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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

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 lt5 years, hospital discharges,

health events - Sampling fraction
- Ratio between sample size and population size
- 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 lt 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

Sampling Population

Sample

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

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

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

Quality of a sampling estimate

Precision validity

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)

179

178

177

176

175

174

173

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

selected - 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 fraction - 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

- Principle
- 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.

neighbourhoods - 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 2

Section 1

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 (homogenous). - 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

isnt one. - Known as the a (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 ß (or type 2 error)
- Power is (1- ß) 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

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

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

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 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

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

Case-controlSTATCALC Sample size

Case-control STATCALC Sample size

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

Case-controlSTATCALC Sample size

Statistical Power of aCase-Control Study for

different control-to-case ratios and odds ratios

(with 50 cases)

Conclusions

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

Conclusions

- If in doubt
- Call a statistician !!!!

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