Sampling in Space and Time. Natural experiment? Analytical Surveys

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1 Sampling in Space and Time Overview of Sampling Approaches Sampling versus Experimental Design Experiments deliberately perturb a portion of population to determine effect objective is to compare the mean response to changes in levels of the factors Sampling see what the population is like without disturbing it objective is to describe the characteristics of the population Natural experiment? Experimental Studies Researcher controls the assignment of the explanatory variables while measuring the response variables Analytical and Observational surveys neither the explanatory nor response variables are under control of the researcher sub-populations are selected and sampled to estimate the observed differences among the subpopulation Analytical Surveys Identify potential explanatory variables (similar to factors in experimental studies) Subdivided into categories depending on the type of stratification the population is pre-stratified by the explanatory variables and sample in each stratum; the population is sampled, and post-stratified by the explanatory variables; the explanatory variables can be used as auxiliary variables in ratio or regression methods. The principle steps in Sampling The principle steps in a survey State the objectives of the survey Define the population to be sampled what is the range of animals or locations to be measured? Establish what data are to be collected you don t t need to collect everything What degree of precision is required examine power needed What is your sample population? the list of all available sampling units often very difficult to establish - e.g. a list of all streams in the central interior of British Columbia; mountain goats in the population under study 1

2 Stratification? Many sampling approaches simple random sampling systematic sample cluster sampling multi-stage design All designs can be improved by stratification Always consider when designing a sampling regime Before you begin? Pre-test try out field methods and questionnaires Probability sampling versus nonprobability sampling Probability sampling units are chosen randomly Non-probability sampling units are chosen in some deliberate fashion Probability Sampling every unit has a known probability of being sampled sample is drawn with some method consistent with these probabilities these selection probabilities are used when making estimates from the sample Advantages sampling biases can be studied standard errors and measures of precision can be obtained Non-probability sampling Quotas select 50 of each sex from the population less expensive than a probability sample may be only option if can t identify sample population Judgement select typical units can perform well if there are a few extreme points which should not be included Convenience select those readily available; useful if is dangerous or unpleasant to sample directly selecting blood samples from grizzly bears Non-probability sampling Haphazard This is often useful if the sampling material is homogeneous and spread throughout the population, e.g. chemicals in drinking water 2

3 Non-probability sampling Disadvantages unable to assess biases no estimates of precision use of formulae from probability sampling is WRONG!. Randomization Remove the possibility of bias in the selection of a sample Sometimes this is simply not possible When this happens and the investigators wish to use the results of a nonrandomized sample, then the final report should discuss the possibility of selection bias and its potential impact on the conclusions. Sampling Where should I take my samples? How should I take my samples? How many samples should I take? Sampling How much work do I need to do? How many samples should I take? How many samples should I take? By previous sampling of a similar population By the results of a pilot study By guesswork By two-stage sampling Simple Random Sample Simple random sampling, or random sampling without replacement, is sampling in which n distinct units are selected from the N units in the population in such a way that every possible combination of n units is equally likely to be sampled. Examples? 3

4 Simple Random Sample Examples? Sampling Does this mean that all units in the population have to always be sampled with the same intensity? In some cases, not possible to randomly select sample units from the population An alternative is to take a systematic sample where every k th unit is selected (after a random starting point) k is chosen to give the required sample size For example, if a stream is 2 km long, and 20 samples are required then k = 100 and samples are chosen every 100 m along the stream after a random starting point If a known trend is present in the sample, this can be incorporated into the analysis e.g., sample follows an elevation gradient that is known to directly influence the response variable. A regression-type correction can be incorporated into the analysis This trend must be known from external sources - it cannot be deduced from the survey. If a systematic sample happens to match a cyclic pattern in the population, values within the systematic sample are highly positively correlated precision is worse than a SRS of the same size Because the units are positively correlated within the sample, the sample variance will underestimate the true variation in the population 4

5 Instead of taking a single systematic sample every k th unit, take 2 or 3 independent systematic samples of every 2k th or 3k th unit, each with a different starting point Rather than taking a single systematic sample every 100 m along the stream, two independent systematic samples can be taken, each selecting units every 200 m along the stream starting at two random starting points. The total sample effort is still the same, but now some measure of the large scale spatial structure can be estimated. This technique is known as replicated sub-sampling But what if you are trying to determine pattern? For temporal pattern could repeatedly sample Apply concepts of stratification to time? Stratified Random Sampling The population is partitioned into regions or strata, and a sample is selected within each strata. The principle of stratification is to partition the strata so that units within a strata are as similar as possible. Stratified Random Sampling Cluster Sampling Units in a population occur naturally in groups or clusters e.g., some animals congregate in herds or family units select a random sample of herds and then measure every animal in the herd not the same as a simple random sample of animals because individual animals are not randomly selected; the herds are the sampling unit 5

6 Cluster Sampling Another example is circular plot sampling; all trees within a specified radius of a randomly selected point are measured, but the sampling unit is the circular plot and trees within the plot are sub-samples. Adaptive Sampling Adaptive sampling refers to sampling designs in which the procedure for selecting units to be included may depend on values of samples during the survey Adaptive Sampling Adaptive Sampling Adaptive Sampling Multi-phase Sampling In some surveys, multiple surveys of the same survey units are performed. In the first phase, a sample of units is selected (SRS); every unit is measured on some variable In subsequent phases, samples are selected ONLY from those units selected in the first phase, not from the entire population. 6

7 Multi-phase Sampling Multi-phase Sampling Multi-phase Sampling Sometimes difficult to stratify a population in advance because the values of the stratification variables are unknown The first phase is used to measure the stratification variable on a random sample of units Selected units are then stratified, and further samples are taken from each stratum as needed to measure a second variable Avoids having to measure the second variable on every unit when the strata differ in importance. Multi-phase Sampling For example, in first phase plots selected and measured for the amount of insect damage Plots then stratified by the amount of damage, and second phase allocation of units concentrates on plots with low insect damage to measure total usable volume of wood Little value to measure the volume of wood on plot with much insect damage Multi-phase Sampling Need to estimate the volume of wood removed from a harvesting area A large sample of logging trucks is weighed (surrogate of volume) Smaller sample of trucks (selected from those weighed) is scaled for volume and the relationship between volume and weight from the second phase sample is used to predict volume based on weight only for the first phase sample Double Sampling Initially a sample of units is selected and then additional data is collected from a subset of the original samples 7

8 Double Sampling Repeated Sampling ACTUAL Separate independent surveys can be conducted at each time point observations are independent over time, but additional variability introduced by having new units sampled at each time point ESTIMATE Repeated Sampling Plot Shape? Permanent study plots can be established that are re-measured for regeneration over time. comparisons over time are free of additional variability introduced by new units being measured at every time point sample size will tend to decline over time Generating Random Variables Uniform Random Variables RAND() returns an evenly distributed random number greater than or equal to 0 and less than 1. A new random number is returned every time the worksheet is calculated. Generating Random Variables To generate a random real number between a and b, use: RAND()*(b-a)+a a)+a.. You can convince yourself of this by seeing that to generate a random number between 0 and 100 this formula simplifies to RAND()*100. In order to get integer random numbers between 0 and 100 use the formula =INT((RAND()*100)+0.5) [if analysis toolpak loaded] Why do we need to add 0.5 before taking the integer of this value? 8

9 Normally Distributed Random Variables An alternate way to generate random numbers between two numbers is using the RANDBETWEEN(bottom,top) function Bottom is the smallest integer RANDBETWEEN will return and top is the largest integer RANDBETWEEN will return Alternatively, the Box-Muller method allows you to generate normal deviates with mean and standard deviation from uniform random numbers by applying the formula NORM (, ) 2 ln( rand()) cos(2* rand()) Random numbers within a radius Random numbers within a radius Random distance and bearing from centre of circle? Random numbers within a radius Random numbers within a radius Overlay square and pick pairs of random points to see if they fall within the circle (distance from centre less than radius) If OK keep, otherwise reject and then get new pair 9

10 Distance Sampling Distance Sampling Estimating the abundance of an item based on its detectability Distance Sampling Nested Designs What happens if we draw our samples from a number of different subpopulations or areas? n D 2 L a Are they still all replicates of each other? Nested Designs Spatial Patterns (Random) within 10

11 Spatial Patterns (Clumped) Spatial Patterns (Uniform) Spatial Patterns s 2 X 11