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					       Sampling and Sample Size
                       Lazereto de Mahón, Menorca,
                                 September 2006
-EPIET Introductory course,
Thomas Grein, Denis Coulombier, Philippe Sudre, Mike Catchpole, Denise Antona
Brigitte Helynck, Philippe Malfait, Institut de veille sanitaire

Modified: Viviane Bremer, EPIET 2004, Suzanne Cotter 2005,
Richard Pebody 2006
    Objectives: sampling

• To understand:
   • Why we use sampling
   • Definitions in sampling
   • Sampling errors
   • Main methods of sampling
   • Sample size calculation
    Why do we use sampling?

Get information from large populations with:
  – Reduced costs
  – Reduced field time
  – Increased accuracy
  – Enhanced methods
  Definition of sampling

  Procedure by which some members
 of a given population are selected as
representatives of the entire population
  Definition of sampling terms

Sampling unit (element)
• Subject under observation on which
  information is collected
  – Example: children <5 years, hospital discharges,
    health events…
Sampling fraction
• Ratio between sample size and population
  – Example: 100 out of 2000 (5%)
  Definition of sampling terms

Sampling frame
• List of all the sampling units from which
  sample is drawn
   – Lists: e.g. children < 5 years of age, households,
     health care units…
Sampling scheme
• Method of selecting sampling units from
  sampling frame
   – Randomly, convenience sample…
                 Survey errors

• Systematic error (or bias)
Sample not typical of population
   – Inaccurate response (information bias)
   – Selection bias

• Sampling error (random error)
    Representativeness (validity)
A sample should accurately reflect distribution of
relevant variable in population

• Person e.g. age, sex
• Place e.g. urban vs. rural
• Time e.g. seasonality

     Representativeness essential to generalise

     Ensure representativeness before starting,

                Confirm once completed
 Sampling and representativeness


               Target Population

Target Population  Sampling Population  Sample
             Sampling error
• Random difference between sample and
  population from which sample drawn
• Size of error can be measured in probability
• Expressed as “standard error”
  – of mean, proportion…
• Standard error (or precision) depends upon:
  – Size of the sample
  – Distribution of character of interest in population
               Sampling error
When simple random sample of size „n‟ is selected
from population of size N, standard error (s) for
population mean or proportion is:

           σ                       p(1-p)
          n                         n

Used to calculate, 95% confidence intervals

   X Z
Estimated 95%
confidence  2
                 s or X  2  s x
           interval x
Quality of a sampling estimate

Precision    No precision   Precision but
& validity                   no validity

                Random        Systematic
                 error        error (bias)
     Survey errors: example

Measuring height:
• Measuring tape held differently by different
  → loss of precision
  – Large standard error
• Tape shrunk/wrong
   → systematic error
  – Bias (cannot be corrected afterwards)
        Types of sampling

• Non-probability samples

• Probability samples
     Non probability samples

• Convenience samples (ease of access)

• Snowball sampling (friend of friend….etc.)

• Purposive sampling (judgemental)
      • You chose who you think should be in the study

     Probability of being chosen is unknown
Cheaper- but unable to generalise, potential for bias
         Probability samples

• Random sampling
   – Each subject has a known probability of being
• Allows application of statistical sampling
  theory to results to:
   – Generalise
   – Test hypotheses
Methods used in probability samples

     •   Simple random sampling
     •   Systematic sampling
     •   Stratified sampling
     •   Multi-stage sampling
     •   Cluster sampling
    Simple random sampling

• Principle
     – Equal chance/probability of drawing each unit

• Procedure
     – Take sampling population
     – Need listing of all sampling units (“sampling frame”)
     – Number all units
     – Randomly draw units
     Simple random sampling

• Advantages
     – Simple
     – Sampling error easily measured

• Disadvantages
     – Need complete list of units
     – Does not always achieve best representativeness
     – Units may be scattered and poorly accessible
       Simple random sampling
    Example: evaluate the prevalence of tooth
    decay among 1200 children attending a school

•   List of children attending the school
•   Children numerated from 1 to 1200
•   Sample size = 100 children
•   Random sampling of 100 numbers between 1
    and 1200

            How to randomly select?
EPITABLE: random number listing
EPITABLE: random number listing

Also possible in Excel
Simple random sampling
         Systematic sampling

• Principle
  – Select sample at regular intervals based on sampling
• Advantages
     – Simple
     – Sampling error easily measured
• Disadvantages
     – Need complete list of units
     – Periodicity
             Systematic sampling

• N = 1200,     and n = 60
      sampling fraction = 1200/60 = 20

• List persons from 1 to 1200

• Randomly select a number between 1 and 20
  (ex : 8)
       1st person selected = the 8th on the list
       2nd person = 8 + 20 = the 28th etc .....
Systematic sampling
         Stratified sampling

• Principle :
  – Divide sampling frame into homogeneous
    subgroups (strata) e.g. age-group, occupation;

  – Draw random sample in each strata.
          Stratified sampling
• Advantages
  – Can acquire information about whole population and
    individual strata
  – Precision increased if variability within strata is less
    (homogenous) than between strata
• Disadvantages
  – Can be difficult to identify strata
  – Loss of precision if small numbers in individual strata
     • resolve by sampling proportionate to stratum population
   Multiple stage sampling
• consecutive sampling
• example :
  sampling unit = household
   – 1st stage: draw neighborhoods
   – 2nd stage: draw buildings
   – 3rd stage: draw households
              Cluster sampling
• Principle
  – Sample units not identified independently but in a
    group (or “cluster”)

  – Provides logistical advantage.
              Cluster sampling
• Principle
  – Whole population divided into groups e.g.

  – Random sample taken of these groups (“clusters”)

  – Within selected clusters, all units e.g. households
    included (or random sample of these units)
            Example: Cluster sampling
     Section 1         Section 2

                                        Section 3

                                    Section 5

Section 4
            Cluster sampling
• Advantages
   – Simple as complete list of sampling units within
     population not required

   – Less travel/resources required

• Disadvantages
   – Potential problem is that cluster members are more
     likely to be alike, than those in another cluster

   – This “dependence” needs to be taken into account in
     the sample size….and the analysis (“design effect”)
Selecting a sampling method

• Population to be studied
   – Size/geographical distribution
   – Heterogeneity with respect to variable
• Availability of list of sampling units
• Level of precision required
• Resources available
      Sample size estimation

• Estimate number needed to
   • reliably measure factor of interest
   • detect significant association
• Trade-off between study size and resources….
• Sample size determined by various factors:
   • significance level (“alpha”)
   • power (“1-beta”)
   • expected prevalence of factor of interest
               Type 1 error
• The probability of finding a difference with our
  sample compared to population, and there
  really isn‟t one….

• Known as the α (or “type 1 error”)

• Usually set at 5% (or 0.05)
                 Type 2 error
• The probability of not finding a difference that
  actually exists between our sample compared
  to the population…

• Known as the β (or “type 2 error”)

• Power is (1- β) and is usually 80%
              A question?
Are the English more intelligent than the Dutch?

• H0 Null hypothesis: The English and Dutch
  have the same mean IQ

• Ha Alternative hypothesis: The mean IQ of
  the English is greater than the Dutch
        Type 1 and 2 errors
Decision        H0 true    H0 false
Reject H0       Type I error   Correct decision

Accept H0       Correct        Type II error
• The easiest ways to increase power are to:
  – increase sample size

  – increase desired difference (or effect size)

  – decrease significance level desired e.g. 10%
Steps in estimating sample size
    for descriptive survey
• Identify major study variable
• Determine type of estimate (%, mean, ratio,...)
• Indicate expected frequency of factor of interest
• Decide on desired precision of the estimate
• Decide on acceptable risk that estimate will fall outside
  its real population value
• Adjust for estimated design effect
• Adjust for expected response rate
                    Sample size for
                   descriptive survey
Simple random / systematic sampling
                 z² * p * q          1.96²*0.15*0.85
           n = --------------        ----------------------     = 544
                      d²                      0.03²

Cluster sampling
                   z² * p * q         2*1.96²*0.15*0.85
           n = g* --------------     ------------------------   = 1088
                       d²                     0.03²

z: alpha risk expressed in z-score
p: expected prevalence
q: 1 - p
d: absolute precision
g: design effect
       Case-control sample size:
          issues to consider
• Number of cases
• Number of controls per case
• Odds ratio worth detecting
• Proportion of exposed persons in source
• Desired level of significance (α)
• Power of the study (1-β)
    – to detect at a statistically significant level a
      particular odds ratio
STATCALC Sample size
     STATCALC Sample size

Risk of alpha error               5%
Power                            80%
Proportion of controls exposed   20%
OR to detect                     >2
STATCALC Sample size
                  Statistical Power of a
                  Case-Control Study
        for different control-to-case ratios and odds ratios
                           (with 50 cases)


        0.6                                               OR=2
        0.4                                               OR=4
              1    2   3   4   5   6   7   8   9   10
                       Control-Case Ratio

• Probability samples are the best

• Ensure
  – Representativeness
  – Precision

• …..within available constraints

• If in doubt…

       Call a statistician !!!!