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Sample size calculation

Ioannis Karagiannis

based on previous EPIET material

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Objectives: sample size

• To understand:

• Why we estimate sample size

• Principles of sample size calculation

• Ingredients needed to estimate

sample size

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The idea of statistical inference

Sample

PopulationConclusions basedon the sample

Generalisation to the population

Hypotheses

3

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Why bother with sample size

• Pointless if power is too small

• Waste of resources if sample sizeneeded is too large

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!uestions in sample size

calculation

•  " national Salmonella outbrea# has occurred with

several hundred cases$

• %ou plan a case&control study to identify ifconsumption of food ' is associated with

infection$

• (ow many cases and controls should you recruit

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•  "n outbrea# of )* cases of a mysteriousdisease has occurred in cohort +,)+$

• %ou suspect e-posure to an activity isassociated with illness and plan to underta#e acohort study under the #ind auspices ofcoordinators$

• With the available cases. how much power willyou have to detect a // of )01

!uestions in sample size

calculation

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Issues in sample size estimation

• 2stimate sample needed to measure the

factor of interest

• Trade&off between study size and resources

• 3ample size determined by various factors:

• significance level 4α5

• power 4)& β 5

• e-pected prevalence of factor of interest

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Which variables should be included

in the sample size calculation

• The sample size calculation should relate to

the study6s primary outcome variable0

• If the study has secondary outcome variableswhich are also considered important. the

sample size should also be sufficient for the

analyses of these variables0

8

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Allowing for response rates and

other losses to the sample

• The sample size calculation should relate to thefinal. achieved sample0

• 7eed to increase the initial numbers in accordancewith: 8 the e-pected response rate

 8 loss to follow up

 8 lac# of compliance

• The lin# between the initial numbers approachedand the final achieved sample size should bemade e-plicit0

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3ignificance testing:

null and alternative hypotheses• Null hypothesis !"#

  There is no difference

 "ny difference is due to chance

• Alternative hypothesis !$#

  There is a true difference

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2-amples of null hypotheses

• %ase&control study

!": O/9)

the odds of e-posure among cases are the same as

the odds of e-posure among controls;

• %ohort study

!": //9)the "/ among the e-posed is the same as the "/

among the une-posed;

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3ignificance level 4p&value5

• probability of finding a difference 4//: a hypothesis is never accepted;

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Type II error and power 

•  βis the type II error 8 probability of not finding a difference. when

a difference really does e-ist• Power is 4)& β5 and is usually set  to

?,= 8

probability of finding a difference when adifference really does e-ist 49sensitivity5

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3ignificance and power 

Truth

(, true

7o difference

(, false

@ifference

'ecision

Aannotreject (,

Aorrect decision Type II error 9  β 

/eject (,Type I error level 9 α 

significance

Aorrect decisionpower 9 )& β 

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(ow to increase power 

• increase sample size

• increase desired difference 4or effect

size5 reBuired 7>: increasing the desired difference in //CO/

means move it away from )D

• increase significance level desired4αerror5

 7arrower confidence intervals

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The effect of sample size

• Aonsider E cohort studies loo#ing at

e-posure to oysters with79),. ),,. ),,,

• In all E studies. F,= of the e-posed are

ill compared to *,= of une-posed4// 9 )015

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Table " 479),5

>ecame ill

%es Total "/

 "teoysters

%es E 1 EC1

7o + 1 +C1

Total 1 ), 1C),

(()$*+, -+. %I/ "*0&+*0, p)"*+1

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Table > 479),,5

>ecame ill

%es Total "/

 "teoysters

%es E, 1, E,C1,

7o +, 1, +,C1,

Total 1, ),, 1,C),,

(()$*+, -+. %I/ $*"&2*1, p)"*"03

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Table A 479),,,5

>ecame ill

%es 7o "/

 "teoysters

%es E,, 1,, E,,C1,,

7o +,, 1,, +,,C1,,

Total 1,, ),,, 1,,C),,,

(()$*+, -+. %I/ $*1&$*4, p5"*""$

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3ample size and power 

• In Table ". with n9), sample. there was

no significant association with oysters.

but there was with a larger sample size0• In Tables > and A. with bigger samples.

the association became significant0

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Aohort sample size:

parameters to consider 

• /is# ratio worth detecting

• 2-pected freBuency of disease in

une-posed population• /atio of une-posed to e-posed

• @esired level of significance 4α5

• Power of the study 4)& β 5

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Aohort:

2pisheet Power calculation

(is6 of α error +.

Population e7posed $""

E7p fre8 disease in une7posed +.

(atio of une7posed to e7posed $/$

(( to detect 9$*+

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Aase&control sample size:

parameters to consider 

• 7umber of cases

• 7umber of controls per case

• O/ ratio worth detecting• = of e-posed persons in source

population

• @esired level of significance 4α5

• Power of the study 4)& β 5

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Aase&control:

Power calculation

α error +.

Number of cases 2""

Proportion of controls e7posed +.

:( to detect 9$*+

No* controls;case $/$

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3tatistical Power of a

Aase&Aontrol 3tudy for different control&to&case ratios and odds ratios 41, cases5

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3tatistical Power of a

Aase&Aontrol 3tudy 

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3ample size for proportions:

parameters to consider • Population size

•  "nticipated p

•   α error 

• @esign effect

 2asy to calculate on openepi.com

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