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Statistical sampling principles for the environment

Marian Scott

August 2013

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Outline

• Variation• General sampling principles• Methods of sampling

– Simple random sampling– Stratified sampling– Systematic sampling– How many samples (power calculations)

• examples- the ECN sites, Ecomags project

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network design-ECN

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Ground sampling locationsEcomags

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Variation

• Natural variation in the attribute of interest, might be due to – feeding habits if measuring sheep, rainfall patterns if

measuring plants

• Also variation/ uncertainty due to analytical measurement techniques.

• Natural variation may well exceed the analytical uncertainty

• Expect therefore that if you measure a series of replicate samples, they will vary and if there is sufficient you may be able to define the distribution of the attribute of interest.

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from Gilbert and Pulsipher (2007)

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Data

Frequency

6.05.55.04.54.03.5

20

15

10

5

0

4.647 0.3815 59

4.704 0.6001 14

Mean StDev N

alllogtlogt2007

Variable

Normal Histogram of log activity

Activity (log10) of particles (Bq Cs-137) with Normal or Gaussian density superimposed

Variation

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What is statistical sampling?

Statistical sampling is a process that allows inferences about properties of a large collection of things (commonly described as the population), to be made from observations made on a relatively small number of individuals belonging to the population (the sample). In conducting statistical sampling, one is attempting to make inferences to the population.

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Statistical sampling

The use of valid statistical sampling techniques increases the chance that a set of specimens (the sample, in the collective sense) is collected in a manner that is representative of the population.

Statistical sampling also allows a quantification of the precision with which inferences or conclusions can be drawn about the population.

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Statistical sampling

• the issue of representativeness is important because of the variability that is characteristic of environmental measurements.

• Because of variability within the population, its description from an individual sample is imprecise, but this precision can be described in quantitative terms and improved by the choice of sampling design and sampling intensity (Peterson and Calvin, 1986).

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Good books

• The general sampling textbooks by Cochran (1977) and Thompson (1992),

• the environmental statistics textbook by Gilbert (1987), and

• papers by Anderson-Sprecher et al. (1994), Crepin and Johnson (1993), Peterson and Calvin (1986), and Stehman and Overton (1994).

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Know what you are setting out to do before you start

 describing a characteristic of interest (usually the average),

describing the magnitude in variability of a characteristic, describing spatial patterns of a characteristic,mapping the

spatial distribution, quantifying contamination above a background or

specified intervention level  detecting temporal or spatial trends,   assessing human health or environmental impacts of

specific facilities, or of events such as accidental releases,

• assessing compliance with regulations

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Rules

• Rule 1: specify the objective • Rule 2: use your knowledge of the

environmental context

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Use your scientific knowledge

   the nature of the population such as the physical or biological material of interest, its spatial extent, its temporal stability, and other important characteristics,

  the expected behaviour and environmental properties of the compound of interest in the population members,

   the sampling unit (i.e., individual sample or specimen),

• the expected pattern and magnitude of variability in the observations .

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What is the population?

• The concept of the population is important. The population is the set of all items that could be sampled, such as all fish in a lake, all people living in the UK, all trees in a spatially defined forest, or all 20-g soil samples from a field. Appropriate specification of the population includes a description of its spatial extent and perhaps its temporal stability

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What is the sampling unit?

• The environmental context helps define the sampling unit. It is not practical to consider sampling units so small that their concentration cannot be easily measured; to consider extremely large sampling units, if they are too difficult to manipulate or process.

• A sampling unit is a unique element of the population that can be selected as an individual sample for collection and measurement.

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Sampling units

• In some cases, sampling units are discrete entities (i.e., animals, trees), but in others, the sampling unit might be investigator-defined, and arbitrarily sized.

• Statistical sampling leads to a description of the sampled members of the population and inference(s) and conclusion(s) about the population as a whole.

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RepresentativityAn essential concept is that the taking of a sufficient number of individual samples should provide a collective sample that is representative of all samples that could be taken and thus provides a true reflection of the population.

A representative collective sample should reflect the population not only in terms of the attribute of interest, but also in terms of any incidental factors that affect the attribute of interest.

Representativeness of environmental samples is difficult to demonstrate. Usually, representativeness is considered justified by the procedure used to select the samples.

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5 step approach

• Define the objectives and questions to be answered• Summarize the environmental context for the

quantities being measured.• Identify the population, including spatial and

temporal extent.• Select an appropriate sampling design.• Document the sampling design and its rationale.

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Methods

Simple random sampling

With simple random sampling, every sampling unit in the population has, in theory, an equal probability of being included in the sample. The resulting estimator based on such a sample will be unbiased, but it may not be efficient, in either the statistical or practical senses. Simple random sampling designs are easy to describe but may be difficult to achieve in practice.

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Population of N units-10 randomly selected

1 2 3 4 5 9

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23 25

31 33

42 45

46 51 54

Random digits: 5,17,23, 25, 31, 33,42, 45,46,51

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Methods

Stratified sampling

The population is divided into strata, each of which is likely to be more homogeneous than the entire population. In other words, the individual strata have characteristics that allow them to be distinguished from the other strata, and such characteristics are known to affect the measured attribute of interest. Some ordinary sampling method (e.g., a simple random sample or systematic sample) is used to estimate the properties of each stratum.

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Methods

Stratified samplingUsually, the proportion of sample observations taken in each stratum is similar to the stratum proportion of the population, but this is not a requirement. If good estimates are wanted for rare strata that have a small occurrence frequency in the population, then the number of samples taken from the rare strata can be increased. Stratified sampling is more complex and requires more prior knowledge than simple random sampling, and estimates of the population quantities can be biased if the stratum proportions are incorrectly specified.

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Methods

Systematic sampling

Systematic sampling is probably the most commonly used method for field sampling. It is generally unbiased as long as the starting point is randomly selected and the systematic rules are followed with care. Line transects and two dimensional grids are specific types of systematic samples that are described in more detail in the spatial section.

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Methods

Systematic sampling

Systematic sampling is often more practical than random sampling because the procedures are relatively easy to implement in practice, but this approach may miss important features if the quantity being sampled varies with regular periodicity and the sampling scheme has similar periodicity.

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Population of N (9x6) units-9 systematically selected

1 2 3 4 5 6

12 18

24

30 36

42

48 54

Systematic selection: 6,12,18,24,30,36,42,48

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So we have sampled, what next?Analyse the resulting data

Two of the most common sampling objectives are:

estimation of the mean, or estimation of a proportion (e.g., the unknown fraction of a

population > a specified value),

We consider how to achieve these under different sampling schemes

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Estimate the population meanSimple random sampling

every sampling unit in the population is expected to have an equal probability of being included in the sample. The first step requires complete enumeration of the population members. In the simple random-sampling scheme, one generates a set of random digits that are used to objectively identify the individuals to be sampled and measured.

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Estimate the population meanThe sampling frame

In simple random sampling, one might assume a population of N units (N 100-cm2 areas), and use simple random sampling to select n of these units. This typically involves generation of n random digits between 1 and N, which would identify the units to sample. If a number is repeated, then one would simply generate a replacement digit.

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Sample mean and variance as estimators of the population

quantities

n

yy i

1

2

2

n

yys

i

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Sampling error

n

fsyVar

1)( 2

the sampling fraction f is usually very small and given by n/N.

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Stratified random sampling

In stratified sampling, the population is divided into two or more strata that individually are more homogeneous than the entire population, and a sampling method is used to estimate the properties of each stratum. Usually, the proportion of sample observations in each stratum is similar to the stratum proportion in the population.

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Stratified random sampling

In stratified sampling, the population of N units is first divided into sub-populations of N1, N2,….NL units. These sub-populations are non-overlapping and together comprise the whole population. The sub-populations are called strata. They need not have the same number of units, but, to obtain the full benefit of stratification, the sub-population sizes or areas must be known. In stratified sampling, a sample is drawn from each of the strata, the size of each sample ideally in proportion to the population size or area of that stratum.

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Sample mean and variance estimators of the population quantities

N

yNl

ll

cA

ll

l

llc f

n

sWAVar 1)(

22

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Systematic sampling

• Systematic sampling differs from the methods of random sampling in terms of practical implementation and in terms of coverage. Again, assume there are N (= nk) units in the population. Then to sample n units, a unit is selected for sampling at random. Then, subsequent samples are taken at every k units. Systematic sampling has a number of advantages over simple random sampling, not least of which is convenience of collection. A systematic sample is thus spread more evenly over the population.

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Systematic sampling

• Data from systematic designs are more difficult to analyze, especially in the most common case of a single systematic sample. Consider first the simpler case of multiple systematic samples. For example, xxx in pond sediment could be sampled using transects across the pond from one shoreline to the other. Samples are collected every 5m along the transect. The locations of the transects are randomly chosen. Each transect is a single systematic sample.

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Systematic sampling

• Each sample is identified by the transect number and the location along the transect. Suppose there are i = 1,.., t systematic samples (i.e. transect in the pond example) and the yij is the jth observation on the ith

systematic sample for j = 1,…, ni. The

average of the samples from the i’th transect is calculated.

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Population mean and variance estimators

t

ii

t

iij

n

j

t

ii

t

iii

sy

n

y

n

yny

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1

1 1

1

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tt

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/1)(

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• How many samples are needed to ?

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Number of sampling units do you need to collect

• state the desired limits of precision for the population inference (how precisely does one want to know the average PCB concentration, or, what size of difference is needed to be detected and with what precision?),

• state the inherent population variability of the attribute of interest, and

• derive an equation which relates the number (n) of samples with the desired precision of the parameter estimator and the degree of significance (the chance of being wrong in the inference).

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Number of samples

What is the power?

Power is a probability, it is the probability that we correctly conclude that the null hypothesis should be rejected. The null would say there is no difference/no effect/no trend.

We want a high power

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Power Curves

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PCB

• estimate the mean concentration with an estimated standard error (e.s.e.) precision of 0.1 mg kg-1. The variation of PCB in salmon flesh is 3.192. Therefore, how many samples would be required? Since the e.s.e. of the sample mean is s/n, then one must solve for n, for example:

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Sample size-too big

2

...

ese

sn 1018

1.0

19.32

Thus this degree of improvement in precision, can only be achieved by increasing the number of samples taken to approximately 1000. This may well be impractical; therefore the only solution may be to accept a lower precision.

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Outline

• Spatial sampling– Grid, transect and cluster sampling

• See also Spatial modelling section

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Know what you are setting out to do before you start

 describing a characteristic of interest (usually the average),

describing the magnitude in variability of a characteristic, describing spatial patterns of a characteristic,mapping the

spatial distribution, quantifying contamination above a background or

specified intervention level  detecting temporal or spatial trends,   assessing human health or environmental impacts of

specific facilities, or of events such as accidental releases,

• assessing compliance with regulations

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Spatial sampling

• In ecology, spatial data usually fall into one of two different general cases:

• Case 1: We assume that there is an attribute that is spatially continuous, where in principle it is possible to measure the attribute at any location defined by coordinates (x, y) over the domain or area of interest.

• Case 2: The attribute is not continuous through space; it exists and can be measured only at specific locations (see point processes in spatial session).

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Random and stratified random sampling

In random sampling, a random sample of locations at which the attribute is to be measured is chosen from the target population of locations. If there is knowledge of different strata over the sampling domain (such as soil type), the use of a stratified sample would be recommended and a random sample of locations would be selected within each strata. The data set is then given by the spatial coordinates of each measurement location and the measured value of the attribute at that location

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systematic sampling

Usually, for systematic sampling the region is considered as being overlaid by a grid (rectangular or otherwise), and sampling locations are at gridline intersections at fixed distance apart in each of the two directions. The starting location is expected to be randomly selected.

Both the extent of the grid and the spacing between locations are important. The sampling grid should span the area of interest (the population). If the goal of the study is to describe spatial correlations, the spacing between locations should be shorter than the range of the correlation.

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Quadrats and transects

• A quadrat is a well-defined area within which one or more samples are taken; it is usually square or rectangular in shape, with fixed dimensions. The position and orientation of the quadrat will be chosen as part of the sampling scheme.

• A line transect is a straight line along which samples are taken, the starting point and orientation of which will be chosen as part of the sampling scheme. In addition, the number of samples to be collected along the transect, and their spacing requires definition.

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Stratified sampling chosen

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Systematic sampling

• Again, assume there are N (= nk) units in the population. Then to sample n units, a unit is selected for sampling at random. Then, subsequent samples are taken at every k units. Systematic sampling has a number of advantages over simple random sampling, not least of which is convenience of collection.

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Transect sampling chosen

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Before-after-control-impact (BACI) designs

• One of the most plausible alternative explanations of a change is that the system changed ‘on its own’. That is, the observed change (from the before-event samples to the after-event samples) would have happened even in the absence of the known impact.

• One simple impact-assessment design evaluates this alternative by estimating the change at a control site presumed to be unaffected by the known event. Data are collected at four combinations of sites and times: affected and unaffected sites, each sampled before the impact and after the impact.

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Before-after-control-impact (BACI) designs

• The impact of the known event is estimated by the interaction between sites and times, i.e., the difference between the change at the impacted site and the change at the control site. The BACI design controls for additive temporal change unrelated to the known event.

• Elaborations on the basic BACI design include using multiple control sites to estimate spatial variability and spatial trends, multiple samples from the impacted area to estimate variability within the impacted area, and very frequent sampling to better characterize the nature of the impact.

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Graph of designs

7654321

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time

Y-D

ata

impactcontrol

Variable

Scatterplot of impact, control vs time

simplest scenario, impact occurs between time point 3 and 4

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Graph of designs

impact occurs between time point 3 and 4, but the impact effect declines with time

7654321

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time

Y-D

ata

impactcontrol

Variable

Scatterplot of impact, control vs time

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Analysis of BACI designs

• before after control impact

• if observations made at the same time at the two sites, then we could analyse the difference (paired design)

• with repeated measurements then would need to consider a repeated measures ANOVA approach.

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summary

• Sampling and monitoring the environment is carried out for many purposes, including estimation of certain characteristics.

• Many experimental and monitoring programs have multiple objectives that must be clearly specified before the sampling program is designed, because different purposes require different sampling strategies and sampling intensities in order to be efficient, and to permit general inferences.

• Good sampling underpins all our statistical modelling