bruno m. cesana1 biomedical statistics brescia prof. bruno mario cesana (sezione di statistica...
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Bruno M. Cesana 1
BIOMEDICAL STATISTICS Brescia
Prof. Bruno Mario Cesana(Sezione di Statistica Medica e Biometria)
Facoltà di Medicina e ChirurgiaUniversità degli Studi di Brescia
SAMPLE SIZE CALCULATIONS
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SAMPLE SIZETHE RESEARCHER’S QUESTIONS
- WHICH IS THE “BETTER” THERAPY ?
- THE (CARDIOVASCULAR) FUNCTION IS DIFFERENT ?
THE STATISTICAL (SCIENTIFICAL) TRANSLATION NULL HYPOTHESIS: H0: S = C o S = C
VS.
ALTERNATIVE HYPOTHESIS UNILATERAL:
HA: S > C o S > C OR HA: S > C o S < C
ALTERNATIVE HYPOTHESIS BILATERAL:
HA: S C o S C
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SAMPLE SIZESAMPLE SIZE CALCULATION FOR:
1)-TESTING / DEMONSTRATING AN EFFECT
(SAMPLE SIZE CALCULATION BASED ON A STATISTICAL SIGNIFICANCE TEST)
EXPERIMENTAL STUDIES (Lab, In Vitro, Animals)
CONTROLLED CLINICAL TRIALS
USUAL APPROACH (ICH E9)
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SAMPLE SIZESAMPLE SIZE CALCULATION FOR:
1)-TESTING / DEMONSTRATING AN EFFECT
(SAMPLE SIZE CALCULATION BASED ON A STATISTICAL SIGNIFICANCE TEST.
CONTROLLED CLINICAL TRIALS
USUAL APPROACH (ICH E9)
Etc., etc…
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SAMPLE SIZE CALCULATIONINGREDIENTS OF THE SAMPLE SIZE CALCULATION (0)
0. CHOICE of the DEPENDENT VARIABLE (QUALITATIVE / QUANTITATIVE) PRINCIPAL EXPRESSION of the PHENOMENON UNDER RESEARCH.
SCELTA della VARIABILE DIPENDENTE (QUALITATIVA / QUANTITATIVA) PRECIPUA ESPRESSIONE DEL FENOMENO OGGETTO DELLA RICERCA.
IT IS THE MAIN OBJECTIVE OF THE STUDY
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SAMPLE SIZEhttp://www.ich.org/fileadmin/Public_Web_Site/ICH_Products/Guidelines/Efficacy/E9/Step4/E9_Guideline.pdf
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SAMPLE SIZE
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SAMPLE SIZE
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SAMPLE SIZE
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SAMPLE SIZE
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SAMPLE SIZE
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SAMPLE SIZE CALCULATIONINGREDIENTS OF THE SAMPLE SIZE CALCULATION (1)
1. = THE MINIMAL DIFFERENCE CLINICALLY RELEVANT (superiority trial):
- % of SUCCESS / EVENT from a BASELINE (Qualitative variables)
- MEAN of QUANTITATIVE (continuous) VARIABLES COMBINED WITH THE PHENOMENON VARIABILITY ( / = EFFECT SIZE) .
2. = THE MAXIMAL DIFFERENCE NOT CLINICALLY RELEVANT (non inferiority trial):
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SAMPLE SIZE CALCULATIONINGREDIENTS OF THE SAMPLE SIZE CALCULATION (1)
2. = THE PROBABILITY OF I TYPE ERROR (SIGNIFICANCE LEVEL)
WRONG CONCLUSION OF A DIFFERENCE WHEN THE «EQUALITY IS ACTUALLY
TRUE» Usually 0.05 (two-tails).
P 0.03 at one tails of the Student’s t distribution is NOT STATISTICALLY SIGNIFICANT !!
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SAMPLE SIZE CALCULATION
3. = THE PROBABILITY OF THE II TYPE ERROR: WRONG CONCLUSION OF NO DIFFERENCE WHEN «A DIFFERENCE IS ACTUALLY TRUE». BETTER THE POWER (1- ) AS THE PROBABILITY OF CORRECTLY CONCLUDING FOR A «TRUE DIFFERENCE» (AT LEAST EQUAL TO THE FIXED/ REQUIRED EFFECT SIZE) Usually 0.80, 0.90.
INGREDIENTS OF THE SAMPLE SIZE CALCULATION (2)
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SAMPLE SIZE CALCULATION
4. THE PARTICULAR KIND OF THE SIGNIFICANCE TEST (all statistical test can be considered for power analysis; otherwise simulation).
STAT. TEST DEPENDS on:
CONSIDERED VARIABLE / OUTCOME
DESIGN / MODEL OF THE STUDY
INGREDIENTS OF THE SAMPLE SIZE CALCULATION (2)
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SAMPLE SIZE
22
1 1n 2(z z )
1
EFFECT SIZE(E.S.)
1
22
1n 2(z z )1
E.S.
A VERY SIMPLE MODEL: TWO GROUPS, LAST OBSERVATION (DIFF. PRE-POST), V.QUANTITATIVE.
APPROXIMATE SAMPLE SIZE for the «UNPAIRED STUDENT’S t TEST»:
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EFFECT SIZE
http://en.wikipedia.org/wiki/Effect_size
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SAMPLE SIZE
21 1
21
2
21
2
(2.3.Bis)n 2(z z )
2(z z )n
n FOR z1- (z1-0.05=1.645 - 1T; z1-0.025=1.96 - 2T);
n FOR z1- (z1-0.20=0.841; z1-0.10 =1.282);
n FOR n FOR
TOTAL SAMPLE SIZE = 2n
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SAMPLE SIZEFOR z1- = z1-0.20 = z0.80 = 0.841: POWER = 0.80
FOR z1-/2 = z1-0.025=z0.975 =1.96: =0.05 (TWO TAILED)
THE DENOMINATOR (FIRST PART) IS 162
1 12
2
2(z z )n
1
MULTIPLIED BY THE RECIPROCAL SQUARED OF THE EFFECT SIZE WE OBTAIN THE SAMPLE SIZE (APPROXIMATED) IN EACH TREATMENT GROUP.
WHEN EFFECT SIZE = 1 …. 16 (+ 1)
WHEN EFFECT SIZE = 0.5: 16 / 0.52 = 64 (+ 2)
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SAMPLE SIZEFOR z1- = z1-0.20 = z0.80 = 1.282: POWER = 0.90
FOR z1-/2 = z1-0.025=z0.975 =1.96: =0.05 (TWO TAILED)
THE DENOMINATOR (FIRST PART) IS 212
1 12
2
2(z z )n
1
MULTIPLIED BY THE RECIPROCAL SQUARED OF THE EFFECT SIZE WE OBTAIN THE SAMPLE SIZE (APPROXIMATED) IN EACH TREATMENT GROUP.
WHEN EFFECT SIZE = 1 …. 21 (+1)
WHEN EFFECT SIZE = 0.5: 21 / 0.52 = 84 (+2)
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SAMPLE SIZE
APPROXIMATE SAMPLE SIZE FOR the «PAIRED STUDENT’S t TEST».
THE SIMPLEST MODEL: ONE GROUP,
COMPARISON of a MEAN of a CONTINUOUS VARIABLE to an «EXPECTED VALUE»:
21 1
2
2
2(z z )n
1
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STUDENT’S TEST for UNPAIRED DATA: =0.05 (TWO TAILS), n IN EACH GROUP
/ 1- = 0.80 1- = 0.900.10 1571 2102
0.20 393 526
0.30 175 234
0.40 99 132
0.50 64 85
0.60 45 59
0.70 33 44
0.80 26 34
0.90 20 27
1.00 17 22
1.10 14 18
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COMPLETELY RANDOMIZED DESIGN (CR-k)Treatment Overall
Mean a1 a2 a3
y11 y12 y13
y21 y22 y23
yn11 yn22 yn33
Means•1y 2•y 3•y ••y
EQUATION OF THE MODEL:
yij = + j + ij con i = 1, 2, …, nj e j = 1, 2, …, k
•1y
1 2 3TO GUESS : AND
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SAMPLE SIZE CALCULATION
b = 0.3450
a=0.05
3.238874
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ONEWAY ANOVAPOWER CALCULATION: 1.5se2; a = 0.05(ONE-SIDED); 5 GROUPS F N SUB DOF_NUM DOF_DEN POWER_1 POWER_2 2.86608 5 4 20 0.54984 0.35672 2.75871 6 4 25 0.66136 0.44191 2.68963 7 4 30 0.75230 0.52264 2.64147 8 4 35 0.82323 0.59696 2.60597 9 4 40 0.87658 0.66372 2.57874 10 4 45 0.91549 0.72248 2.55718 11 4 50 0.94315 0.77327 2.53969 12 4 55 0.96236 0.81648 2.52522 13 4 60 0.97543 0.85274 2.51304 14 4 65 0.98418 0.88279 2.50266 15 4 70 0.98994 0.90740 2.49370 16 4 75 0.99367 0.92736 2.48588 17 4 80 0.99606 0.94340 2.47902 18 4 85 0.99757 0.95616 2.47293 19 4 90 0.99852 0.96625 2.46749 20 4 95 0.99910 0.97416
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FACTORIAL DESIGN (CR-ab)
A1 a2
b1 y111, y211,
y311, ... , yr11
y121, y221,
y321, ... , yr21
b2 y112, y212,
y312, ... , yr12
y122, y222,
y322, ... , yr22
TREATMENT AT
RE
AT
ME
NT
B
11•(y ) 12•(y )
21•(y ) 22•(y )
1••(y )
•1•(y ) •2•(y ) •••(y )
2••(y )
1 2
3 3
TO GUESS :
AND
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REPEATED MEASURES DESIGNS
TIME0 1 2 3 4 5
* * * * *DRUG A
DRUG A
UnderH0
* * * * * DRUG A
0 1 2 3 4 5 TIME
DRUG AUnder
HA
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MIXED FACTORIAL ANOVA FOR REPEATED MEASUREMENTS - TABLE
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REPEATED MEASURES DESIGN
T1 T2 T3 Tj T LAST
TREATMENT "S" m11 m12 m13 … m1J
TREATMENT "C" m21 m22 m23 … m2J
TABLE OF THE MEANS UNDER HA (to be guessed)
21 12 1J
221
2 2 2
2
1 2 J 1
2 2J
2J
2 2
2
2
21
2J2
J
1
2J xJ
J JJ
.
.
. . . .
.
T T . T T T . T
T
T
.
.
.
. . . .
.T
PATTERN OF THE VARIANCE-COVARIANCE MATRIX UNDER HA (to be guessed)
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SAMPLE SIZE - PROPORTION
THE SIMPLEST MODEL: COMPARISON BETWEEN A PROPORTION AND AN «EXPECTED VALUE».
BINOMIAL TEST (EXACT TEST)
APPROXIMATED TEST: TEST Z:
H0: = 0 vs. HA: = A - A =
2
1 0 0 1 A A
0 A
z 1 z 1n
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SAMPLE SIZEA VERY SIMPLE MODEL: COMPARISON BETWEEN TWO PROPORTIONS
EXACT FISHER’s TEST
APPROXIMATED TEST: TEST Z:
EQUAL SAMPLE SIZE: n1 = n2)
H0: 1 = 2 1 - 2 = 0
vs. HA: 1 - 2 =
2
1 1 1 1
1
2 21 2
(For n n)
z 2 1 z 1
n
n
1
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SAMPLE SIZESAMPLE SIZE: n in each group
2
1 1 1 1 2 2
21 2
1 2
z 2 1 z 1 1n
2
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2 1 n (1-)0.05 0.10 466 .8009
0.05 0.15 151 .8006
0.05 0.20 82 .8053
0.05 0.25 55 .8012
0.20 0.25 1131 .8001
0.20 0.30 311 .8005
0.20 0.35 150 .8027
0.20 0.40 90 .8017
0.30 0.35 1414 .8000
0.30 0.40 375 .8010
0.30 0.45 174 .8017
0.30 0.50 102 .8061
0.40 0.45 1577 .8002
0.40 0.50 404 .8012
0.40 0.55 183 .8002
0.40 0.60 102 .8008
0.45 0.50 1606 .8001
0.45 0.55 404 .8000
FISHER’ s EXACT TEST: 0.05 (two tails), Power (1 - ) 0.80
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http://invivostat.co.uk/
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