1 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Helmut SchützBEBAC
Helmut SchützBEBAC
Wik
imed
iaW
ikim
edia
Com
mon
s C
omm
ons
••20
07
2007
Suj
it K
umar
Suj
it K
umar
••C
reat
ive
Com
mon
s A
ttrib
utio
nC
reat
ive
Com
mon
s A
ttrib
utio
n --S
hare
Alik
eS
hare
Alik
e3.
0 3.
0 U
npor
ted
Unp
orte
d
Introduction to Biostatistics
Part I: Basic Concepts
Introduction to Introduction to BiostatisticsBiostatistics
Part I: Part I: Basic Basic ConceptsConcepts
2 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
BiometryBiometry , , BiometricsBiometrics ,,and and BiostatisticsBiostatistics
�Introduced in 1947 by R.A Fisheras ‘Biometry’ and later ‘Biometrics’
‘Biometry, the active pursuit of biologicalknowledge by quantitative methods.’
�The International Biometric Society‘The terms “Biometrics” and “Biometry” have been used since early in the 20th century to refer to the field of development of statistical and mathematical methods applicable to data analysis problems in the biological sciences. Recently, the term “Biometrics” has also been used to refer to the emerging field of technology devoted to identification of individuals […]’
�‘Biostatistics’ was introduced as a new term…
3 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
BiometryBiometry , , BiometricsBiometrics ,,and and BiostatisticsBiostatistics
StatisticsStatistics. . A subject which most A subject which most statisticians find difficult but in which nearly statisticians find difficult but in which nearly all physicians are expert.all physicians are expert.
BiostatisticianBiostatistician. . One who has neither theOne who has neither theintellect for mathematicsintellect for mathematics nor the commitment for nor the commitment for medicine but likes to dabble in both.medicine but likes to dabble in both.
Medical Medical sstatisticiantatistician. . One who will not accept that One who will not accept that Columbus discovered America…Columbus discovered America… because he said because he said he was looking for India in the trial plan.he was looking for India in the trial plan.
Stephen Stephen SennSenn
4 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
TerminologyTerminology IIhigh bias low bias
low variance
high variance
bias
5 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
TerminologyTerminology IIIIdata
continuousdiscrete
nominal scale ordinal scale interval scale ratio scale
distictness distictness +
rank order
distictness +
rank order +
interval
distictness +
rank order +
interval +
ratio
increasing information
6 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
DataData II�Nominal scale (aka categorial)
�Sex, ethnicity,…�Statistics: mode, χ² test�Transformations: equality
�Ordinal scale�School grades, disease states,…
�Statistics: median, percentile, sign test,Wilcoxon test
�Transformations: monotonic increasing order
7 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
DataData IIII�Interval scale
�Calendar dates, temperature in °C, IQ,…�Statistics: mean, variance (standard
deviation), correlation, regression,ANOVA
�Transformations: linear
�Ratio scale�Measures with true zero point, temperature in K,…
�Statistics: all of the above, geometric and harmonic mean, coefficient ofvariation
�Transformations: multiplicative, logarithm
8 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Examples fromExamples from PKPK�Ordinal scale
�tmax, tlag
�Statistics: median, percentile, sign test,Wilcoxon test
�Transformations: monotonic increasing order
�Ratio scale�AUC, Cmax, λz,…
�Statistics: mean, variance (standarddeviation), correlation, regression,ANOVA, geometric and harmonicmean, coefficient of variation
�Transformations: multiplicative, logarithm
9 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Bell Bell curvecurve –– and and beyondbeyond� Abraham de Moivre (1667–1754),
Pierre-Simon Laplace (1749–1827)Central limit theorem 1733, 1812
� Frank Wilcoxon (1892–1965)Nonparametric tests 1945
� Carl F. Gauß (1777–1855)Normal distribution 1795
� William S. Gosset, aka Student (1876–1937)t-distribution 1908
� Ronald A. Fisher (1890–1962)Analysis of variance 1918
10 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
normal distr.: mean = 100 CV = 30 %
n = 12
Den
sity
0 50 100 150 200
0.00
00.
005
0.01
00.
015
0.02
0normal distr.: mean = 100 CV = 30 %
n = 48
Den
sity
0 50 100 150 200
0.00
00.
005
0.01
00.
015
0.02
0
normal distr.: mean = 100 CV = 30 %
n = 128
Den
sity
0 50 100 150 200
0.00
00.
005
0.01
00.
015
0.02
0
normal distr.: mean = 100 CV = 30 %
n = 1024
Den
sity
0 50 100 150 200
0.00
00.
005
0.01
00.
015
0.02
0
Statistical DistributionsStatistical Distributions
11 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Statistical DistributionsStatistical Distributionslognormal distr.:
mean = 100 CV = 30 %
n = 12
Den
sity
0 50 100 150 200 250
0.00
00.
005
0.01
00.
015
0.02
0lognormal distr.:
mean = 100 CV = 30 %
n = 48
Den
sity
0 50 100 150 200 250
0.00
00.
005
0.01
00.
015
0.02
0
lognormal distr.: mean = 100 CV = 30 %
n = 128
Den
sity
0 50 100 150 200 250
0.00
00.
005
0.01
00.
015
0.02
0
lognormal distr.: mean = 100 CV = 30 %
n = 1024
Den
sity
0 50 100 150 200 250
0.00
00.
005
0.01
00.
015
0.02
0
12 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Statistical DistributionsStatistical Distributions�Normal Distribution
�Defined by location (aka central tendency)and dispersion
�Population�Location: population mean�Dispersion:population variance
�Sample�Location: sample mean�Dispersion:sample variance
�Probability = 1 within -∞ and +∞
x2s
2σµ
21
21( )
2
x
f x eµ
σ
σ π
− − =
2121
( )2
tx
F x e dtµ
σ
σ π
− −
−∞= ∫
13 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Statistical DistributionsStatistical Distributions�Lognormal Distribution
�Defined by location and dispersion
�Population�Location: population mean�Dispersion:population variance
�Sample�Location: sample mean�Dispersion:sample variance
�Probability = 1 within 0 and +∞
x2s
2σµ
( )2
2
ln
210( ) 2
0 0
x
e xf x xx
µσ
σ π
−−
>= ≤
( )2
2
ln
2
0
1 1( )
2
tx
F x e dtt
µσ
σ π
−−
= ∫
14 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Statistical DistributionsStatistical Distributionsnormal distr.: mean = 100
sd = 20 CV = 20 %
n = 1000
Den
sity
0 50 100 150 200
0.00
00.
010
0.02
0
normal distr.: mean = 120 sd = 24 CV = 20 %
n = 1000
Den
sity
0 50 100 150 200
0.00
00.
010
0.02
0
normal distr.: mean = 110 sd = 22.09 CV = 20.08 %
n = 2000
Den
sity
0 50 100 150 200
0.00
00.
010
0.02
0
normal distr.: mean = 100 sd = 20 CV = 20 %
n = 1000
Den
sity
0 50 100 150 200
0.00
00.
010
0.02
0
normal distr.: mean = 100 sd = 30 CV = 30 %
n = 1000D
ensi
ty0 50 100 150 200
0.00
00.
010
0.02
0
normal distr.: mean = 100 sd = 25.5 CV = 25.5 %
n = 2000
Den
sity
0 50 100 150 200
0.00
00.
010
0.02
0
15 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Statistical DistributionsStatistical Distributionspopulation: mean = 100 sd = 20.08 CV = 20.08 %
N = 1e+06
Den
sity
0 50 100 150 200
0.00
00.
010
0.02
0sample 1: mean = 101
sd = 15.94 CV = 15.78 %
n = 36
Den
sity
0 50 100 150 200
0.00
00.
010
0.02
0
sample 2: mean = 100.1 sd = 20.41 CV = 20.38 %
n = 36
Den
sity
0 50 100 150 200
0.00
00.
010
0.02
0
sample 3: mean = 100.2 sd = 19.61 CV = 19.57 %
n = 36D
ensi
ty0 50 100 150 200
0.00
00.
010
0.02
0
sample 4: mean = 96.69 sd = 19.31 CV = 19.97 %
n = 36
Den
sity
0 50 100 150 200
0.00
00
.010
0.02
0
sample 5: mean = 102.5 sd = 19.31 CV = 18.85 %
n = 36
Den
sity
0 50 100 150 200
0.00
00
.010
0.02
0
16 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Statistical DistributionsStatistical Distributionspopulation: mean = 100 sd = 20.03 CV = 20.03 %
N = 1e+06
Den
sity
0 50 100 150 200
0.00
00.
005
0.01
00.
015
0.02
00.
025
20 samples drawnfrom population
n = 36
Den
sity
0 50 100 150 200
0.00
00.
005
0.01
00.
015
0.02
00.
025
17 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Central Limit TheoremCentral Limit Theorem�If samples are drawn by a random process from a population with a normal distribution, distribution of sample means is also normal.
�The mean of the distribution of sample means is identical to the mean of the ‘parent population’ – the population from which the samples are drawn.
�The higher the sample size that is drawn,the ‘narrower’ will be the dispersion of the distribution of sample means.
18 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Normal Distribution INormal Distribution I
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Standard normal distribution: µ = 0, σ = 1
z
Den
sity
±1σ p 68.27%
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Standard normal distribution: µ = 0, σ = 1
z
Den
sity
±2σ p 95.45%
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Standard normal distribution: µ = 0, σ = 1
z
Den
sity
±3σ p 99.73%
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Standard normal distribution: µ = 0, σ = 1
z
Den
sity
±4σ p 99.99%
19 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Normal Distribution IINormal Distribution II
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Standard normal distribution: µ = 0, σ = 1
z
Den
sity
p 90% ±1.645σ
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Standard normal distribution: µ = 0, σ = 1
z
Den
sity
p 95% ±1.960σ
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Standard normal distribution: µ = 0, σ = 1
z
Den
sity
p 99% ±2.576σ
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Standard normal distribution: µ = 0, σ = 1
z
Den
sity
p 99.9% ±3.291σ
20 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Normal Distribution IIINormal Distribution III
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Standard normal distribution: µ = 0, σ = 1
z
Den
sity
[±1.645] 90%
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Standard normal distribution: µ = 0, σ = 1
z
Den
sity
[-∞, +1.645] 95%
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Standard normal distribution: µ = 0, σ = 1
z
Den
sity
[-1.645, +∞] 95%
21 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Confidence Interval Confidence Interval II�If we have drawn a sample from a population, we get the sample mean and the sample standard deviation s.
�Can we make a prediction about the population mean?
�Yes. That’s called a Confidence Interval (CI).�If is known:σ
[ ] x zn
σµ = ±
x
22 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Confidence IntervalConfidence Interval IIII�Example from previous slides: 100, 20Sample sizes 36, z0.05 1.960
�But generally we don’t know !
�Help is on the way…
σµ
109.095.97102.5
103.290.1696.69
106.793.67100.2
106.693.57100.1
107.594.47101.0
Confidence IntervalSamples’ means
σ
23 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Student’sStudent’s tt DistributionDistribution�Depends on one parameter, the‘degrees of freedom ν ’. In themost simple case df = n – 1.
�The t Distribution is ‘heavy tailed’ compared tothe normal distribution. Small sample sizesare penalized.
�Approaches quickly the normal distribution for df >≈30.
�Allows calculation of a CI of the sample mean based on the sample standard deviation s.
( )
12 2
1
0
12( ) 1
2
x t
xf x
x t e dt
νν
ν ννπ
+−
+∞− −
+ Γ = + Γ
Γ = ∫
24 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Student’sStudent’s tt DistributionDistribution
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Student’s t distribution: ν = 1
x
Den
sity
n = 2
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Student’s t distribution: ν = 5
x
Den
sity
n = 6
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Student’s t distribution: ν = 11
x
Den
sity
n = 12
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
Student’s t distribution: ν = 35
x
Den
sity
n = 36
25 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Confidence Interval Confidence Interval IIIIII�Example from previous slides: 100, 20Sample sizes 36, z0.05 1.960, t36-1,0.05 2.030
σµ
based on tbased on zstand. dev.mean
19.31
19.31
19.61
20.41
15.94
95.97
90.16
93.57
93.19
95.61
109.0
103.2
106.8
107.0
106.4
109.095.97102.5
103.290.1696.69
106.793.67100.2
106.693.57100.1
107.594.47101.0
Confidence IntervalsSamples’
[ ] sx t
nµ = ±[ ] x z
n
σµ = ±
26 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Location parametersLocation parameters�x = [91,72,141,119,92,124,92,101,90,145]
ranks [3, 1, 9, 7, 4.5, 8, 4.5, 6, 2, 10]ordered [72,90,91,92,92,101,119,124,141,145]
�Mode: 92 (most frequent number)
�Median: 96.5 (middle value) � If n=odd: value at xn/2
� If n=even: value (xn/2+xn/2+1)/2 = (92+101)/2
27 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Location parametersLocation parameters�Harmonic mean: 101.9516
�Geometric mean: 104.2814
�Arithmetic mean: 106.7
1
1ln
1 21
i n
ii
i n xn
nngeom i n
i
x x x x x e
=
=
=
=
∑= = ⋅ =∏ ⋯
1 21
1 1 11harm i n
ii i
n nx
x x xx
=
=
= =+∑ ⋯
1 2
1
1 i ni
arithm ii
x x xx x
n n
=
=
+= =∑⋯
28 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
A A notenote on on thetheharmonic meanharmonic mean
�Driving at 100 km/h from A to B;distance is 100 km.
�Driving back at 50 km/h.
�What is the average speed for the round-trip?�75 km/h � 70.71 km/h � 66.67 km/h
�1 h for 100 km (A→B) and 2 h for 100 km (A←B); 200 km/3 h = 66.67 km/h.
�Harmonic meanfor rates!
2 266.6
1 1 0.01 0.02100 50
harmx = = =++
ɺ
����
29 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Location parametersLocation parameters�Application of any location parameter always (!)
implies an underlying distributional assumption.�Median: discrete (or unknown)�Arithmetic mean: normal distribution�Geometric mean: lognormal distribution�Harmonic mean: rates
�Example from above sampled from a lognormal distribution
�Arithmetic mean: 106.7 (too high!)�Geometric mean:104.3 (correct)
30 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
6080
100
120
140
Boxplot
linear scale
4.2
4.4
4.6
4.8
5.0
Boxplot
logarithmic scale
Location parametersLocation parameters
31 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Nitpicking terminologyNitpicking terminology�If we are estimating parameters of a distribution,
we are using Estimators ; e.g., the arithmetic mean is the unbiased estimator of the central tendency of the normal distribution.
�The numerical outcomes (i.e., values one give in the report) are Estimates .
�Don’t write something like‘The point estimator was 95.34 %.’
… when it was actually a maximum likelihood estimator based on least squares means in log-scale ☺☺☺☺
32 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Dispersion Dispersion parametersparameters�ordered [72,90,91,92,92,101,119,124,141,145]
�Quartiles (25%, 75%): Be cautious! Different methods implemented in software…�90.00, 124.00: SAS�91.25, 122.75: S, R, M$-Excel�90.75, 128.25: Minitab, SPSS, Phoenix/WinNonlin�90.92, 125.42: Hyndman & Fan (1996)�… and many others!
33 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Dispersion Dispersion parametersparameters�Standard deviation (SD) of arithmetic mean: 24.2400
�SD of geometric mean:23.8000
( )2
1
1
1
i n
arithm ii
SD x xn
=
=
= −− ∑
( )2
1
1ln ln
1
i n
i geomi
x xn
geomSD e
=
=
−− ∑=
34 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Dispersion Dispersion parametersparameters�SD of harmonic mean: 22.8519
( ) ( )2
1
1
1
1
1
1
1 1
i n
harm ii
i n
ii
i i n
j j i
SD n H H
H Hn
nH
x x
=
=
=
=
=
=
= − −
=
−=
−
∑
∑
∑
35 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Dispersion Dispersion parametersparameters�Coefficient of Variation(sometimes given in percent of mean):
% 100CV SD x= ⋅
22.83
23.80
104.28
22.72
24.24
106.70
20.00CV%
20.00σ
100.00µ
Sample (n=36), parametersPopulation (N=106),
parameters
arithmx
arithmSD
%CV
geomx
geomSD
%CV
36 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
A A remark remark on on VariancesVariances�Whilst means and variances are additive, standard deviations (and CVs as well) are not!
400201001
√650=25.5!!∑s²/2
900
s²
25??(100+100)/21+2
301002
smeansample
mean = 100 sd = 20 CV = 20 %
n = 1000
Den
sity
0 50 100 150 200
0.00
00.
010
0.02
0
mean = 100 sd = 30 CV = 30 %
n = 1000
Den
sity
0 50 100 150 200
0.00
00.
010
0.02
0
mean = 100 sd = 25.5 CV = 25.5 %
n = 2000
Den
sity
0 50 100 150 200
0.00
00.
010
0.02
0
37 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
0
5
10
15
20
25
concentration
0 4 8 12 16 20 24
time
Arithmetic mean (95% CI)
0
5
10
15
20
25
concentration
0 4 8 12 16 20 24
time
Geometric mean (95% CI)
.1
1
10
concentration
0 4 8 12 16 20 24
time
Arithmetic mean (95% CI)
.1
1
10concentration
0 4 8 12 16 20 24
time
Geometric mean (95% CI)
ArithmArithm . . vs.vs. geomgeom . . meansmeans
38 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Median and Median and quantilesquantiles
0
5
10
15
20
25
concentration
0 4 8 12 16 20 24
time
Median (5%, 25%, 75%, 95% quantile)
.1
1
10
concentration
0 4 8 12 16 20 24
time
Median (5%, 25%, 75%, 95% quantile)
39 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
DegreesDegrees of of freedomfreedom ……�For every estimated parameter in a statistical model one degree of freedom is ‘lost’ from the number of samples (df = n – p).�Any model becomes useless if df=0, and impossible
to fit if df<0 (p>n). Example:�Linear regression: Two parameters are fit (slope,
intercept; since any line is definedby two points (x1/y1|x2,y2) at leastthree data points are needed(df=1).
40 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
0 50 100 150 200
050
150
mean = 100, sd = 20
n = 100
y
0 50 100 150 200
050
150
sample # 1
n = 12 , df = 10
ys
0 50 100 150 200
050
150
sample # 2
n = 12 , df = 10
ys
0 50 100 150 200
050
150
sample # 3
n = 12 , df = 10
ys
0 50 100 150 200
050
150
sample # 4
n = 12 , df = 10
ys
0 50 100 150 200
050
150
sample # 5
n = 12 , df = 10
ys
0 50 100 150 200
050
150
sample # 6
n = 12 , df = 10
ys
0 50 100 150 200
050
150
sample # 7
n = 12 , df = 10
ys
0 50 100 150 200
050
150
sample # 8
n = 12 , df = 10
ys
0 50 100 150 200
050
150
sample # 9
n = 12 , df = 10
ys
0 50 100 150 200
050
150
sample # 10
n = 12 , df = 10
ys
0 50 100 150 200
050
150
sample # 11
n = 12 , df = 10
ys0 50 100 150 200
050
150
sample # 12
n = 12 , df = 10
ys
0 50 100 150 200
050
150
sample # 13
n = 12 , df = 10
ys
0 50 100 150 200
050
150
sample # 14
n = 12 , df = 10
ys
0 50 100 150 200
050
150
sample # 15
n = 12 , df = 10ys
DegreesDegrees of of freedomfreedom ……
41 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
0 50 100 150 200
050
150
mean = 100, sd = 20
n = 100
y
0 50 100 150 200
050
150
sample # 1
n = 2 , df = 0
ys
0 50 100 150 200
050
150
sample # 2
n = 2 , df = 0
ys
0 50 100 150 200
050
150
sample # 3
n = 2 , df = 0
ys
0 50 100 150 200
050
150
sample # 4
n = 2 , df = 0
ys
0 50 100 150 200
050
150
sample # 5
n = 2 , df = 0
ys
0 50 100 150 200
050
150
sample # 6
n = 2 , df = 0
ys
0 50 100 150 200
050
150
sample # 7
n = 2 , df = 0
ys
0 50 100 150 200
050
150
sample # 8
n = 2 , df = 0
ys
0 50 100 150 200
050
150
sample # 9
n = 2 , df = 0
ys
0 50 100 150 200
050
150
sample # 10
n = 2 , df = 0
ys
0 50 100 150 200
050
150
sample # 11
n = 2 , df = 0
ys0 50 100 150 200
050
150
sample # 12
n = 2 , df = 0
ys
0 50 100 150 200
050
150
sample # 13
n = 2 , df = 0
ys
0 50 100 150 200
050
150
sample # 14
n = 2 , df = 0
ys
0 50 100 150 200
050
150
sample # 15
n = 2 , df = 0ys
DegreesDegrees of of freedomfreedom ……
42 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
0 5 10 15 20
0
5
10
15
20
Anscombe1
0 5 10 15 20
0
5
10
15
20
Anscombe2
0 5 10 15 20
0
5
10
15
20
Anscombe3
0 5 10 15 20
0
5
10
15
20
Anscombe4
Visualize your dataVisualize your data !!Anscombe’sQuartet (1973)
All datasets:meanx 9.0, s²x 10meany 7.5, s²y 3.75Corryx 0.898Regryx y = 3 + 0.5x
correct wrong model:quadratic!
correct model, but biased by outlier
nonsenseDon’t rely solely on numerical results.
43 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Data Data Transformation?Transformation?MPH, 405 subjects
Shapiro-Wilk p= 3.2854e-14AUC [ng×h/mL]
Den
sity
0 50 100 150
0.00
00.
010
0.02
00.
030
MPH, 405 subjects
Shapiro-Wilk p= 0.26008ln(AUC [ng×h/mL])
Den
sity
2.5 3.0 3.5 4.0 4.5 5.0
0.0
0.5
1.0
1.5
-3 -2 -1 0 1 2 3
2040
6080
100
120
Normal Q-Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
-3 -2 -1 0 1 2 3
3.0
3.5
4.0
4.5
Normal Q-Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
Clearly in favor of a lognormal distribution. Shapiro-Wilk test highly signi-ficant fornormal distribution(rejected).
44 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Data Data Transformation!Transformation!MPH, 12 subjects
Shapiro-Wilk p= 0.29668AUC [ng×h/mL]
Den
sity
0 50 100 150
0.00
00.
010
0.02
00.
030
MPH, 12 subjects
Shapiro-Wilk p= 0.85764ln(AUC [ng×h/mL])
Den
sity
2.5 3.0 3.5 4.0 4.5 5.0
0.0
0.5
1.0
1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
2030
4050
60
Normal Q-Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
3.0
3.2
3.4
3.6
3.8
4.0
Normal Q-Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
Data set from a real study. Both tests notsignificant(assumed distributions not reject-ed).
Tests not acceptable accordingto GLs; log-transforma-tion basedon prior knowledge(PK)!
45 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
DataData TransformationTransformation�BE testing started in the early 1980s with an acceptance range of 80% – 120% of the reference based on the normal distribution.
�Was questioned in the mid 1980s�Like many biological variables AUC and Cmax do not
follow a normal distribution� Negative values are impossible� The distribution is skewed to the right� Might follow a lognormal distribution
�Serial dilutions in bioanalytics lead to multiplicative errors
46 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
,T T R RT R
T R
AUC CL AUC CLF F
D D
⋅ ⋅= =
DataData Transformation:Transformation: PKPK
Assumption 1: D1=D2 (D1/D2=1*)
Assumption 2: CL1=CL2
( ) Trel
R
AUCF BA
AUC=
47 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
DataData TransformationTransformation�‘Problems’ with logtransformation
�If we transform the ‘old’ acceptance limits of80% – 120%, we get –0.2231, +0.1823.
�These limits are not symetrical around 100% any more, the maximum power is obtained at
e0.1823–0.2231 = 96%…
�Solution:lower limit = 1 – 0.20, upper limit = 1/lower limit
ln(0.80) = –0.2231 and ln(1.25) = +0.2231.Symetrical around 0 in the log-domain and around 100% in the backtransformed domain (e0=1).
48 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
DataData TransformationTransformation�‘Problems’ with logtransformation
�Discussion, whether more bioinequivalent formula-tions will pass due to ‘5% wider’ limitslower limit = 1 – 0.20, upper limit = 1/lower
80.00% – 125.00% (width 45.00%)instead of keeping the ‘old’ widthlower limit = 1 – 0.1802, upper limit = 1/lower
81.98% – 121.98% (width 40.00%)or even become more strict by settingupper limit = 1 + 0.20, lower limit = 1/upper
83.33% – 120.00% (width 36.67%)80% – 125% was chosen for convenience (!)
49 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
FF DistributionDistribution�Allows comparison of variances (depending onν ) of two distributions. We will need that in ANOVA.
�Note that if one of the degrees of freedom = 1, there is a relationshop to the t distribution:
( )
( )
1
1 2
1 2
1 2 12
2 21 2
1 11 2 21 2
1
0
2 2 0( | , )
2 2
0 0
x t
xx
F x x
x
x t e dt
νν ν
ν ν
ν ν
ν νν νν ν ν ν
−
+
+∞− −
Γ + ⋅ ⋅ ≥= +Γ Γ <
Γ = ∫
( )2
1 2( 1, ) ( )F tν ν ν ν= = =
50 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Significance testsSignificance tests�In statistics (as well as in science in general)it is not possible to prove something.
�We can only state a hypothesis and try to rejectthis so called null hypothesis by evaluating data from an experiment.
�Example:�H0: µ1 = µ2 (no difference in means, null hypothesis)
vs.�Ha: µ1 ≠ µ2 (different means; alternative hypothesis)
51 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
αααααααα-- vs.vs. ββββββββ--ErrorError�All formal decisions are subjected to two typesof error:�Error Type I (α-Error, Risk Type I)�Error Type II (β-Error, Risk Type II)
Example from the justice system:
Error type IICorrectPresumption of innocence accepted(not guilty)
CorrectError type I Presumption of innocence not accepted (guilty)
Defendant guiltyDefendant innocentVerdict
52 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
αααααααα-- vs.vs. ββββββββ--ErrorError�… in more statistical terms:
�In BE-testing the null hypothesis is bioinequivalence (µ1 ≠ µ2)!
Error type IICorrect (H 0)Failed to reject null hypothesis
Correct (H a)Error type I Null hypothesis rejected
Null hypothesis falseNull hypothesis trueDecision
Producer’s riskCorrect (not BE)Failed to reject null hypothesis
Correct (BE)Patients’ riskNull hypothesis rejected
Null hypothesis falseNull hypothesis trueDecision
53 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
95% one-sided CI
particular patient
0.6 0.8 1 1.25 1.67
95% one-sided CI
particular patient
0.6 0.8 1 1.25 1.67
90% two-sided CI= two 95% one-sided
population of patients
0.6 0.8 1 1.25 1.67
αααααααα-- vs.vs. ββββββββ--ErrorError�α-Error: Patients’ Risk to be treated with a bioinequivalent formulation (H0 falsely rejected)
�BA of the test compared to reference in a particularpatient is risky either below 80% or above 125%.
�If we keep the risk of particular patients at 0.05 (5%), the risk of the entire population of patients(<80% and >125%) is 2×α (10%) is:90% CI = 1 – 2×α = 0.90
54 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
αααααααα-- vs.vs. ββββββββ--ErrorError�β-Error: Producer’s Risk to get no approval fora bioequivalent formulation (H0 falsely not rejected)
�Set in study planning to ≤0.2, wherepower = 1 – β = ≥80%
�If power is set to 80 %One out of five studies will fail just by chance!
ββββ 0.20not BE
BEαααα 0.05
55 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Significance test: α, β
β Scrit.α
Significance Significance test (test ( αααααααα-- vs.vs. ββββββββ))
56 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
Part I: Basic Part I: Basic ConceptsConcepts
Helmut SchützBEBAC
Consultancy Services forBioequivalence and Bioavailability Studies
1070 Vienna, [email protected]
57 • 57
Introduction to Introduction to BiostatisticsBiostatistics (1/3: Basic (1/3: Basic ConceptsConcepts ))
BiostatisticsBiostatistics : Basic concepts & applicable principles for variou s designs: Basic concepts & applicable principles for variou s designsin in bioequivalencebioequivalence studiesstudies and data analysisand data analysis | | MumbaiMumbai , , 29 29 –– 3030 January January 20112011
ππππππππεεεεεεεεχχχχχχχχεεεεεεεεππππππππ Pharma Edge
To bear in Remembrance...To bear in Remembrance...
It is a good morning exercise for a researchIt is a good morning exercise for a research scientistscientistto discard a pet hypothesis every day beforeto discard a pet hypothesis every day beforebreakfast.breakfast.It keeps him young.It keeps him young. Konrad LorenzKonrad Lorenz
The theory of probabilities is at bottomThe theory of probabilities is at bottomnothing butnothing but common sense reduced to calculuscommon sense reduced to calculus.
PierrePierre--Simon Simon LaplaceLaplace
In these matters the only certainty isIn these matters the only certainty isthat nothing is certain.that nothing is certain.
Gaius Plinius SecundusGaius Plinius Secundus ((Pliny the ElderPliny the Elder))
Top Related