PROCEDURE FOR CALCULATING SAMPLE SIZE

1. SAMPLE SIZE CALCULATION
SPEAKER:- Dr.Swati Singh
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2. THE OUTLINE
• What is sample size?
• Basic information needed for sample size calculation.
• Why to determine sample size?
• How large a sample do we need?
• What are the methods of determining it?
• What are the factors that affect it?
• Types of measurement in research.
• How do we determine sample size?
• Conclusion
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3. WHAT IS A SAMPLE?
• This is the sub-population, to be studied in order to draw a
inference from a reference population (a population to which
the findings of the Study are to be generalized).
• In Census, the sample size is equal to the population size.
However, in research, because of time constraints and budget,
a representative sample is normally used.
• Larger the sample, more accurate will be the findings from a
Study.
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4. • Availability of resources sets upper limit of the sample size.
• Required accuracy sets lower limit of sample size.
• Thus, an optimum sample size is an essential component of
any research.
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5. BASIC INFORMATION NEEDED FOR
SAMPLE SIZE CALCULATION
The approach to sample size calculation can be arrived at by
thinking through the following set of questions:
• What type of study is this?
Single sample (prevalence survey)
Comparison of two groups (cross-sectional, case-control, cohort
study)
• What is the main (primary) outcome?
Mean of a measurement (mean blood pressure)
Proportion
Ordered scale (pain scores)
• What is the expected variability between the subjects?
• How large a difference would be considered clinically important and
reasonable?
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6. WHAT IS SAMPLE SIZE DETERMINATION
 Sample size determination is the mathematical estimation of the
number of subjects/units to be included in a study.
 When a representative sample is taken from a population, the
finding are generalized to the population.
 Optimum sample size determination is required for the following
reasons:
1. To allow appropriate analysis
2. To provide desired level of accuracy
3. To allow validity to the significance test.
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7. HOW LARGE A SAMPLE DO WE NEED?
 If the sample is too small:
1. Even a well conducted Study may fail to answer it’s
research question.
2. It may fail to detect important effects or
associations.
3. It may associate this effect or association
imprecisely.
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8. CONVERSELY
 If the sample size is too large:
1. The Study will be difficult and costly.
2. Time constraint.
3. Loss of accuracy.
Hence, optimum sample size must be determined
before commencement of a Study.
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9. Types of Measurement in Research
• Random error
• Systematic error (bias)
• Precision (reliability)
• Accuracy (Validity)
• Power
• Effect size
• Design effect
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10.  Random error: Errors that occur by chance. Sources are sample
variability, subject to subject differences & measurement errors.
These can be reduced by averaging, increasing sample size,
repeating the experiment.
 Systematic error: Deviations not due to chance alone. Several
factors, e.g. patient selection criteria may contribute. It can be
reduced by good study design and conduct of the experiment.
 Precision: The degree to which a variable has the same value
when measured several times. It is a function of random error.
 Accuracy: The degree to which a variable actually represent the
true value. It is function of systematic error.
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12.  Power: This is the probability that the test will correctly
identify a significant difference, effect or association in the
sample should one exist in the population. Sample size is
directly proportional to the power of the study. The larger the
sample size, the study will have greater power to detect
significance difference, effect or association.
 Effect size: Is a measure of the strength of the relationship
between two variables in a population. The bigger the size of
the effect in the population, the easier it will be to find out.
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13. • Design effect: Geographic clustering is generally used
to make the study easier & cheaper to perform.
The effect on the sample size depends on the
number of clusters & the variance between & within
the cluster.
In practice, this is determined from previous
studies and is expressed as a constant called ‘design
effect’ often between 1.0 & 2.0. The sample sizes for
simple random samples are multiplied by the design
effect to obtain the sample size for the cluster
sample.
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14. AT WHAT STAGE CAN SAMPLE SIZE BE
ADDRESSED?
It can be addressed at two stages:
1. Calculation of the optimum sample size is required
during the planning stage, while designing the
Study and information on some parameters.
2. At the stage of interpretation of the result.
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15. APPROACHES FOR ESTIMATING
SAMPLE SIZE
 Approaches for estimating sample size depend primarily
on:
1. The study design &
2. The main outcome measure of the study
There are distinct approaches for calculating sample size
for different study designs & different outcome
measures.
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16. Sample Size Formula
• The formula requires that we (i)specify the amount of
confidence we wish to have, (ii) estimate the variance
in the population, and (iii) specify the level of desired
accuracy we want.
• When we specify the above, the formula tells us what
sample size we need to use….n
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17. PROCEDURE FOR CALCULATING SAMPLE
SIZE
There are 3 procedures that could be used for
calculating sample size:
1. Use of formulae
2. Ready made tables
3. Computer softwares
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18. USE OF FORMULAE
• Requirements for sample size calculations
µ/p = mean/proportion of interest.
µo/po = null hypothesis mean/proportion.
d = range of confidence interval(CI).
u = one –sided percentage point of normal
distribution corresponding to 100%-power.
v = two-sided percentage point of normal
distribution corresponding to required significance level.
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19. TYPICAL VALUES FOR SIGNIFICANCE LEVEL
AND POWER
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20. STATISTICAL FORMULA FOR SAMPLE
SIZE
• Sample size for The Mean :
n= Z² (var)²/ (e)²
Where
Z=confidence level at 95% (standard value of 1.96)
var = Variance of population
e=Allowable error
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21. EXAMPLE
• A health officer wishes to estimate the mean
hemoglobin in a defined community. Preliminary
information is that this mean is about 150mg/l with a
SD of 32mg/l. If a sampling error of up to 5mg/l in
the estimate is to be tolerated, how many subjects
should be included in the study?
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22. SOLUTION
• SD=32mg/l
• e=5mg/l
• Z=1.96
1762
5
32*96.1
2
22


n
n
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23. • Sample Size for Proportions & Prevalence :
n= Z² p(1-p)/ (e)²
Where
Z=confidence level at 95% (standard value of 1.96)
P=Estimated prevalence or proportions of project
area
e=range of CI
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24. EXAMPLE
• Suppose the prevalence of brucella infection is 2%
and the absolute difference to be detected is 0.25%
with a 95% confidence, what is the sample size
required?
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25. SOLUTION
• P=0.02 q=1-p = 1-0.02
• q=0.98
• Z=1.96
• e=0.0025
12047
0025.0
98.0*02.0*96.1
2
2


n
n
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26. SAMPLE SIZE FOR TWO MEANS
• (u+v)²(σ² 1+ σ² 2)/(µ1-µ2) ²
• µ1-µ2-Difference between means
• σ 1+ σ 2- Standard deviation
• u- one –sided percentage point of normal distribution
corresponding to 100%-power.
• v- two-sided percentage point of normal distribution
corresponding to required significance level.
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27. EXAMPLE
• A study was planned to find out whether food
supplementation during pregnancy increases birth weight of
child. Pregnant women were randomly assigned to cases and
control . To calculate sample size ,we need:
• Size of difference between mean birth weight that was
considered appreciable:
=(µ1-µ2) : decided as 0.25 kg by investigators.
• Standard deviation of distributions in each group : ROL
suggested it to be 0.4kg .
Assuming σ 1, σ 2 = 0.4kg
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28. • Power required : decided to be kept at 95%
1-power= 5%
u=1.64
• Significance level desired : decided to be kept at 1%
v=2.58
• Applying formula
n>(1.64+2.58) ²x(0.4²+0.4²)/0.25²
=17.8084x0.32/0.0625 = 91.2
Therefore 92 subjects are needed in both the groups.
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29. COMPARISON OF TWO PROPORTION
• 2x(u+v) ² ]px(1-p)]/(p1-p2)
• p= p2+p1/2
• p1,p2 - Proportions
• u -one –sided percentage point of normal
distribution corresponding to 100%-power.
• v- two-sided percentage point of normal distribution
corresponding to required significance level
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30. EXAMPLE
• A study was planned to record difference in mortality among
cases of road traffic injuries graded AIS score 4 and 5 in the
month of July. Results from previous study shows 18 deaths
among 72 patients graded AIS score 4. To calculate sample
size, we need:
• Proportion mortality in previous study: p1 =18/72=0.25 or
25%
• Size of difference between proportion mortalities that would
be considered appreciable:
• =( p1-p2) : decided as 3% by investigators.
• Thus expected proportion mortality in grade 5 cases:
• p2=28%=28/100=0.28
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31. p=(0.25+0.28)/2= 0.265
• Power required : decided to be kept at 95%
1-power= 5%
u=1.64
• Significance level desired : decided to be kept at 1%
v=2.58
• Applying formula
• N = 2x (1.64+2.58)²x(0.265)(1-0.265)/(0.28-0.25)
=7707
Therefore 7707 subjects are needed to be studied in each group.
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32. FORMULAE USED WITH RATE
• Estimation of single rate
µ v²/d²
µ - Rate
d – Range of CI
v - two-sided percentage point of normal distribution
corresponding to required significance level.
• Comparison of two rates
(u+v) ² (µ1+µo) / (µ1-µo) ²
µ1,µo - Rates
u - one –sided percentage point of normal distribution
corresponding to 100%-power.
v - - two-sided percentage point of normal distribution
corresponding to required significance level
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33. EXAMPLE-ESTIMATION OF SINGLE RATE
• A study was planned to find out mean number of viral
influenza incidence per child per annum in 0-5 year old
children in Orrisa , India. To calculate sample size , we need:
• Average number of influenza incidence expected in 0-5year
olds per annum . Review of existing literature states it to be 4
approximately.
• 95% confidence interval we would like to have for our desired
average : Decided to be ± 0.2 by investigators .
i.e. 95% CI = 3.8-4.2
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34. • Two-sided percentage point of normal distribution
corresponding to required significance level:
v for 95% CI = 1.96(˜2)
• Applying formula
n> µ v²/d² = 4(2) ²/(0.2) ²
=400
• Thus, at least 400 subjects need to be studied to obtain mean
influenza of 4 per child per annum with 95% CI of ± 0.2
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35. EXAMPLE-COMPARISON OF RATES
• A study was planned to find out whether KAP improvement tools
for driving skills decrease injuries per annum in school going
children . School children were randomly assigned to cases who
received special education and controls who didn’t . To calculate
sample size , we need:
• Size of difference between mean road traffic accident rates that
was considered appreciable:
=(µ1-µo) : decided as 2 injuries per child per annum by
investigators
• Rate of injuries per child per annum among controls:
suggest it to be 4 injuries per child per annum .
Therefore µo=4 , µ1 =2
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36. • Power required : decided to be kept at 95%
1-power = 5%
u = 1.64
• Significance level desired : decided to be kept at 1%
v = 2.58
• Applying the formula :
n> (1.64+2.58) ²x(2+4)/(2-4) ²
=17.8084x6/4 = 26.71
Therefore 27 subjects are needed in both the groups.
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37. SAMPLE SIZE CALCULATION FOR ODD RATIO
• Unmatched case-control
This formula is same as the formula used for comparing the two
proportions.
2x(u+v) ² ]px(1-p)]/(p1-p2)
Where p= p2+p1/2
p- Proportions of controls exposed
OR- Odds ratio
p1- Proportion of cases exposed
p1=p2 OR/1+p2 (OR-1)
u - one –sided percentage point of normal distribution corresponding to 100%-
power.
v - two-sided percentage point of normal distribution corresponding to required
significance level
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38. SAMPLE SIZE CALCULATION FOR RISK RATIO
• Unmatched cohort study
This formula is same as the formula used for comparing the two
proportions
2x(u+v) ² ]px(1-p)]/(p1-p2)
Where p= p2+p1/2
p1- Risk of disease among non-exposed
RR- Risk ratio
p2- Risk of disease among exposed
p2 = p1 x RR
u - one –sided percentage point of normal distribution corresponding to 100%-
power.
v - two-sided percentage point of normal distribution corresponding to required
significance level
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39. USE OF COMPUTER SOFTWARE FOR SAMPLE
SIZE CALCULATION
The following softwares can be used for calculating
sample size .
Epi-info (epiinfo.codeplex.com)
nQuerry (nquery.codeplex.com)
STATA (www.stata.com)
SPSS (www.spss.co.in)
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40. USE OF READYMADE TABLES FOR
SAMPLE SIZE CALCULATION
 How large a sample of patients should be followed up if an
investigator wishes to estimate the incidence rate of a disease
to within 10% of it’s true value with 95% confidence?
 The table show that for e=0.10 & confidence level of 95%, a
sample size of 385 would be needed.
 This table can be used to calculate the sample size making the
desired changes in the relative precision & confidence level
.e.g if the level of confidence is reduce to 90%, then the
sample size would be 271.
 Such table that give ready made sample sizes are available for
different designs & situation.
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42. CONCLUSIONS
• Sample size determination is one of the most
essential components of every research Study.
• The larger the sample size, the higher will be the
degree of accuracy, but this is limited by the
availability of resources.
• It can be determined using formulae, readymade
tables and computer softwares.
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