Sampling and Sample Size Calculation

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Sampling and Sample Size Calculation
Lazereto de Mahón, Menorca, Spain September 2006
Sources -EPIET Introductory course, Thomas
Grein, Denis Coulombier, Philippe Sudre, Mike
Catchpole, Denise Antona -IDEA Brigitte
Helynck, Philippe Malfait, Institut de veille
sanitaire Modified Viviane Bremer, EPIET 2004,
Suzanne Cotter 2005, Richard Pebody 2006
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Objectives sampling

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

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Why do we use sampling?

• Get information from large populations with
• Reduced costs
• Reduced field time
• Increased accuracy
• Enhanced methods

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Definition of sampling

• Procedure by which some members
• of a given population are selected as
  representatives of the entire population

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Definition of sampling terms

• Sampling unit (element)
• Subject under observation on which information is
  collected
• Example children lt5 years, hospital discharges,
  health events
• Sampling fraction
• Ratio between sample size and population size
• Example 100 out of 2000 (5)

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Definition of sampling terms

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

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Survey errors

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

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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

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Sampling and representativeness
Sampling Population
Sample
Target Population
Target Population ? Sampling Population ? Sample
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Sampling error

• Random difference between sample and population
  from which sample drawn
• Size of error can be measured in probability
  samples
• Expressed as standard error
• of mean, proportion
• Standard error (or precision) depends upon
• Size of the sample
• Distribution of character of interest in
  population

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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 s
p(1-p)
? n
n Used to calculate, 95 confidence intervals
Estimated 95 confidence interval
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Quality of a sampling estimate
Precision validity
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Survey errors example

• Measuring height
• Measuring tape held differently by different
  investigators
• ? loss of precision
• Large standard error
• Tape shrunk/wrong
• ? systematic error
• Bias (cannot be corrected afterwards)

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Types of sampling

• Non-probability samples
• Probability samples

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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
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Probability samples

• Random sampling
• Each subject has a known probability of being
  selected
• Allows application of statistical sampling theory
  to results to
• Generalise
• Test hypotheses

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Methods used in probability samples

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

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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

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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

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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?
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EPITABLE random number listing
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EPITABLE random number listing
Also possible in Excel
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Simple random sampling
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Systematic sampling

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

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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 .....

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Systematic sampling
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Stratified sampling

• Principle
• Divide sampling frame into homogeneous subgroups
  (strata) e.g. age-group, occupation
• Draw random sample in each strata.

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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

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Multiple stage sampling

• Principle
• consecutive sampling
• example sampling unit household
• 1st stage draw neighborhoods
• 2nd stage draw buildings
• 3rd stage draw households

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Cluster sampling

• Principle
• Sample units not identified independently but in
  a group (or cluster)
• Provides logistical advantage.

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Cluster sampling

• Principle
• Whole population divided into groups e.g.
  neighbourhoods
• Random sample taken of these groups (clusters)
• Within selected clusters, all units e.g.
  households included (or random sample of these
  units)

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Example Cluster sampling
Section 2
Section 1
Section 3
Section 5
Section 4
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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 (homogenous).
• This dependence needs to be taken into account
  in the sample size.and the analysis (design
  effect)

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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

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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

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Type 1 error

• The probability of finding a difference with our
  sample compared to population, and there really
  isnt one.
• Known as the a (or type 1 error)
• Usually set at 5 (or 0.05)

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Type 2 error

• The probability of not finding a difference that
  actually exists between our sample compared to
  the population
• Known as the ß (or type 2 error)
• Power is (1- ß) and is usually 80

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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

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Type 1 and 2 errors

• Truth
• Decision H0 true H0 false
• Reject H0 Type I error Correct decision
• Accept H0 Correct Type II error
• decision

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Power

• The easiest ways to increase power are to
• increase sample size
• increase desired difference (or effect size)
• decrease significance level desired e.g. 10

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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

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Sample size fordescriptive survey
Simple random / systematic sampling
z² p q
1.96²0.150.85
-------------- ----------------------
544
n
d²
0.03²
Cluster sampling
z² p q
21.96²0.150.85
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
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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
  population
• Desired level of significance (a)
• Power of the study (1-ß)
• to detect at a statistically significant level a
  particular odds ratio

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Case-controlSTATCALC Sample size
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Case-control STATCALC Sample size

• Risk of alpha error 5
• Power 80
• Proportion of controls exposed 20
• OR to detect gt 2

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Case-controlSTATCALC Sample size
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Statistical Power of aCase-Control Study for
different control-to-case ratios and odds ratios
(with 50 cases)
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Conclusions

• Probability samples are the best
• Ensure
• Representativeness
• Precision
• ..within available constraints

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Conclusions

• If in doubt
• Call a statistician !!!!