected randomly. Each first-stage unit is subdivided into M second- stage units. A second-stage sample of m of these is selected randomly. The m elements of the second-stage sample are concentrated within n first-stage samples.

Multistage Sampling Statistics When the primary units are of equal size, the population mean of a multi-stage sample is given by the arithmetic mean of the nm measurements x ij : 1 n m 1 n nm i=1 j=1 n i=1 where 1 m m j=1 selected subunits in the i th primary unit is the mean of the m selected subunits in the ith primary unit. Formulae for are provided by Gilbert (1987) and Philip (1994). x =   x ij =  x i x i =  x ij is the mean of the m

Multistage Sampling To estimate the total amount I of the measured variable (e.g. the total amount of a pollutant), I = N M x and s I 2 = (N M) 2 s x 2

Multistage Sampling When the primary units are of unequal size, the population mean of a multi-stage sample is given by  M i x i x =  M i where n i=1 n i=1 x i =  x ij 1 m m j=1 i

Multistage Sampling The total amount of the variable is given by I =  M i x i Gilbert ( Statistical Methods for Environmental Pollution Monitoring) provides formulae for allocating samples among sampling units, for estimating variances, and for including costs in the sample allocation protocol. N n n i=1

Sampling Methods revisited simple random sampling stratified random sampling two-stage sampling cluster sampling systematic sampling random sampling within segments 2° units cluster 1° unit stratum

Double Sampling (multiphase sampling) Use the easiest (and least accurate) method to measure all samples ( n' samples). Use the more accurate technique to measure a relatively small proportion of samples ( n samples, where n  n' ). Correct the relatively inaccurate measurements, using the relationship between the measurements made with both techniques. When two or more techniques are available to measure a variable, double sampling may improve the efficiency of the measurement protocol.

Double Sampling Examples GIS interpretation Chemical assays Wildlife surveys Inventories Monitoring plots

Double Sampling the underlying relationship between the methods is linear optimum values of n and n' are used (Gilbert, 1987) C A (1 +  1 -  2 ) 2 C I  2 where C A is the cost of an accurate measurement, C I is the cost of an inaccurate measurement, and  is the correlation coefficient between the methods. > Double sampling will be more efficient than simple random sampling if

Example of Double Sampling Contaminated soil at a nuclear weapons test facility in Nevada (Gilbert 1987) 241 Am (nCi/m 2 ) ,240 Pu (nCi/m 2 ) y = (x )  = 0.998