Introduction to Biostatistics - bebac.at · Introduction to Biostatistics (1/3: Basic Concepts ) ... used to refer to the emerging field of technology devoted to identification of

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

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Introduction to Biostatistics

Part I: Basic Concepts

Introduction to Introduction to BiostatisticsBiostatistics

Part I: Part I: Basic Basic ConceptsConcepts

2 • 57




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




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




TerminologyTerminology IIhigh bias low bias

low variance

high variance

bias

5 • 57




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




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




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




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




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




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


n = 128

Den

sity

0 50 100 150 200

0.00

00.

005

0.01

00.

015

0.02

0


n = 1024

Den

sity

0 50 100 150 200

0.00

00.

005

0.01

00.

015

0.02

0

Statistical DistributionsStatistical Distributions

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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




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




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




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


n = 1000

Den

sity

0 50 100 150 200

0.00

00.

010

0.02

0


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

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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

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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




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




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


z

Den

sity

±2σ p 95.45%

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4


z

Den

sity

±3σ p 99.73%

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4


z

Den

sity

±4σ p 99.99%

19 • 57




Normal Distribution IINormal Distribution II

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4


z

Den

sity

p 90% ±1.645σ

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4


z

Den

sity

p 95% ±1.960σ

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4


z

Den

sity

p 99% ±2.576σ

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4


z

Den

sity

p 99.9% ±3.291σ

20 • 57




Normal Distribution IIINormal Distribution III

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4


z

Den

sity

[±1.645] 90%

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4


z

Den

sity

[-∞, +1.645] 95%

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4


z

Den

sity

[-1.645, +∞] 95%

21 • 57




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




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




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




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


x

Den

sity

n = 6

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4


x

Den

sity

n = 12

-4 -2 0 2 4

0.0

0.1

0.2

0.3

0.4


x

Den

sity

n = 36

25 • 57




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




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




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




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




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




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




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




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




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




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




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




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




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




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




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




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




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




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




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


Sam

ple

Qua

ntile

s

Clearly in favor of a lognormal distribution. Shapiro-Wilk test highly signi-ficant fornormal distribution(rejected).

44 • 57




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


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


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)!

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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

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,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=

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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).

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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 (!)

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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= = =

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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)

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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

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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

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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

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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

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Significance test: α, β

β Scrit.α

Significance Significance test (test ( αααααααα-- vs.vs. ββββββββ))

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Part I: Basic Part I: Basic ConceptsConcepts

Helmut SchützBEBAC

Consultancy Services forBioequivalence and Bioavailability Studies

1070 Vienna, [email protected]

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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))

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Introduction to Biostatistics - bebac.at · Introduction to Biostatistics (1/3: Basic Concepts ) ... used to refer to the emerging field of technology devoted to identification of