Bruno M. Cesana1 BIOMEDICAL STATISTICS Brescia Prof. Bruno Mario Cesana (Sezione di Statistica Medica e Biometria) Facolt à di Medicina e Chirurgia Universit à degli Studi di Brescia SAMPLE SIZE CALCULATIONS

Bruno M. Cesana2 SAMPLE SIZE THE RESEARCHER’S QUESTIONS - WHICH IS THE “BETTER” THERAPY ? - THE (CARDIOVASCULAR) FUNCTION IS DIFFERENT ? THE STATISTICAL (SCIENTIFICAL) TRANSLATION NULL HYPOTHESIS: H 0 :  S =  C o  S =  C VS. ALTERNATIVE HYPOTHESIS UNILATERAL: H A :  S >  C o  S >  C OR H A :  S >  C o  S <  C ALTERNATIVE HYPOTHESIS BILATERAL: H A :  S   C o  S   C

Bruno M. Cesana3 SAMPLE SIZE SAMPLE 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)

Bruno M. Cesana4 SAMPLE SIZE SAMPLE 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…

Bruno M. Cesana5 SAMPLE SIZE CALCULATION INGREDIENTS OF THE SAMPLE SIZE CALCULATION (0) CHOICE of the DEPENDENT VARIABLE (QUALITATIVE / QUANTITATIVE) PRINCIPAL EXPRESSION of the PHENOMENON UNDER RESEARCH. 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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Bruno M. Cesana12 SAMPLE SIZE CALCULATION INGREDIENTS OF THE SAMPLE SIZE CALCULATION (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)  = THE MAXIMAL DIFFERENCE NOT CLINICALLY RELEVANT (non inferiority trial):

Bruno M. Cesana13 SAMPLE SIZE CALCULATION INGREDIENTS OF THE SAMPLE SIZE CALCULATION (1)  = 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 !!

Bruno M. Cesana14 SAMPLE SIZE CALCULATION  = 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, INGREDIENTS OF THE SAMPLE SIZE CALCULATION (2)

Bruno M. Cesana15 SAMPLE SIZE CALCULATION 4. 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)

Bruno M. Cesana16 SAMPLE SIZE 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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Bruno M. Cesana18 SAMPLE SIZE n  FOR z 1-   (z = T; z = T); n  FOR z 1-   (z =0.841; z =1.282); n  FOR  n  FOR  TOTAL SAMPLE SIZE = 2n

Bruno M. Cesana19 SAMPLE SIZE FOR z 1-  = z = z 0.80 = 0.841: POWER = 0.80 FOR z 1-  /2 = z =z =1.96:  =0.05 (TWO TAILED) THE DENOMINATOR (FIRST PART) IS  16 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 / = 64 (+ 2)

Bruno M. Cesana20 SAMPLE SIZE FOR z 1-  = z = z 0.80 = 1.282: POWER = 0.90 FOR z 1-  /2 = z =z =1.96:  =0.05 (TWO TAILED) THE DENOMINATOR (FIRST PART) IS  21 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 / = 84 (+2)

Bruno M. Cesana21 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»:

Bruno M. Cesana22 STUDENT’S TEST for UNPAIRED DATA:  =0.05 (TWO TAILS), n IN EACH GROUP // 1-  =  =

Bruno M. Cesana23 COMPLETELY RANDOMIZED DESIGN (CR-k) TreatmentOverall Mean a1a1 a2a2 a3a3 y 11 y 12 y 13 y 21 y 22 y 23              yn11yn11 yn22yn22 yn33yn33 Means EQUATION OF THE MODEL: y ij =  +  j +  ij con i = 1, 2, …, n j e j = 1, 2, …, k

Bruno M. Cesana24 SAMPLE SIZE CALCULATION  =  =

Bruno M. Cesana25 ONEWAY ANOVA POWER CALCULATION: 1.5   2 ;  = 0.05(ONE-SIDED); 5 GROUPS F N SUB DOF_NUM DOF_DEN POWER_1 POWER_

Bruno M. Cesana26 FACTORIAL DESIGN (CR-ab) A1a2 b1y 111, y 211, y 311,..., y r11 y 121, y 221, y 321,..., y r21 b2y 112, y 212, y 312,..., y r12 y 122, y 222, y 322,..., y r22 TREATMENT A TREATMENT B

Bruno M. Cesana27 REPEATED MEASURES DESIGNS TIME * * * * * DRUG A Under H 0 * * * * * DRUG A TIME DRUG A Under H A

Bruno M. Cesana28 MIXED FACTORIAL ANOVA FOR REPEATED MEASUREMENTS - TABLE

Bruno M. Cesana29 REPEATED MEASURES DESIGN T1T2T3TjT LAST TREATMENT "S"  11  12  13 …  1J TREATMENT "C"  21  22  23 …  2J TABLE OF THE MEANS UNDER H A (to be guessed) PATTERN OF THE VARIANCE-COVARIANCE MATRIX UNDER H A (to be guessed)

Bruno M. Cesana30 SAMPLE SIZE - PROPORTION THE SIMPLEST MODEL: COMPARISON BETWEEN A PROPORTION AND AN «EXPECTED VALUE». BINOMIAL TEST (EXACT TEST) APPROXIMATED TEST: TEST Z: H 0 :  =  0 vs. H A :  =  A   -  A = 

Bruno M. Cesana31 SAMPLE SIZE A VERY SIMPLE MODEL: COMPARISON BETWEEN TWO PROPORTIONS EXACT FISHER’s TEST APPROXIMATED TEST: TEST Z: EQUAL SAMPLE SIZE: n 1 = n 2 ) H 0 :  1 =  2   1 -  2 = 0 vs. H A :  1 -  2 = 

Bruno M. Cesana32 SAMPLE SIZE SAMPLE SIZE: n in each group

Bruno M. Cesana33 22 11 n (1-  ) FISHER’ s EXACT TEST:  0.05 (two tails), Power (1 -  )  0.80

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