Chapter-7 Sampling Design

7/30/2019 Chapter-7 Sampling Design

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Sampling Design

M.Mariappan


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Outline

Introduction

Sampling methods

Sampling design


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What Do We Want to Know in Sampling?

Census and Sample Survey

A complete enumeration of all items in thepopulation is known as a census of inquiry

Government is the only institution which can get

the compete enumeration carried out. But many a time it is not possible to examine

every item in the population, and sometimes it ispossible to obtain sufficiently accurate results

by studying only a part of total population. So sample become necessary in the research

study


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Why Sampling?

What is a sample?

It is a representative sub-set of the population

Why not study the whole population directly?

The sampling dilemma: time, cost vs. accuracy

Sampling is not always possible Small population

Example: Survey sampling

To study a population by asking structured questions

Used in social science, policy study, marketing


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Steps in Sampling Design (Concepts)

Type of universe: The universe can be finite orinfinite. In finite universe the number of items is

certain, but in case of an infinite universe (city

population, factory workers etc,) the number ofitems is infinite, i.e., we cannot have any idea

about the total number of items ( number of stars).

Sampling Unit: Sampling Unit may be

geographical one such as state, district, village,

etc., or a construction uit such as house flat., it

may be individual.

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Source list: It is also known as sampling frame

from which sample is to be drawn. It contains the

names of all items of a universe (incase of finite

universe only). If the source list is not available,the researcher has to prepare it. Such list should

be comprehensive, correct, reliable and

appropriate. It is extremely important for thesource list to be as representative of the

population as possible.

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Steps in Sampling Design (Concepts) Size of sample: This refers to the number of items to be

selected from the universe to constitute a sample. The

sample size should neither be excessively large, nor too

small. It should be optimum. An optimum sample is one

which fulfills the requirements of efficiency, appropriate

representation, reliability and flexibility. While deciding the

sample size, researcher must determine the desired precision

as also an acceptable confidence level for the estimate. The

size of population variance needs to be considered as incase

of larger variance usually a bigger sample is needed. Thesize of population must be kept in view for this also limits

the sample size. The parameters in a research study must be

kept in view, while deciding the sample size


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Parameters of interest: In determining the sampledesign, one must consider the question of the specific

population parameters which are of interest. For

instance, we may be interested in estimating the

proportion of persons with some characteristic in thepopulation, or we may be interested in knowing some

average or the other measure concerning the population.

There may be also be important sub-groups in thepopulation about whom we would like to make

estimates. All this has strong impact upon the sample

design we would accept.

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Sampling procedure: Finally, the researchermust decide the type of sample he will use i.e., he

must decide about the technique to be used in

selecting the items for the sample. In fact, thistechnique or procedure stands for the sample

design itself. There are several sample design out

of which the researcher must choose one for his

study. Obviously, he must select that designwhich, for a given sample size and for a given

cost, has a smaller error.

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Sampling Validity

External validity is the generalizability of findings

from a sample to its target population

Affected by both sampling design & execution

Statistical validity is the ability to reach

conclusions about relationships that appear in the

sample data

Affected by sample size. Also called power. Affected by reliability of measures


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Sources of Error in Sampling

Total

Error

Target PopulationTo whom we want to

generalize findings

Study Population

Operational definition of

target population &measurement instrument

Sample Distribution

The distribution of an

estimator, e.g., sample

average

Sample

The subset of subjects or units

for which data is obtained

Non-sampling bias

Listing & frame

Non-response

Measurement error

Sampling bias

Selection bias

Estimation bias

Sampling variability Sample size

Sample Homogeneity


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Sample Design Decisions

Presampling choices

Define the target population

Justify the sampling method

Sampling choices Define the sampling frame

Select the sampling method

Set the sample size

Postsampling choices Evaluate non-response and other bias

Measure the quality of the estimates


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Presampling Choices

What is the nature of the studyexploratory,descriptive, or analytical?

What are the variables of greatest interest?

What is the target population for the study Are subpopulations or special groups important for

the study?

How will the data be collected?

Is sampling appropriate?


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Sampling Choices

What listing of the target population can be used

for the sampling frame?

What is the tolerable error or estimated effect

size?

What type of sampling technique will be used?

Will the probability of selection be equal or

unequal?

How many units will be selected for the sample?


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Element selection

technique

Unrestricted

sampling

Representation basis

Probability sampling Non- Probability

sampling

Simple Random

sampling

Haphazard sampling or

convenience sampling

Restricted

sampling

Complex random

sampling (such ascluster sampling,

systematic sampling

stratified sampling,

etc.)

Purposive sampling

(such as quotasampling, judgment

sampling)

CHART SHOWING BASIC SAMPLING DESIGNS


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Sampling Methods

Two Broad Categories

Non-probability sampling

Depends on researchers subjective judgment

Based on convenience or systematically employed

criteria. Some units will surely not be selected

Probability sampling

Randomization w/o subjective judgment Each unit has a known, nonzero probability to be

selected


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Nonprobability Sample Design

Convenience samples

Most similar/most dissimilar samples

Typical case samples

Critical case samples

Snowball samples

Quota samples


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Quota Sampling

Quota sampling is a method of sampling widely used inopinion polling and market research. Interviewers are

each given a quota of subjects of specified type to

attempt to recruit for example, an interviewer might be

told to go out and select 20 adult men and 20 adultwomen, 10 teenage girls and 10 teenage boys so that

they could interview them about their television

viewing.

It suffers from a number of methodological flaws, the

most basic of which is that the sample is not a random

sample and therefore the sampling distributions of any

statistics are unknown. 19


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Spatial Sampling

This is an area of survey sampling concerned with

sampling in two (or more) dimensions. For

example, sampling of fields or other planar areas.

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Snowball sampling Snowball sampling is a special nonprobability

method used when the desired sample characteristic

is rare. It may be extremely difficult or cost

prohibitive to locate respondents in these situations.

Snowball sampling relies on referrals from initialsubjects to generate additional subjects. While this

technique can dramatically lower search costs, it

comes at the expense of introducing bias because the

technique itself reduces the likelihood that the

sample will represent a good cross section from the

population.

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Convenience sampling

Convenience sampling is used in exploratory

research where the researcher is interested in

getting an inexpensive approximation of the truth.

As the name implies, the sample is selectedbecause they are convenient. This nonprobability

method is often used during preliminary research

efforts to get a gross estimate of the results,

without incurring the cost or time required to

select a random sample.

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Judgment sampling

Judgment sampling is a common nonprobability

method. The researcher selects the sample based

on judgment. This is usually and extension of

convenience sampling. For example, a researchermay decide to draw the entire sample from one

"representative" city, even though the population

includes all cities. When using this method, the

researcher must be confident that the chosen

sample is truly representative of the entire

population.

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Nonprobability Sampling

Pros

Expedient and economic method

The only applicable method in some cases.

Good for exploratory research/pilot study

Cons

External validity

Consequently, the findings may not be valid


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Probability Sampling

Probability sampling implies random selection

mechanism.

Randomness means the independence of each

selection

Equal probability sampling

Every unit has same probability to be included. But why?

Unequal probability sampling Some units are more likely to be selected

The estimate needs to be adjusted later


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Probability Sampling Designs

Simple random sampling

Systematic sampling

Stratified sampling

Cluster sampling

Multistage sampling


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A sampling procedure that assures that each

element in thepopulation has an equal chance of

being selected is referred to assimple random

sampling .Let us assume you had a school with a1000 students, divided equally into boys and girls,

and you wanted to select 100 of them for further

study. You might put all their names in a drum

and then pull 100 names out.

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Not only does each person have an equal chance

of being selected, we can also easily calculate the

probability of a given person being chosen, since

we know the sample size (n) and the population(N) and it becomes a simple matter of division:

n/N x 100 or 100/1000 x 100 = 10%

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This means that every student in the school as a

10% or 1 in 10 chance of being selected using this

method.

The sample can be selected by many computerstatistical packages, including SPSS, are capable

of generating random numbers and some phone

systems are capable of random digit dialing.

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Simple Random Sampling

Assuming selection without replacement

p=f=n/N

p: probability of selection

f: sampling fraction

n: sample size

N: population size


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Systematic Sampling- Procedure If a systematic pattern is introduced into random

sampling, it is referred to as "systematic (random)sampling". For instance, if the students in our

school had numbers attached to their names

ranging from 0001 to 1000, and we chose a

random starting point, e.g. 533, and then pick

every 10th name thereafter to give us our sample

of 100 (starting over with 0003 after reaching

0993). In this sense, this technique is similar tocluster sampling, since the choice of the first unit

will determine the remainder.

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Systematic Sampling- Procedure

Determining the sample size

Determining the selection interval, i=N/n rounded downto an integer

Obtaining a listing or physical representation of studypopulation

Selecting a random start r between 1 and i to select thefirst member

Selecting member r+i, r+2i

If the sample size so selected is larger than n, discardthe excess number of unit(s) with a random selection

process


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Distinction between a random sample

and a simple random sample

In a simple random sample, not only does eachelement have the same probability of being selected

for the sample, but furthermore every possible

sample has the same probability. The latter, stronger

property distinguishes the notion of "simple random

sample" from the notion of just "random sample".

Consider a sample of 4 integers from the population

of integers 1 through 8. One way to draw thissample is to flip a coin. If the coin comes up heads,

then select the four even integers.

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Distinction between a random sample

and a simple random sample

If the coin comes up tails, then select the four oddintegers. This is a random sample because the

probability that any one element appears in the

sample is exactly 50%. But it is not a simple

random sample because one possible sample

(1,3,5,7) has 50% probability while another possible

sample (1,2,3,4) has 0% probability. Since there are

8!/(4!4!)=70 different samples of size 4 from apopulation of 8, for a sample to be a simple random

sample, each sample has to have probability 1/70.

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Pros and Cons

Pros

Pseudo-simple random samples, with same statistical

properties as random samples.

Ease of selection in field settings. E.g. selecting 300invoices out of 15,222.

No list to compile if there is physical presentation

No need for random number table

Cons

Not good for cyclical data

Limited accuracy as with SRS


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Stratified sampling When sub-populations vary considerably, it is advantageous

to sample each subpopulation (stratum) independently.Stratification is the process of grouping members of the

population into relatively homogeneous subgroups before

sampling. The strata should be mutually exclusive : every

element in the population must be assigned to only onestratum. The strata should also be collectively exhaustive:

no population element can be excluded. Then random or

systematic sampling is applied within each stratum. This

often improves the representativeness of the sample by

reducing sampling error. It can produce a weighted mean

that has less variability than the arithmetic mean of a simple

random sample of the population. 36
http://en.wikipedia.org/wiki/Weighted_meanhttp://en.wikipedia.org/wiki/Arithmetic_meanhttp://en.wikipedia.org/wiki/Arithmetic_meanhttp://en.wikipedia.org/wiki/Weighted_mean


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Stratified sampling

Randomized stratification can also be used to

improve population representativeness in a study.

Advantages

focuses on important subpopulations but ignores

irrelevant ones

improves the accuracy of estimation

efficient

sampling equal numbers from strata varying widely

in size may be used to equate the statistical powerof

tests of differences between strata.

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http://en.wikipedia.org/wiki/Statistical_powerhttp://en.wikipedia.org/wiki/Statistical_testshttp://en.wikipedia.org/wiki/Statistical_testshttp://en.wikipedia.org/wiki/Statistical_power


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Stratified sampling

Disadvantages can be difficult to select relevant stratification

variables

not useful when there are no homogeneous

subgroups

can be expensive

requires accurate information about the

population, or introduces bias.

looks randomly within specific sub headings.

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Choice of sample size for each stratum In general the size of the sample in each stratum

is taken in proportion to the size of the stratum.This is called proportional allocation. Suppose

that in a company there are the following staff:

male, full time: 90

male, part time: 18

female, full time: 9

female, part time: 63 Total: 180

and we are asked to take a sample of 40 staff,

stratified according to the above categories.39


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Choice of sample size for each stratum

The first step is to find the total number of staff (180)

and calculate the percentage in each group.

% male, full time = (90 180) x 100 = 0.5 x 100 = 50

% male, part time = ( 18 180 ) x100 = 0.1 x 100 = 10

% female, full time = (9 180 ) x 100 = 0.05 x 100 = 5

% female, part time = (63 180) x 100 = 0.35 x 100 = 35

40


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Choice of sample size for each stratum

This tells us that of our sample of 40,

50% should be male, full time.

10% should be male, part time.

5% should be female, full time.

35% should be female, part time.

50% of 40 is 20.

10% of 40 is 4.

5% of 40 is 2. 6+

35% of 40 is 14.


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Stratified sampling

Sometimes there is greater variability in some

strata compared with others. In this case, a larger

sample should be drawn from those strata with

greater variability. All people in sampling frame are divided into

"strata" (groups or categories). Within each

stratum, a simple random sample or systematicsample is selected.

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St tifi d li


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Stratified sampling Example of stratified sampling - If we want to

ensure that a sample of 5 students from a group of50 contains both male and female students in same

proportions as in the full population (i.e. the group

of 50), we first divide that population into male and

female. In this case, there are 22 male students and28 females. To work out the number of males and

females in the sample........

No. of males in sample = (5 / 50) x 22 = 2.2 No. of females in sample = (5 / 50) x 28 = 2.8 We obviously can't interview .2 of a person or .8 of a person, and have to

"round" the numbers. Therefore we choose 2 males and 3 females in the

sample. These would be selected using simple random orsystematic samplemethods43

S ifi d li
http://www.tardis.ed.ac.uk/~kate/qmcweb/s3.htmhttp://www.tardis.ed.ac.uk/~kate/qmcweb/s4.htmhttp://www.tardis.ed.ac.uk/~kate/qmcweb/s4.htmhttp://www.tardis.ed.ac.uk/~kate/qmcweb/s3.htm


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Stratified samplingActivity

A company employs 200 part-time staff and 800full-time staff, and you want to undertake depth

interviews with 20 of these. If you were to take a

random sample, you might find that all of the names

you selected were part-time staff. For this reason

you have decided to undertake a stratified sample.

Work out how many part-time and how many full-

time employees you should interview so as toaccurately reflect the proportions of the two groups

in the whole workforce.

Answer = ___ Part-time and _ Full-time 44


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Proportional Stratification

Using same sampling fraction in each stratum

When to be used?

Unit/case/element can be easily classified

Population list is available, and the proportions for

strata are available

Units in each stratum are homogeneous and are easy

for sampling (cost-wise)


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Proportional Stratification-properties

2/122 ])/([

/)(

kkkx

i

nsws

nxx

n: total sample size

wk: Nk/N, stratum proportion in population

sk: std error of stratum sample

xi

: unit in any of the kstrata

Noticesk2/nk is the variance of mean for a stratum

Demo
http://localhost/var/www/apps/conversion/tmp/scratch_4/sampling.xlshttp://localhost/var/www/apps/conversion/tmp/scratch_4/sampling.xls


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The Design Effect

Stratification reduces standard errors (for mean

estimate)

Why?

Variance of the mean consists of only within-stratumvariance

Between-stratum variance is ignored, because the

classification information is known


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Pros and Cons

Pros

Proportional stratification reduces error. The magnitude

of gain is depending on:

variability between strata and

homogeneity within stratum

Ensures proportional representation of stratifying

groups

Cons

Classifying members could be expensive or infeasible


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Disproportional Stratification

Unequal probability of selection for each stratum

Used when:

sub-population is a subject of study.

Variance of sub-population needs is too high to give

accurate estimate of mean.

Estimates need to be adjusted/weighted by

proportion of each stratum. This is also calledpost-stratification.


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Statistical Properties

Give more proportion to high variance stratank=n(NkSk)/(NkSk)

stratumaofmeantheoferrorstd:

populationinproportionstratum,:

stratumforsizesample:

])/([ 2/122

k

kk

k

kkkx

kk

s

/NNw

kn

nsws

xwx


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CLUSTER SAMPLING Cluster sampling is a sampling technique

where the entire population is divided intogroups, or clusters, and a random sample of

these clusters are selected. All observations in

the selected clusters are included in thesample.

Cluster sampling is typically used when the

researcher cannot get a complete list of themembers of a population they wish to study

but can get a complete list of groups or

'clusters' of the population.51


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CLUSTER SAMPLING

It is also used when a random sample would

produce a list of subjects so widely scattered

that surveying them would prove to be far too

expensive, for example, people who live indifferent postal districts in the State.

This sampling technique may well be more

practical and/or economical than simplerandom sampling or stratified sampling.

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CLUSTER SAMPLING

For Example

Suppose that the Department of Agriculture wishes to

investigate the use of pesticides by farmers in England.

A cluster sample could be taken by identifying the

different counties in England as clusters. A sample ofthese counties (clusters) would then be chosen at

random, so all farmers in those counties selected would

be included in the sample. It can be seen here then that

it is easier to visit several farmers in the same county

than it is to travel to each farm in a random sample to

observe the use of pesticides.

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CLUSTER SAMPLING

It is sometimes expensive to spread your sample

across the population as a whole. For example,

travel can become expensive if you are using

interviewers to travel between people spread all over

the country. To reduce costs you may choose acluster sampling technique.

Cluster sampling divides the population into groups,

or clusters. A number of clusters are selectedrandomly to represent the population, and then all

units within selected clusters are included in the

sample. 54


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CLUSTER SAMPLING

No units from non-selected clusters are

included in the sample. They are represented

by those from selected clusters. This differs

from stratified sampling, where some units are

selected from each group.

Examples of clusters may be factories, schools

and geographic areas such as electoral sub-

divisions. The selected clusters are then usedto represent the population.

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C ST SA G


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CLUSTER SAMPLING

Suppose an organisation wishes to find out which sports

Year 11 students are participating in across Australia. It

would be too costly and take too long to survey every

student, or even some students from every school. Instead,

100 schools are randomly selected from all over Australia.

These schools are considered to be clusters. Then, every

Year 11 student in these 100 schools is surveyed. In effect,

students in the sample of 100 schools represent all Year 11students in Australia.

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CLUSTER SAMPLING


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CLUSTER SAMPLING

Cluster sampling has several advantages: reduced costs,

simplified field work and administration is moreconvenient. Instead of having a sample scattered over the

entire coverage area, the sample is more localised in

relatively few centers (clusters).

Cluster samplings disadvantage is that less accurate resultsare often obtained due to higher sampling error than for

simple random sampling with the same sample size. In the

above example, you might expect to get more accurate

estimates from randomly selecting students across allschools than from randomly selecting 100 schools and

taking every student in those chosen.

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Cl S li


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Cluster Sampling

Randomly select clusters (as primary selectionunit, PSU) and all its members

e.g. customers at regional stores

Use

when list of clusters is available but not the list ofthe population

when data collection involves visits togeographically dispersed locations

Less expensive

Precision in estimate is lower as the selection ofeach unit is not all independent

P i i P bl


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Precision Problem

The precision problem

Differences between the cluster means and the overall means

(Difference->precision)

Heterogeneity of the clusters (Heterogeneity->precision )

Within-cluster difference ->precision )

Cluster number-> precision

The design effect >1

Cl S li i


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Cluster Sampling -- properties

K: number of clusters selected

Mk: cluster size

A: total number of clusters

xkbar: is the cluster mean

x-bar: is the overall mean

(1-K/A): finite population correction

)2(])/1[(

)1(])/1[(

/

2/1

1

221

2/1

1

)1(

)(

11 1

2

2

K

k

kkKMx

K

k

KK

xx

x

K

i

i

K

k

M

i

ki

sMAKs

AKs

Mxx

k

k

M l i S li


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Multistage Sampling

Simple multistage sampling (two stage sampling)

selection of clusters as PSUs and

randomly sampling members of the selected clusters

There could be more than two levels and other sampling

methods can be also involved at each level.

E.g., to study students attitude towards races,

School districtSchoolClassroomindividuals

To select PSU, probability proportionate to size(PPS)method can be used.

Statistical packages are used to calculate the estimators.

M l i S li


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Multistage Sampling

Multi-stage sampling is like cluster sampling,

but involves selecting a sample within each

chosen cluster, rather than including all units

in the cluster. Thus, multi-stage samplinginvolves selecting a sample in at least two

stages. In the first stage, large groups or

clusters are selected. These clusters aredesigned to contain more population units than

are required for the final sample.

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M l i S li


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Multistage Sampling

In the second stage, population units are chosenfrom selected clusters to derive a final sample.

If more than two stages are used, the process of

choosing population units within clusterscontinues until the final sample is achieved. An example of multi-stage sampling is where, firstly,

electoral sub-divisions (clusters) are sampled from a city

or state. Secondly, blocks of houses are selected fromwithin the electoral sub-divisions and, thirdly, individual

houses are selected from within the selected blocks of

houses.

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M l i S li


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Multistage Sampling

The advantages of multi-stage sampling are

convenience, economy and efficiency. Multi-stage

sampling does not require a complete list of members

in the target population, which greatly reduces samplepreparation cost. The list of members is required only

for those clusters used in the final stage. The main

disadvantage of multi-stage sampling is the same as

for cluster sampling: lower accuracy due to highersampling error

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M l i S li


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Multistage Sampling A multistage random sampleis constructed by taking a

series of simple random samples in stages. This type ofsampling is often more practical than simple random

sampling for studies requiring "on location" analysis, such

as door-to-door surveys.

In a multistage random sample, a large area, such as a

country, is first divided into smaller regions (such as states),

and a random sample of these regions is collected.

In the second stage, a random sample of smaller areas (suchas counties) is taken from within each of the regions chosen

in the first stage.

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M l i S li


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Multistage Sampling

Then, in the third stage, a random sample of

even smaller areas (such as neighborhoods) istaken from within each of the areas chosen in

the second stage.

If these areas are sufficiently small for thepurposes of the study, then the researcher

might stop at the third stage.

If not, he or she may continue to sample fromthe areas chosen in the third stage, etc., until

appropriately small areas have been chosen.

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Multistage Sampling


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Multistage SamplingMulti-stage cluster sampling

As the name implies, this involves drawing several different

samples. It does so in such a way that cost of finalinterviewing is minimised.

Basic procedure: First draw sample of areas. Initially large

areas selected then progressively smaller areas within larger

area are sampled. Eventually end up with sample ofhouseholds and use method of selecting individuals from

these selected households.

You are employed by a market research organisation and

have been asked to undertake a study across the whole of the

British Isles to determine attitudes towards the National

Lottery.

What are the possible sampling frames for this study? 67

Multistage Sampling


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g p g

You have been asked to interview 4,000 people in

total. Consider what would happen if you undertooka simple random survey....... Of these 4,000 you

might, for example, have to interview 2 people in X

place, 3 in the Y Place, a dozen scattered across the

Borders, and so on. As this is uneconomic, both in

terms of time and money, it would be considered

more efficient to undertake a Multi-stage Cluster

Sample. Think of some ways in which you coulddivide the country up into progressively smaller

sections to undertake this type of sample.

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E l


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Example

School District Students Cumulative Students

Albermarle 2,318 2,316

Roanoke 11,538 13,586

Amherst 1,217 15,073

Nottoway 548 15,657

Fairfax 46,154 61,811

Amelia 3,121 324,071

Winchester 929 325,000

To select districts based on PPS:

1. Decide the number of PSUs

=30.

2. Compute sampling

interval=total student

count/30=325000/30=10,833

3. Select a random start rbetween 1 and 10,833

4. The PSU containing the

r+10,833u students is

selected, u=129

5. Go on to second levelsampling

M lti t S li


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Multistage Sampling

The advantages to multi-stage cluster sampling

plans are really the same as for one-stage cluster

sampling.

Including stratification in some levels maymitigate (but never eliminate) the loss of

precision

Cost is reduced for multistage sampling

P t pli Ch i


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Postsampling Choices

How can non-response be evaluated?

Is weighting necessary?

What are the standard errors and related

confidence intervals for the study estimates?

S mpl Siz


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Sample Size

Factors Tolerable error

Cost

Variation in subject of interest

Design effect

Subpopulation concern

Ineligibles and non-responses

For large population, sample size is notdependenton population size. You dont have to have 5% of

the population!

S bpop lation Concern


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Subpopulation Concern

If 5% significance level is required for both

population and subpopulation, an efficient sample

size calculated for the population may not be

sufficient for subpopulation if the standarddeviation of the subpopulation is larger than that

of the population.

Solution?

Review


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Review

Sampling

What is/why sampling? The validity issue

Sampling methods

Nonprobability methods

Probability methods

Practical design issues

Population Sampling method

Sampling size

Chapter-7 Sampling Design

Documents

Transcript of Chapter-7 Sampling Design