Stratified sampling technique.

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welcome
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MAJOR CREDIT SEMINAR
ON

• STRATIFIED RANDOM SAMPLING
• By -Shashank kshandakar
  M.V.Sc(1st year)
  Division
  of lesit

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Content....

• Definitions and concepts in sampling
• Principles of sampling
• Advantages of sampling
• Types of sampling
• Stratified random sampling its properties

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Important definitions

• Population population is an aggregates of
  object under study.
• Sample- A finite sub-set of statistical
  individual in a population is called a sample.
• Sampling Unit Sets of units considered for
  selection in some stage of sampling.
• Sampling scheme-Method of selecting sampling
  units from population

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Important definitions.

• Census-The process of complete enumeration of
  all the elements or units in the population
• Sample size-The number of elements or units in a
  sample is known as sample size.
• Accuracy-The amount of deviation of the
  estimate from the true value.
• Precision-The deviation of the estimate from the
  average value. It is reciprocal of the variance

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Sampling Frame

• Sampling Frame- A complete list, map or other
  acceptable materials which serve as a guide to
  the population to be covered is known as frame.
• A sampling frame which has the property that we
  can identify every single element and include any
  in our sample
• The sampling frame must be representative of the
  population
• A sampling frame must be up-to-date.

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Important definitions

• Sampling fraction- Ratio of sample size(n) to
  the population size(n) i.e. (n/N)
• Finite population correction-1- (Sampling
  fraction)
• Sample space-A space consisting of all possible
  sample is called sample space
• Relation between cost (per unit of sample
  selection) precision

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Important definitions

• Sampling distribution-The aggregates of the
  various value of the statistic under
  consideration so obtained(one from each sample)
  may be grouped into a frequency distribution
  which is known as sampling distribution of the
  statistic.
• Standard error-The standard deviation of the
  sampling distribution of a statistic is known as
  standard error.

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Important definitions

• Parameter-A descriptive measure computed from
  the data of a population is called as parameter.
• Statistic-A descriptive measure computed from a
  sample data is called as statistic.
• Estimator-An estimator is a rule, function or
  formula of variates for estimating the population
  parameter.
• Estimates -A particular value of an estimator
  from a fixed set of value of a random sample is
  known as estimate.

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It is very easy and convenient to draw the
sample from homogenous population
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The population having significant variations
(Heterogeneous), observation of multiple
individual needed to find all possible
characteristics that may exist
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When sampling is not necessary?

• When population is very small.
• When we have extensive resources.
• When we dont expect a very high response.

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Sampling

• Sampling is the process of selecting units or
  elements from the study population in such a way
  that the units or elements selected represent the
  whole population

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Principles of Sampling Theory

• Principle of Statistical Regularities.
• Principle of Inertia of large number.
• Principle of Validity.
• Principle of Optimization.

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Principal Steps in a Sample Survey

• Statement of objectives.
• Definition of population to be studied.
• Determination of sampling frame and sampling
  units.
• Selection of proper sampling design.
• Organization of field work.
• Summary and analysis of data.

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The Sampling Process
Plan procedure for selecting sampling units
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Determine if a probability or non-probability
sampling method will be chosen
Determine sample size
3
5
Select actual sampling units
Select a Sampling Frame
2
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Define the Target population
Conduct fieldwork
1
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Advantage of sampling

• 1. Reduced cost of survey.
• 2. Greater speed of getting results.
• 3. Greater accuracy of results.
• 4. Greater scope.(When its impossible to study
  the whole population)
• 5. Adaptability.

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Type of sampling method

• Probability sampling
• Non-probability sampling.

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Probability sampling

• The term probability sampling is used when the
  selection of the sample is purely based
  on chance. 
• Every unit of the population has known nonzero
  probability of being selected for the sample. The
  probability of selection may be equal or unequal
  but it should be non-zero and should be
  known. The probability sampling is also called
  the random sampling

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Types Probability sampling

• Probability sampling includes
• Simple Random Sampling,
• Systematic Sampling,
• Stratified Random Sampling,
• Cluster Sampling
• Multistage Sampling.
• Multiphase sampling

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Simple random sampling

• Applicable when population is small, homogeneous
  readily available but there is no guarantee
  that all segment of population will be
  represented in SRS
• All unit within the frame have an equal
  probability.
• It provides for greatest number of possible
  samples.
• Simplest common method of sampling
• Every sample are drawn unit by unit with equal
  probability of selection on the basis of sample
  drawn SRS may be divide into two group

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Simple random sampling with replacement

• Simple random sampling with replacement
• If a unit is selected noted then
  returned back to the population before the next
  selection is made this procedure is repeated n
  times is known as Simple random sampling with
  replacement
• Number of sample required Z2a pq
  /(S.E)2
• Var(yn)
•  

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Simple random sampling without replacement

• If a unit is selected noted not returned back
  to the population (selected unit is not available
  for further selection) this procedure is known as
  Simple random sampling with replacement
• Number of sample requiredn0/1( n0-1)/N
•  
• Var(yn)

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Advantage of SRS

• Minimal knowledge of population needed
• Economical required less time.
• Easy to analyze data
• Disadvantage of SRS
• High cost low frequency of use
• Requires sampling frame
• Does not use researchers expertise
• Larger risk of random error than stratified

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STEPS IN COMPUTING THE SIZE OF A SAMPLE

• Determine the size of the target population.
• Decide on the margin of error. As much as
  possible the margin of error should not be higher
  than 5.
• Use the formula
• n N
• 1 Ne2 (Pagoso , et
  al.)
• n sample size
• N the size of the population
• e the margin of error
• Compute the sample proportion by dividing the
  result in number 3 by the population.

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STEPS IN COMPUTING THE SIZE OF A SAMPLE

• Population is 5,346
• Margin of error is 3
• Using the formula
• n ___5,346_
• 1 5346(.03)2
• n 920
• Sample proportion () 920 / 5346
• 17

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Procedure of selection of a random sample

• Lottery method
• Use of random number table
• Tippets random number table
• Fisher Yates random number table
• Kendall smith random number table
• A million random digit number table.

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Test for randomness of selected random number

• Frequency test
• Serial test
• Gap test
• Poker test.

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Stratified random sampling

• Most commonly used sampling technique
• Stratum (Strata.pl) - Clear division into which
  some thing is separated
• The population (N) unit is divided into K group
  or sub-population called strata
• The sample is drawn by first dividing the
  population into homogeneous sub-populations
  (strata) and then drawing samples from each of
  the sub-population (stratum).
• It ensures proportionate representation of
  character under study when drawing a sample from
  a heterogeneous population and hence increases
  the quantity of information for a given cost.

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How to Stratify and How Many Strata?

• Strata should be homogeneous to increase
  efficiency
• Usually 5-10 Strata are taken. If too many strata
  then the sample size within strata is too low.

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Principle of stratification

• Strata should be non-overlapping should
  together comprise the whole population.
• Strata are homogenous themselves with respect to
  characters under study.
• Population are divided into strata on the basis
  of sex , age, purpose ( milch, drought, dual )
  breed, land holding capacity of livestock owners,
  or any auxiliary information.

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Auxiliary information

• Past data or some other information related
  to the character on which we divide the
  population into strata such that
• Within strata -Variable are homogenous.
  
  Between strata -Variable
  are heterogeneous.

Sampling technique Use of auxiliary information
Stratified sampling technique Construction of strata
PPS Sample selection, to get more efficient estimator of the population parameter
Ratio regression estimation For estimation purpose.
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Advantage of stratification

• Administrative convenience
• Improving sampling design
• Stratification makes it possible to use different
  sampling design in different strata
• Provide better cross section of population i.e.
  adequate representation from various group of the
  population
• Gain in precision in the estimation of population
  parameter.

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Main problem in stratification

• What should be the number of strata?
• How to allocate n sample from different strata
• How we distribute population into various strata
• How to determine the boundary of the strata.

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Allocation of sample size in different strata

• Allocation of the sample to different strata is
  done by considering of 3 factors
• Total no. of unit in the strata
• Variability within the stratum
• The cost of taking observation per sampling unit
  in the stratum
• (N.B-a good allocation is one where maximum
  precision is obtained with minimum recourses)

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Method of allocation

• Equal allocation
• Proportional allocation
• Neyman allocation
• Optimum allocation.

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Equal allocation

• Most convenience
• Equal no. of sample are drawn from each stratum.
• Not usually sensible unless all strata are equal
  size in terms of overall estimates precision.
  However, maybe good if you want to compare
  stratum means as your primary focus

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Proportional allocation

• Given by Bowley (1926)
• Very common in practice
• Sampling fraction of each stratum is same

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Proportional allocation
Allocate proportional to the size of the
strata-very widely used
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Neyman allocation

• Also known as minimum variance allocation
• 1st discovered by Tschuprow (1923), but
  rediscovered by J.Neyman (1934)
  (Assumption sampling cost per unit
  among different strata is same size of the
  sample is fixed)
• Allocation of sample among different strata is
  based on joint consideration of stratum size
  stratum variation

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Optimum allocation

• Optimum allocation is mean to choose sample so as
  to
• Minimise the variance(maximise the precision) of
  the estimate for fixed sample size(n).
• Minimise the variance(maximise the precision) of
  the estimate at fixed cost.
• Minimise the total cost for fixed desired
  precision

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Cluster Sample

• A Cluster Sample is obtained by first grouping
  the elements of the population into clusters and
  then simple random sampling or other type of
  sampling is used to select the clusters.
• This type of sampling is used when a sampling
  frame cannot be prepared for individual units in
  the population but can be prepared for some
  cluster of them or when substantial time or
  expense can be saved by collecting data from a
  modest number of clusters.

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.

• When drawing a cluster sample, the first task is
  to specify appropriate clusters. In doing so,
  consideration has to be made about the level of
  heterogeneity of elements within clusters.
  
• If clusters are generally heterogeneous, then
  few large clusters may be selected to constitute
  the sample but if they are homogeneous, then many
  small-sized clusters should be used.

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Difference between strata cluster

• Although strata and clusters are both
  non-overlapping subsets of the population, they
  differ in several ways.
• All strata are represented in the sample but
  only a subset of clusters are in the sample.
• With stratified sampling, the best survey results
  occur when elements within strata are internally
  homogeneous. However, with cluster sampling, the
  best results occur when elements within clusters
  are internally heterogeneous.

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Difference Between Cluster and Stratified Sampling
Population of L strata, stratum l contains nl
units
Population of C clusters
Take simple random sample in every stratum
Take srs of clusters, sample every unit in chosen
clusters
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Drawbacks

• Sampling frame of entire population has to be
  prepared separately for each stratum
• When examining multiple criteria, stratifying
  variables may be related to some, but not to
  others, further complicating the design, and
  potentially reducing the utility of the strata.
• In some cases (such as designs with a large
  number of strata, or those with a specified
  minimum sample size per group), stratified
  sampling can potentially require a larger sample
  than would other methods

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Post-stratification

• Stratification is sometimes introduced after the
  sampling phase in a process called
  "poststratification.
• This approach is typically implemented due to a
  lack of prior knowledge of an appropriate
  stratifying variable or when the experimenter
  lacks the necessary information to create a
  stratifying variable during the sampling phase.
  Although the method is susceptible to the
  pitfalls of post hoc approaches, it can provide
  several benefits in the right situation.
  Implementation usually follows a simple random
  sample. In addition to allowing for
  stratification on an ancillary variable,
  poststratification can be used to implement
  weighting, which can improve the precision of a
  sample's estimates.

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Oversampling

• Choice-based sampling is one of the stratified
  sampling strategies. In this, data are stratified
  on the target and a sample is taken from each
  strata so that the rare target class will be more
  represented in the sample. The model is then
  built on this biased sample. The effects of the
  input variables on the target are often estimated
  with more precision with the choice-based sample
  even when a smaller overall sample size is taken,
  compared to a random sample. The results usually
  must be adjusted to correct for the oversampling

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References

• Cochran, W. G., 1977. Sampling Techniques, Third
  Edition. New York John Wiley SonsCochran, W.
  G., 1977. Sampling Techniques, Third Edition. New
  York John Wiley Sons
• Des Raj and Chandhok, P. (1998). Sampling Survey
  Theory. Narosa Publishing House, New Delhi
• Horvitz, D. G., and D.J. Thompson, 1952. A
  generalization of sampling without replacement
  from a finite universe. The Journal of the
  American Statistical Association 47663-685.
• Murthy, M.N. (1977). Sampling Theory and Methods.
  Statistical Publishing Society, Calcutta.
• Sukhatme, P.V., Sukhatme, B.V., Sukhatme, S and
  Ashok, C. (1984). Sampling Theory of Surveys with
  Applications. Indian Society of Agricultural
  Statistics, New Delhi

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Thank you