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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$*+
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23
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Aase&control sample size:
parameters to consider
• 7umber of cases
• 7umber of controls per case
• O/ ratio worth detecting• = of e-posed persons in source
population
• @esired level of significance 4α5
• Power of the study 4)& β 5
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Aase&control:
Power calculation
α error +.
Number of cases 2""
Proportion of controls e7posed +.
:( to detect 9$*+
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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29
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
30
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