Topic VI: Sampling Theory

1
Topic VISampling Theory Sampling Methods

• Concepts Definitions
• Sampling With Without replacement
• Probability Non-Probability Sampling
• Types of Sampling Methods
• Determining Sample Size

2
Concepts Definitions

• Population
• Sampling Frame
• Unit of Analysis
• Sampling Units
• Principal Information

• Auxiliary Information
• Sampling Error
• Non-Sampling Error
• Sampling Fraction
• Bias

3
Population

• This a collection of all the units of a specified
  type defined over a given space or time
• It is defined by
• Content this refers to who or what exactly are
  the subjects of interest. Eg. All persons above
  aged 18 and over
• Units this refers to how the subjects are
  grouped. Eg. Within households
• Extent this refers to the spatial feature of
  the population. Eg. The subjects can only be
  living in Jamaica.
• Time this refers to what period of time that
  your subjects must possess the particulars named
  above. Eg. June October 1998

4
Sampling Frame

• This is a list of the all the units in the target
  population from which the sample is to be chosen
• A subset of subjects for a survey should only be
  taken from a sampling frame

5
Sampling Frames II

• When conducting a national survey there are two
  types of sampling frames that can be used (note
  there are others)
• Electoral registers this lists electors in each
  polling district by street streets in
  alphabetical order
• Postcode Sectors as their primary sampling unit

6
Sampling Frames III

• Postal Sectors are determined from a Postcode
  Address File
• Postal Areas (CT) 121
• Postal Districts (CT2) 2700
• Postal Sectors (CT2 7) 8900
• Postcodes (CT2 7PE) 1.6mn
• Delivery Points 26mn

7
Sampling Frames IV
8
Sampling Frames - Notes
Post Code Sectors
Electoral Register

• not all adults are electors (exclude felons, non
  EU or Commonwealth, ...)
• limited corrections
• includes institutions (colleges)
• coverage high but incomplete
• compiled annually
• compiled locally with variety of software
• in force until 18 months from data collection

• does not give names
• no indication of household size
• multi occupancy indicator available
• will contain some small business addresses
• updated quarterly
• national system
• computerised format, so lower selection cost

9
Unit of Analysis

• Sometimes referred to as Sampling Units
• This is the items/units being investigated
• Eg Individuals, households, hospitals

10
Sampling Units

• This refers to the items/units selected for
  inclusion in the sample
• Eg If John Brown was selected to be included in
  the sample then he is a sampling unit

11
Principal Information

• This refers to information on the central
  variable of the study
• Also known as principal variable or principal
  data
• Eg. For a household budget survey, the principal
  variable would be considered to be expenditure on
  food

12
Auxiliary Information

• This refers to any other information other than
  the principal data
• In the example on the previous slide

13
Sampling Error I

• This is a measure of the departure of all the
  possible estimates of a probability sampling
  procedure from the population quantity being
  measured
• In other words, it refers to the difference
  between the estimate derived from a sample survey
  and the 'true' value that would result if the
  whole population was studied (using the same
  conditions)
• It is also known sometimes as the bias

14
Sampling Error II

• The standard error, variance coefficient of
  variation (C.V.) are all measures of the sampling
  error
• Note A census does not have a sampling error
• The sampling error is therefore equal to

Population Parameter
Sample Statistic
15
Sampling Error III - Characteristics

• The sampling error
• generally decreases as the sample size increases
  (but not proportionally)
• depends on the size of the population under study
  
• depends on the variability of the characteristic
  of interest in the population
• can be accounted for and reduced by an
  appropriate sample plan
• can be measured and controlled in probability
  sample surveys

16
Sampling Error - IV

• Although the Margin of Error and Sampling Error
  are sometimes used interchangeably they are TWO
  different concepts
• Sampling error
• .

Margin of Error
17
Non-Sampling Errors

• These are errors resulting from some imperfection
  in the research design that causes response error
  from a mistake in the execution of the research
• Examples
• sample bias
• errors in recording responses
• nonresponses
• Also know as Systematic Errors

18
Sampling Fraction

• This is the size of the sample as a proportion of
  the population from which it was drawn
• It is equal to n/N
• If n/N gt 1 then there is sampling with
  replacement

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Bias I

• This means that results based on the sample do
  not (even on average) reflect the same answers as
  would come from a census
• They are caused by both sampling non-sampling
  factors

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Bias II
21
Bias III
This is an adaptation of the diagram in Kish pg
519
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Sampling with without Replacement

• Sampling with Replacement
• occurs when a unit sampled is placed back into
  the population
• A particular unit is can be included more than
  once in the sample
• It is possible that n gt N
• Sampling without Replacement
• Occurs when a unit sampled is not placed back
  into the population
• A particular unit can only occur ONCE in the
  sample
• In some cases sampling without replacement from
  an infinite population can be equal to sampling
  from a small population with replacement

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Probability Non-Probability Sampling I

• Probability Samples
• Aka Random Samples (though sample units are not
  chosen haphazardly)
• The probabilities for selecting different samples
  are specified
• For each unit of the population the probability
  of it appearing any sample is known
• It provides an estimate for the unknown
  population quantity
• It also allows for the assessment of the standard
  error which can be used to obtain confidence
  intervals

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• There are 3 main steps involved in choosing a
  probability sample
• Decide on the population of interest
• Establish a sampling frame
• Select units from the frame using a probabilistic
  algorithm

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Probability Non-Probability Sampling II

• Non-Probability Sampling
• This involves the selection of a units by
  arbitrary methods
• The probability of selection for each unit is
  unknown
• It is dangerous to make inferences about the
  target population
• It is often used to test aspects of a survey such
  as questionnaire design, processing systems etc.
  rather than make inferences about the target
  population

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Probability Non-Probability Sampling III

• Choosing between the two types depends on
• the objectives and scope of the survey
• the method of data collection suitable to those
  objective
• the precision required of the results and whether
  that precision needs to be able to be measured
• the availability of a sampling frame
• the resources required to maintain the frame
• the availability of extra information about the
  units in the population

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Sampling Methods
Probability Sampling
Non Probability Sampling

• Simple Random
• Stratified
• Systematic
• Cluster
• Multi-stage

• Purposive
• Quota
• Snowball
• Convenience

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Simple Random Sampling I

• Each member of the population has the same
  probability of being a part of the sample
  independent of whether another subject is in the
  sample
• AKA equal probability selection method
• It is the simplest sampling method

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Simple Random Sampling II

• n units are selected from N possible units in the
  population
• Every combination of n units is equally likely to
  be the sample selected
• The selection process can either be
• sampling without replacement, which is more
  common
• unrestricted sampling / sampling with
  replacement

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Simple Random Sampling III

• It is important because it possesses simple
  mathematical properties which are useful for
  statistical theory and the computations are
  relatively easy
• All other probability sampling methods are
  restrictions of SRS (usually where some
  combinations of population elements are
  suppressed)
• For the mathematical properties to hold we must
  assume an infinite population

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Simple Random Sampling IV

• There are three main ways of choosing a simple
  random sample
• Table of Random Numbers
• Lottery Method
• Computer Generated Numbers

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Stratified Random Sampling I

• A population is subdivided or partitioned
• Each subdivision is called a stratum
• All the subdivisions are the strata
• The idea is to ensure that the observations of
  the units of a stratum are closer to each other
  than to units of another stratum

33
Stratified Random Sampling II

• SRS does not produce good results in cases where
  the population to be sampled contains easily
  recognisable subpopulations or strata
• Strata do not overlap and any member of the
  population can belong to only ONE stratum

34
Stratified Random Sampling III
35
Stratified Random Sampling IV

• Examples
• Household income or expenditure surveys
• urban rural
• Business surveys
• employee size
• Production
• sales
• industrial classification
• Agricultural surveys
• Stratification depends on purpose of survey

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Stratified Random Sampling V

• Why Stratify?
• In situations where there is foreknowledge of
  some non-homogeneity in the population,
  proportional stratified sampling ensures a
  representative sampling across the non-homogenous
  population
• Used for administrative or scientific reasons,
  where each stratum needs to be reported
  separately
• Eg. crop yield in each agro climatic stratum
  where the results for each stratum will have its
  own meaning

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Stratified Random Sampling VI

• Why Stratify? Contd
• Stratification has the advantage of
  administrative convenience
• May be due to practical constraints of access to
  the population or cost
• it may be easier to have each province/parish
  conduct the survey
• Survey problems may be different in different
  strata
• Eg. A financial survey of businesses maybe done
  differently for small companies who are not
  required to pay a certain tax in comparison with
  larger companies who are

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Stratified Random Sampling VII

• Why Stratify? Contd
• Each stratum is more homogenous than the
  population when taken as a whole
• stratified sample would provide relatively
  precise estimates within each stratum
• yield more precise population estimates than if
  simple random sampling was used

39
Stratification The Procedure I

• The population is divided into H strata
• Each strata doesnt overlap and is exhaustive
• A SRS of size nh is taken from each stratum with
  population Nh

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Stratification The Procedure II

• There are three types of Allocation
• Equal Allocation
• used when the main interest is to compare strata
  parameters
• and/or the population is thought to have a
  homogenous variance within each stratum  (ie the
  variances are similar)
• The same number of elements are taken from each
  strata

41
Stratification The Procedure III

• Proportional Allocation
• Used in cases where the sample is supposed to
  reflect the population with respect to the
  stratification variable
• The number of units sampled within a given
  stratum is proportional to the size of the
  stratum
• It is best to use proportional allocation in
  situations where the variances of each stratum
  are approximately equal

42
Stratification The Procedure IV

• Optimal Allocation
• Used in cases where the variances for the strata
  differ greatly
• Also used when
• primary interests are the estimates for the
  entire population
• it is assumed that there is unequal variance
  between each stratum
• It produces estimates for the population mean or
  total with the lowest variance for a fixed total
  sample size, n

43
Stratification - Advantages

• Estimates for each stratum can be evaluated
  separately
• Differences among the strata can be evaluated
• Total, means and proportion can be estimated with
  high precision using appropriate weights
• Savings in time and cost (convenience)

44
Stratification - Disadvantages

• The proportion of the total population that
  belongs to each stratum needs to be known
• It may be complex and time consuming

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Systematic Random Sampling I

• Most widely known method of selection
• Simple to apply
• Consists of taking every kth sampling unit after
  a random start
• AKA Pseudo-Random selection
• Often used jointly with stratification with
  cluster sampling

46
Systematic Random Sampling II

• The first element is based on random selection
  but subsequent elements are not
• Procedure
• The population is divided into k groups of size n
  N/k in each
• One unit is chosen randomly from the first k
  units
• Every kth unit following is included in the
  sample
• It is possible that N/k is not an integer

47
Systematic Random Sampling III

• Examples
• Agricultural Survey Selection of every 10th
  farm from 500 farms in an area (would produce 50
  farms)
• Industrial Quality Control every 30 minutes or
  every 10th batch
• Marketing or Political Surveys every 10th
  person passing a particular location
• Surveys to supplement censuses
• Large multistage surveys samples are selected
  systematically at the different stages

48
Systematic Random Sampling - Advantages

• Operationally convenient
• Flexible
• Convenient to use when the sampling frame is not
  available
• It is spread out more evenly over the population
  so that it is more likely to produce a more
  representative sample

49
Systematic Random Sampling - Disadvantages

• More precise than SRS when units within the
  sample are heterogeneous and imprecise when the
  units are homogeneous
• Generally, it is not possible to gain suitable
  estimates of the variance of the estimator from
  one sample. The approximate variance can be
  calculated

50
Cluster Sampling I

• Cluster sampling divides the population into
  groups, or clusters
• A number of clusters are selected randomly to
  represent the population, and then all units
  within selected clusters are included in the
  sample
• 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

51
Cluster Sampling II

• The unit of selection contains more than one
  population element
• Examples of possible clusters

52
Cluster Sampling III

• Advantages
• The cost per element is lower due to the lower
  cost of listing or of location. Cost is also
  lower because sampling is done within clusters
• All elements are in one cluster, then there is
  the convenience of reaching each members
• Disadvantages
• Combining the variance from two separately
  homogenous clusters may cause the variance of the
  entire sample to be higher when compared with SRS
• less accurate results are often obtained due to
  higher sampling error than for simple random
  sampling with the same sample size

53
Multi-Stage Sampling I

• Multi-stage sampling is like cluster sampling
• It involves selecting a sample within each chosen
  cluster, rather than including all units in the
  cluster
• It is sometimes referred to as sub-sampling

54
Multi-Stage Sampling II

• Multi-stage sampling involves selecting a sample
  in at least two stages
• 1st stage, large groups or clusters are selected
  
• These clusters are designed to contain more
  population units than are required for the final
  sample
• 2nd stage, population units are chosen from
  selected clusters to derive a final sample
• This is called TWO-STAGE SAMPLING
• If more than two stages are used, the process of
  choosing population units within clusters
  continues until the final sample is achieved
  this would be considered MULTI-STAGE SAMPLING

55
Multi-Stage Sampling -Advantages

• Useful when there is no sampling frame
• If sub-units within a selected unit give similar
  results, it is uneconomical to measure all the
  second stage units
• Lists are prepared for a small portion of the
  total populations of second stage units so it is
  considered economical
• No need for sampling procedures at each stage to
  be the same

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Multi-Stage Sampling -Disadvantages

• The sampling of compact clusters may present
  practical difficulties

57
Summary
58
Purposive/Judgemental Sampling

• The sample is hand-picked
• The researcher exercises deliberate subjective
  choice in drawing what he/she regards as a
  representative sample
• Often used for case study research
• It may also be used to eliminate anticipated
  sources of distortion

59
Quota Sampling

• Participants are selected from certain subgroups
  in the population
• In most cases, participants are chosen just
  before the interview begins although the aim is
  to be as random as possible
• Usually used in market surveys opinion polls
• A proper statistical design is used to determine
  what numbers are needed for each subgroup

60
Snowball

• Members of the sample name other persons which
  can (and usually is) included in the sample
• Used mainly for populations which do not have a
  proper or adequate sampling frame
• Researcher identifies a few key participants who
  then identify other relevant participants

61
Convenience

• Participants are selected because they are
  readily available
• Considered to be the most unreliable method of
  sampling

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Sampling Rare Populations I

• Problems arise if there is no relevant, accurate
  sampling frame for a rare group
• Special methods need to be used to estimate
• Prevalence / incidence of occurrence
• Characteristics of the population
• Population Means, Totals etc
• Examples
• Medical Conditions
• Social Conditions

63
Sampling Rare Populations II

• There are 6 methods used to sample rare
  populations
• Screening
• Disproportionate Sampling
• Multiplicity Sampling
• Snowballing
• Multiple Frames
• Sequential Sampling

64
Screening I

• This involves double or two phase sampling
• Procedure
• Survey general population
• Identify potential members of the group
• Detail survey of the potential members

65
Screening II

• Problems
• High cost
• Should apparent non-members also be sampled
• High costs can be cut by
• Telephone interviews (which can be
  unrepresentative)
• Postal questionnaires (which can have low
  response)
• Sharing costs with other surveys
• Sampling more intensively in cluster with
  relatively high concentration of the rare group
  (eg sample cluster only if 1st selected element
  is a member of the rare population)

66
Disproportionate Sampling

• Sampling more intensively in cluster with
  relatively high concentration of the rare group
• eg sample cluster only if 1st selected element
  is a member of the rare population
• Gains are only high when stratum to be
  oversampled does have a high prevalence relative
  to other strata
• Optimal allocation theory can be used to
  determine sampling fractions in each stratum

67
Multiplicity

• Sample all close neighbours and / or relatives of
  selected sample members
• May use proxy information to estimate prevalence

68
Snowballing

• Creation of a sampling frame through other
  relevant contacts as suggested by group members

69
Multiple Frame

• This involves using many sampling frames
• Overlaps are dealt with by
• merging the files
• cleaning the data file
• use of weights related to the probability of
  selection

70
Sequential Sampling

• Continue sampling until a large enough sample of
  the rare population is achieved

71
Choice of Sample Size I

• An increase in sample size leads to an increase
  in the precision of the sample mean as an
  estimator of the population mean
• An increase in sample size leads typically to an
  increase in sampling costs

72
Choice of Sample Size II

• The trade off between cost and precision is key
• Sample too large waste of resources
• Sample too small an estimator with inadequate
  precision
• Choose either precision required OR maximum cost
  which can be expended and then choose sample size

73
Choice of Sample Size III

• The main method presupposes that the population
  variance is known
• In practice, most times the population variance
  is unknown
• Usually the sample variance is used to replace
  the population variance but there is no sample
• Solution Can be chosen by using one of the
  following methods

74
Choice of Sample Size IV

• From Pilot Studies
• If the pilot study uses SRS then its results may
  give some indication of the value of the
  population variance
• NB A pilot study is limited to a certain part of
  the population so the estimate of the variance
  will be biased
• From Previous Surveys
• Usually the study of a population with similar
  characteristics in a similar population has been
  previously conducted
• The measure of variability from earlier surveys
  can be used to estimate the variance of the
  population currently under study
• NB Caution must be taken in using the
  information

75
Choice of Sample Size V

• From a Preliminary Sample
• Most reliable approach
• May not be feasible because of administration or
  cost
• A preliminary SRS is taken and used to estimate
  the population variance
• Procedure
• Preliminary sample of size n1 is chosen and used
  to estimate the population variance by the sample
  variance s12
• n1 is inadequate in producing the necessary
  precision so another sample of (n-n1) is chosen
  by using s12 as the preliminary estimate of the
  population variance

76
Choice of Sample Size VI

• From Practical Considerations of the Structure of
  the Population
• It might be able to determine what kind of
  distribution an event may have
• The variance is estimated using the formulas for
  the specified distribution
• eg if it is assumed that a specific event might
  follow a Possion distribution then we can assume
  that the mean and the variance are equal

77
Choice of Sample Size VII- Large Populations
Table 1
Source Parker Rea, Designing and Conducting
Research
78
Choice of Sample Size VIII- Small Populations
Table 2
Source Parker Rea, Designing and Conducting
Research
79
Choice of Sample Size IX Interval Variables

• Large Populations
• Small Populations
• N population
• n sample
• C confidence interval (Z times std deviation)
• Z Z score for level of confidence
• s standard error for the distribution of sample
  means

80
Choice of Sample Size X

• If interested in BOTH proportions and intervals
  then choose the higher sample size
• Tables 1 and 2 should be adequate to cover sample
  sizes for both