1. The document discusses sample size calculation for various study designs including cross-sectional studies, case-control studies, and clinical trials. It provides the key formulas and parameters involved in sample size calculation including confidence level, precision, power, and Z-values. 2. Several examples are provided to demonstrate how to calculate the minimum required sample size given information about the study design, variables, expected outcomes, confidence level, and power. 3. Key factors that determine sample size are confidence level, precision or minimum detectable difference, power, and values of outcomes in pilot or previous studies. The appropriate formulas are selected based on the study design and scale of measurement of variables.

1. Sample SizeSample Size
Dr. A.P. KulkarniDr. A.P. Kulkarni
drapkulkarni@gmail.comdrapkulkarni@gmail.com

2. The NecessityThe Necessity
 Results become invalid if study doneResults become invalid if study done
on smaller size than requiredon smaller size than required
 Un-necessary large size avoidedUn-necessary large size avoided

3. Pre-requisitesPre-requisites
 Statistician requires some informationStatistician requires some information
 Some of this can be given from records/Some of this can be given from records/
literature/ pilotliterature/ pilot
 Some information requires knowledge ofSome information requires knowledge of
basic statisticsbasic statistics
 In analytical / experimental studiesIn analytical / experimental studies
sample size is different for hypothesissample size is different for hypothesis
testing and for testing the differencetesting and for testing the difference

4. αα errorerror
 αα is Significance level of a test. It isis Significance level of a test. It is
probability of rejecting a true nullprobability of rejecting a true null
hypothesis. Also called “Type-I Error”hypothesis. Also called “Type-I Error”
 Commonly accepted level of alphaCommonly accepted level of alpha
error is: 0.05 or 0.10error is: 0.05 or 0.10
= 5% or 10%= 5% or 10%
In clinical trial a largeIn clinical trial a large αα errorerror
would lead to bringing a uselesswould lead to bringing a useless
drug in marketdrug in market

5. Confidence LevelConfidence Level
 The probability of correctly accepting aThe probability of correctly accepting a
TRUE NH Denoted by 1-TRUE NH Denoted by 1- αα or 100 (or 100 ( 1-1- αα))
WhenWhen αα is decided, Conf. Level isis decided, Conf. Level is
automatically fixed.automatically fixed.
AlphaAlpha CLCL
5%5% 95%95%
10%10% 90%90%

6. Confidence LevelConfidence Level
 Higher the confidence level, higher isHigher the confidence level, higher is
the sample size.the sample size.
Confidence Level
Sample
Size

7. BetaBeta ((ß)
 The probability of failing to reject falseThe probability of failing to reject false
null hypothesis. It is also called “Type-null hypothesis. It is also called “Type-
II Error”II Error”
 Commonly accepted levels of BetaCommonly accepted levels of Beta
error are: 0.1 , 0.2error are: 0.1 , 0.2
 In terms of % : 10%, 20%In terms of % : 10%, 20%
In a clinical trial a largeIn a clinical trial a large ß errorerror
would lead to preventing a goodwould lead to preventing a good
drug from marketdrug from market

8. Power of A TestPower of A Test
 The probability of correctly rejecting aThe probability of correctly rejecting a
false null hypothesis. It is denoted byfalse null hypothesis. It is denoted by
1-1- ßß In terms of % it is 100 x (1-In terms of % it is 100 x (1- ß)ß)
 WhenWhen ßß is selected, Power of Test isis selected, Power of Test is
automatically fixed.automatically fixed.
BetaBeta PowerPower
10%10% 90%90%
20%20% 80%80%

9. Power of A TestPower of A Test
 Higher the Power of Test, Higher isHigher the Power of Test, Higher is
the Sample Sizethe Sample Size
Power of Test
Sample
Size

10. Z valuesZ values
zz 1-1- αα /2/2 , z, z 1-1- αα , z, z 1-1- ßß
Represent the number of standardRepresent the number of standard
errors from the mean .errors from the mean .
zz 1-1- αα /2/2 and z zand z z 1-1- αα are the functions of theare the functions of the
confidence level, whileconfidence level, while
zz 1-1- ßß is the function of the power of theis the function of the power of the
test.test.
In this presentation Z1, Z2 will be usedIn this presentation Z1, Z2 will be used
for Zfor Z αα andand ZZ ßß respectively.respectively.

11. Alpha
error
Confidenc
e
Z -alpha Term
used
here
One
sided
Two
sided
0.01 99% 2.33 2.58
Z10.05 95% 1.65 1.95
0.10 90% 1.28 1.65
Beta
Error
Power
of test
Z -beta Term
used
here
0.10 90% 1.28
Z2
0.20 80% 0.84

12. PrecisionPrecision
 Difference between Guestimate of aDifference between Guestimate of a
variable and minimum/ maximumvariable and minimum/ maximum
valuevalue
 When expressed as absolute value itWhen expressed as absolute value it
is called “Absolute Precision”is called “Absolute Precision”
GuestimateGuestimate

13. Relative PrecisionRelative Precision
 When “Absolute Precision” is
expressed not in absolute terms, but
as % of Guestimate, it is called
“Relative Precision”
 Other things equal, lesser the
value of precision, more is the
sample size required

14. Common situationsCommon situations
Study
type
Objective Variabl
e
Informati
on
required
Cross
sectional
1.Estimating population PP is a
nominal
variable
P, d, Z1
2. Estimating population MM is on
ratio
scale
M, µ, d,
Z1

15. Common situationsCommon situations
Study
type
Objective Variable Informati
on
required
Case
control
3.
Testing hypothesis OR=1Exposure :
P1 on
NOMINAL
scale
P1,P2,
OR, Z1,
Z2
4.
Estimating OR

16. Study type Objective Variable Information
required
Clinical trial
(Superiority
design)
5. Test
hypothesis
P1=P2
P1 and P2
measured
on
NOMINAL
Scale
P1, P2, Z1,
Z26. Estimate
difference
P1 , P2
7. Test
Hypothesis
M1=M2
M1 and M2
measured
on RATIO
scale
M1, M2, µ1,
µ2, Z1, Z28. Estimate
difference
M1=M2

19. 1. Epi Info
2. Open EPI
3. Power & Sample size
Calculator

23. 1. Estimate Population P1. Estimate Population P
Cross sectional studyCross sectional study
Z1 is the corresponding value for setZ1 is the corresponding value for set
level of confidencelevel of confidence
P is guestimate of population proportionP is guestimate of population proportion

24. Example-1Example-1
 In an epidemic, investigator wants toIn an epidemic, investigator wants to
know proportion of persons affectedknow proportion of persons affected
with 2 percent absolute precision andwith 2 percent absolute precision and
95% confidence. The guestimated P is95% confidence. The guestimated P is
4%4%
 Here : P = 4%Here : P = 4%
 Confidence is 95% so: Z = 1.96Confidence is 95% so: Z = 1.96
 Absolute precision: d = 2Absolute precision: d = 2

25. Calculations: Example-1Calculations: Example-1
1.962
x {4 (100-4)}
= -------------------------
22
= 368.79

26. 2.: Estimate Population2.: Estimate Population
MM
Cross sectional studyCross sectional study
 M is Mean, aM is Mean, a
quantitative variablequantitative variable
measured onmeasured on
RATIO scaleRATIO scale
 d is absoluted is absolute
precision.precision.
 S is SD of MS is SD of M
 Guestimate of MGuestimate of M
and its SD (S) isand its SD (S) is
requiredrequired

27. Example-2Example-2
 We want to
estimate mean
Hb% of female
employees of
an organization
based on
sample.
Information
available:
Guestimate M =10
Guestimate of SD: (S)
= 3.0
Confidence level
95%, so: Z =1.96
Precision expected:
0.5

28. Calculation: Example-2Calculation: Example-2
1.961.9622
x 3x 322
n = -----------------n = -----------------
0.50.522
= 138.29= 138.29
Note: Mean of the
variable is NOT
required for
calculation of sample
size here. But is
required to arrive at
“d”

29. 3. Test NH OR=1: Case Control Study3. Test NH OR=1: Case Control Study
Outcome: NOMINAL variableOutcome: NOMINAL variable
Where:Where:
r = Ratio of controls tor = Ratio of controls to
casescases
Z1: Z value associatedZ1: Z value associated
with set level alphawith set level alpha
errorerror
Where:Where:
Z2= Z value for setZ2= Z value for set
level of beta errorlevel of beta error
P1, P2= ProbabilityP1, P2= Probability
exposure cases/exposure cases/
ControlsControls
P = P1+P2/ 2P = P1+P2/ 2

30. Example-3Example-3
In a pilot studyIn a pilot study
(hypothetical) it was(hypothetical) it was
found that 8% diabeticsfound that 8% diabetics
had family history,had family history,
while 2% % non-while 2% % non-
diabetics had familydiabetics had family
history. A large study ishistory. A large study is
planned What will beplanned What will be
minimum sample sizeminimum sample size
required?required?
Here:Here:
r =1r =1
Alpha = 0.05; so:Alpha = 0.05; so:
Z1= 1.96Z1= 1.96
Beta = 0.2 so; so: Z2Beta = 0.2 so; so: Z2
= 0.84= 0.84
P1 =0.08, P2=0.02P1 =0.08, P2=0.02
P = 0.05P = 0.05

31. Calculation: Example-3Calculation: Example-3
Objective-A: Test NH=0Objective-A: Test NH=0
Putting values in equation:Putting values in equation:
n= 207. So, Cases=Controls = 207n= 207. So, Cases=Controls = 207

32. 44. Estimate OR: Case Control Study. Estimate OR: Case Control Study
Outcome: NOMINAL variableOutcome: NOMINAL variable
 Example-4Example-4: In example-3 (Family: In example-3 (Family
H/O diabetes), if objective is EstimateH/O diabetes), if objective is Estimate
OR, we use:OR, we use:
 ε is relative precision (=0.5 forε is relative precision (=0.5 for
example)example)
 Putting values n = 517Putting values n = 517

33. 5. Testing hypothesis P1=P25. Testing hypothesis P1=P2
Clinical trialClinical trial
 P1 and P2 are outcome variablesP1 and P2 are outcome variables
measured on NOMINAL scale. P =measured on NOMINAL scale. P =
(P1+P2)/2(P1+P2)/2
 Z1 is Z value corresponding to ConfidenceZ1 is Z value corresponding to Confidence
levellevel
 Z2 is Z value corresponding to power ofZ2 is Z value corresponding to power of

34. Example-5Example-5
 Pilot: Complication rates: new methodPilot: Complication rates: new method
(5%) and control surgical method (15%)(5%) and control surgical method (15%)
What would be required minimum sampleWhat would be required minimum sample
size to test NH of P1= P2 againstsize to test NH of P1= P2 against
alternate hypothesis of P1<P2alternate hypothesis of P1<P2
 Z1 = 1.65 (for one sided hypothesis)Z1 = 1.65 (for one sided hypothesis)
 Z2 = 0.84 (Power of test = 80%)Z2 = 0.84 (Power of test = 80%)

35. Information: Example-5Information: Example-5
 P1, P2: Proportion of outcome in test andP1, P2: Proportion of outcome in test and
control intervention. 5 % (0.05) and 15%control intervention. 5 % (0.05) and 15%
(0.15) respectively(0.15) respectively
 P =(P1+P2)/ 2= 10% = (0.10)P =(P1+P2)/ 2= 10% = (0.10)
 Confidence level: for Z1: 95% (OneConfidence level: for Z1: 95% (One
sided) so,sided) so, αα = 0.05= 0.05 ,, so: Z1= 1.65so: Z1= 1.65
 (If two sided, Z will be =1.96)(If two sided, Z will be =1.96)
 Power of test 80% So, Z2 = 0.84Power of test 80% So, Z2 = 0.84

36. Calculation: Example-5Calculation: Example-5
A. Testing NH P1=P2A. Testing NH P1=P2
Using formula given below:Using formula given below:
n = Study group=Control group = 30n = Study group=Control group = 30

37. Example-6Example-6
Objective: Estimating difference P1, P2Objective: Estimating difference P1, P2
With absolute precision of 2 %With absolute precision of 2 %
n= study group=control group= 642n= study group=control group= 642