envir.stat lec 3.2ppt

7/27/2019 Envir.stat Lec 3.2ppt

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Lecture-4Sampling Methods2. Stratified Random Sampling.Engr. Dr. Attaullah Shah


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Simple Random Sampling

Used when there is inadequate

information for developing a conceptual

model for a site or for stratifying a site

Any sample in which the probabilities of

selection are known

Sampling units are chosen by usingsome method using chance to determine

selection


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Simple random sampling is the basis

for all probability sampling techniques

and is the point of reference from whichmodifications to increase sampling

efficiency may be made

Alone, simple random sampling may notgive the desired precision


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Simple Random Sampling Advantages

Prior information about population is not necessary Easy to perform, easy to analyze

Disadvantages

May not give desired precision

Need a sampling frame.

One way to overcome this problem while still keeping theadvantages of random sampling is to use stratifiedrandom sampling.

This involves dividing the units in the population into nonover lapping strata, and selecting an independent simplerandom sample from each of these strata.


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One way to overcome this problem while

still keeping the advantages of random

sampling is to use stratified randomsampling. This involves dividing the units

in the population into non over lapping

strata, and selecting an independentsimple random sample from each of these

strata.


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Stratified Random Sampling

Prior knowledge of the sampling area

and information obtained from

background data may be used toreduce the number of observations

necessary to attain specified precision

Goal is to increase precision and controlsources of variability in the data


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Variability between strata must be larger

than variability with strata for any benefit

to be seen Sampling within each stratum is done

with a Simple Random Sample


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AdvantagesGives estimates for subgroups

Can be more precise than Simple

Random SamplingCan be more convenient to

implement

DisadvantagesRequires prior information about the

population

More complicated computation


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Potential gains of Stratified Sampling

First, if the individuals within strata are more similarthan individuals in general, then the estimate of theoverall population mean will have a smallerstandard error than can be obtained with the samesimple random sample size.

Second, there may be value in having separateestimates of population parameters for the differentstrata.

Third, stratification makes it possible to sampledifferent parts of a population in different ways,which may make some cost savings possible.


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Assume that Kstrata have been chosen, ith the ithof these having size Niand the total populationsize being Ni= N.

Then if a random sample with size niis taken fromthe ith stratum, the sample meanyiwill be anunbiased estimate of the true stratum mean i,with estimated variance as:

Where siis the sample standard deviation withinthe stratum.

In terms of the true strata means, the overallpopulation mean is the weighted average.


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And the corresponding sample estimate is

with estimated variance

The estimated standard error of is , the square root of the

estimated variance, and an approximate 100(1 )% confidence

interval for the population mean is given by:

If the population total is of interest, then this can be estimated by

The estimated standard error of population total:

Again, an approximate 100(1 )% confidence interval takes the

form


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When a stratified sample of points in a spatial region is

carried out, it will often be the case that there are an

unlimited number of sample points that can be takenfrom any of the strata, so thatNi andNare infinite.

Equation can then be modified to and the

equation becomes

Where wi, the proportion of the total study area within

the ith stratum, replacesNi/N.


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Example 2.3: Bracken Density in Otago

As part of a study of the distribution of scrub weeds in New Zealand,data were obtained on the density of bracken on 1-hectare (ha, 100 100m) pixels along a transect 90-km long and 3-km wide, running from

Balclutha to Katiki Point on the South Island of New Zealand, as shownin Figure 2.2 (Gonzalez and Benwell 1994).

This example involves a comparison between estimating the density(the percentage of the land in the transect covered with bracken) using(a) a simple random sample of 400 pixels, and (b) a stratified randomsample with five strata and the same total sample size.

There are altogether 27,000 pixels in the entire transect, most of whichcontain no bracken. The simple random sample of 400 pixels was foundto contain 377 with no bracken, 14 with 5% bracken, 6 with 15%

bracken, and 3 with 30% bracken. The sample mean is thereforey =0.625%, the sample standard deviation iss = 3.261, and the estimatedstandard error of the mean is

The approximate 95% confidence limits for the true population meandensity is therefore 0.625 1.96 0.162, or 0.31% to 0.94%.


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The strata for stratified sampling were five stretches of thetransect, each about 18-km long, and each containing 5400 pixels.The sample results and some of the calculations for this sampleare shown in Table 2.4.

The estimatedpopulation mean

density from equation

given equation is

0.613%, with an

estimated variance of

0.0208 from equationThe estimated

standard error is

therefore 0.0208 =

0.144, and an

approximate 95%

confidence limits for

the true population

mean density is 0.613

1.96 0.144, or 0.33%

to 0.90%.


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Post Stratification

Can be used when stratification is

appropriate for some key variable, but

cannot be done until after the sample isselected

Often appropriate when a simple

random sample is not properly balancedaccording to major groupings


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A simple random sample is expected to placesample units in different strata according to the

size of those strata. Therefore, post-stratification should be quite similar to stratifiedsampling with proportional allocation, providingthat the total sample size is reasonably large.

It therefore has some considerable potentialmerit as a method that permits the method ofstratification to be changed after a sample has

been selected. This may be particularlyvaluable in situations where the data may beused for a variety of purposes, some of whichare not known at the time of sampling.

envir.stat lec 3.2ppt

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