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Basic Statistical Concepts
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Chapter 2 Reading
instructions• 2.1 Introduction: Not very important• 2.2 Uncertainty and proaility: Read
• 2.! Bias and variaility: Read
• 2." Con#ounding and interaction: Read• 2.$ %escriptive and in#erential statistics:
Repetition
• 2.& 'ypothesis testing and p(values: Read
• 2.) Clinical signi*cance and clinical e+uivalence:Read
• 2., Reproduciility and generali-aility: Read
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Bias and variaility
Bias: Systemtic deviation #rom the true value
[ ] µ µ −ˆ E %esign Conduct /nalysis 0valuation
ots o# eamples on page "3(
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Bias and variaility
arger study does not decrease ias
ϕ µ µ +→n
ˆ 4 ∞→n
%rog 5 ( 6laceo
() ("(17 mm 'g () ("(17
%rog 5 ( 6laceo
mm 'g mm 'g
%rog 5 ( 6laceo
() ("(17
n8"7 n8277 N82777
%istriution o# sample means: 8 population mean
µ
ϕ
6opulation mean
ias
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Bias and variaility 9here is a multitude o# sources #or ias
6ulication ias
Selection ias
0posure ias
%etection ias
/nalysis ias
Interpretation
ias
6ositive results tend to e pulished hile negative o#inconclusive results tend to not to e pulished
9he outcome is correlated ith the eposure. /s aneample treatments tends to e prescried to thosethought to ene*t #rom them. Can e controlled y
randomi-ation%i;erences in eposure e.g. compliance to treatmentcould e associated ith the outcome e.g. patents
ith side e;ects stops ta
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Bias and variaility
/mount o# di;erence eteen oservations
9rue iological:
9emporal:
>easurement error:
?ariation between su@ectdue to iological #actorsAcovariates including thetreatment.?ariation over time Aand space#ten within su@ects.
Related to instruments or oserver
%esign Conduct /nalysis 0valuation
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Ra Blood pressure data
Baseline , ee
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Bias and variaility
ε β += X Y
Uneplained variation
?ariationin
oservations
80plained
variation
D
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Bias and variaility
%rug /
%rug B
utcome
Is there any di;erence eteen drug / and drug BE
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Bias and variaility
Y=μA+βx
Y=μB+βx
μA
μB
x=age
Model: ijijiij xY ε β µ ++=
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Con#ounding
Predictors
of
treatment
Predictors
of outcome
Confounder
s
9reatmentallocation
/
B
utcome
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0ampleSmoortality rateF 27.2 27.$ !$.$
Cochran Biometrics 13&,F per 1777 person(years G
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0ampleSmoortality rateF 27.2 27.$ !$.$
/verage age $".3 $7.$ &$.3
Cochran Biometrics 13&,F per 1777 person(years G
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0ampleSmoortality rateF 27.2 27.$ !$.$
/verage age $".3 $7.$ &$.3
/d@usted
mortality rateF
27.2 2&." 2".7
Cochran Biometrics 13&,F per 1777 person(years G
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Con#ounding
9he e;ect o# to or more #actors can not e separated
0ample: Compare survival #or
surgery and drug
R
i#e long treatment ith drug
Surgery at time 7
•Surgery only i# healty enough•6atients in the surgery arm may taa e
oo
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Con#ounding
Can e sometimes e handled in the design
0ample: %i;erent e;ects in males and #emales
Imalance eteen genders a;ects result
Strati#y y gender
R
/
B
Hender
>
R
R
/
/
B
B
Balance on average /lays alance
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Interaction
9he outcome on one varialedepends on the value o# anothervariale.
0ample Interaction eteen to drugs
R
/
/
B
B
Jashout
/8/K%12!"
B8/K%12!" D
Clarithromycin
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Interaction
Mean
0
1
2
3
4
5
0 4 8 12 16 20 24
Time after dose
P l a s m a c o n c e n t r a t i o n ( µ m o l / !
l i n e a r s c a l e
A"#$%&' alone
(om)ination of clarit*romcinand A"#$%&'
13.)$ ALmolFhM
!&.&2 ALmolFhM
/UC /K%12!":
/UC /K%12!" D Clarithromycin:
Ratio: 7.$$ 7.$1 7.&1O
/K%12!"
/K%12!"
0ample: %rug interaction
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Interaction
0ample: 9reatment y center interaction 9reatment di;erence in diastolic lood pressure
-25
-20
-15
-10
-5
0
5
10
15
0 5 10 15 20 25 30
Ordered center number
m m H g
/verage treatment e;ect: (".!3 (&.! (2."O mm'g
9reatment y center: p87.71
Jhat can e said aout the treatment e;ectE
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%escriptive and in#erential
statistics 9he presentation o# the results #rom a clinicaltrial can e split in three categories:
•%escriptive statistics•In#erential statistics
•0plorative statistics
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%escriptive and in#erential
statistics%escriptive statistics aims to descrie variousaspects o# the data otained in the study.
•istings.•Summary statistics A>ean Standard %eviationP.•Hraphics.
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%escriptive and in#erential
statisticsIn#erential statistics #orms a asis #or aconclusion regarding a prespeci*ed o@ective
addressing the underlying population.
'ypothesis Results
Con*rmatory analysis:
Conclusion
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%escriptive and in#erential
statistics0plorative statistics aims to *nd interestingresults that
Can e used to #ormulate neo@ectivesMhypothesis #or #urther investigation in#uture studies.
Results 'ypothesis
0plorative analysis:
ConclusionE
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'ypothesis testing p(values
and con*dence intervals@ectives?ariale
%esign
Statistical >odelNull hypothesis
0stimatep(value
Con*dence interval
ResultsInterpretation
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'ypothesis testing p(
valuesStatistical model: servations ( ) nn R X X ∈= ,-X#rom a class o# distriution #unctions
{ }Θ∈=℘ θ θ : P
'ypothesis test: Set up a null hypothesis: '7: $Θ∈θ
and an alternative '1: -Θ∈θ
Re@ect '7 i# ( ) α θ θ
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Con*dence intervals
/ con*dence set is a random susetcovering the true parameter value ithproaility at least .
( ) Θ⊆XC
α −-
et ( )re/ectednot:if $
re/ected:if -,
0
$
0
$0
θ θ
θ θ θ δ
=
==
H
H X Acritical #unction
Con*dence set: ( ) ( ){ }$,: == θ δ θ XXC
9he set o# parameter values correponding tohypotheses that can not e re@ected.
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0ample
y ij 8 μ D τi D β A x ij ( x ·· D εij
?ariale: 9he change #rom aseline to end o# study in sitting %B6 Asitting SB6 ill e descried ith an /NC?/ modelith treatment as a #actor and aseline lood pressureas a covariate
Null hypoteses Asusets o# :'71: τ1 8 τ2 A%B6
'72: τ1 8 τ2 ASB6
'7!: τ2 8 τ3 A%B6
'7": τ2 8 τ3 ASB6
@ective: 9o compare sitting diastolic lood pressure A%B6 loering e;ecto# hypersartan 1& mg ith that o# hypersartan , mg
>odel:treatment e;ecti 8 12!Q1& mg , mg " mg
6arameter space:1 R
1 R
( )$23-
=++ τ τ τ
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0ample contined
Hypothesis Variable LS Mean CI (95%) p!al"e
#$ # mg !s & mg Sitting'
*+, mmHg -.+/ 0+&1 23+33#
0$ # mg !s & mg SittingS
,+ mmHg -9+0/ +#1 23+33#
*$ & mg !s . mg Sitting'
3+9 mmHg -#+&/ 3+31 3+355
. $ & mg !s . mg SittingS
0+# mmHg -*+/ 3+1 3+335 9his is a t(test here the test statistic #ollos a t(distriution
Re@ection region: ( ){ }cT >XX :
$$-4$
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6(value says nothing aoutthe si-e o# the e;ect
No. of patients per group Estimation of effect p-value
10 1.94 mmg 0.3!"
100 -0."5 mmg 0.3!#
1000 0.33 mmg 0.129
10000 0.2# mmg $0.0001
100000 0.30 mmg $0.0001
/ statistical signi*cant di;erence does N9
need to e clinically relevant
0ample: Simulated data. 9he di;erence eteen treatmentand placeo is 7.! mm'g
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Statistical and clinical
signi*cance
Statistical signi*cance:
Clinical signi*cance:
'ealth ecominical relevance:
Is there any di;erence
eteen the evaluatedtreatmentsE%oes this di;erence haveany meaning #or thepatientsEIs there any economicalene*t #or the society inusing the ne treatmentE
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Statistical and clinicalsigni*cance
/ study comparing gastropra-ole "7 mg and myglopra-ole!7 mg ith respect to healing o# erosived eosophagitis a#ter, ee
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