MPH CHAPTER 6 Power and sample size.pptx

Sample size and power analysis for analytic epidemiology

1. CHAPTER SIX: SAMPLE SIZE AND POWER ANALYSIS FOR ANALYTIC EPIDEMIOLOGY

2. W HAT I S S AM PL E S IZ E?  T h i s i s the s u b- p opu l at i on t o b e s t ud i e d i n o r d e r to m a ke an i n fe r e n c e to a re f e r e n c e pop u l a t i o n (A b r oad e r pop u l a t i on t o w h i c h t he f i n d i n g s f r o m a s t udy a r e to b e g e n er a l i z e d ) I n c e n s u s , the s amp l e s i ze i s e q u a l t o t he p o pu l at i o n si z e . H o w ev er , i n re s e a r c h, b e c a u s e of t i m e c o n s t r a i nt a nd b ud ge t,  a r e p re s e n t at i v e T h e l a r ge r the s t ud y . s amp l e s amp l e a r e n o r ma l l y us e d. s i ze the mo r e a cc u r a t e  t he f i nd i n g s f r om a 4

3.  A v a i l a b i l i t y of r e s ou r c e s s e ts the u pp e r li m i t of the s a m p l e si z e . W h i l e the r e q u ire d a cc u r acy s e ts the l o we r l i m i t of samp l e si z e   T h er e f o r e , a n o pt i m u m s a m p le s ize i s an e ss e nt i a l c om p on e n t of a ny re s e a r c h. 5

4. W H AT IS S A M P LE SI Z E DETER M I N ATION  S a m p l e s i z e d ete r m i na t i o n is t h e ma t h em a t i c a l e s t i m a t i o n o f t h e n u m b e r o f s ub j ect s / u n i t s t o b e i n c lu d e d in a s t u d y . W h e n a r e p r e s e n t a t i v e s a m pl e is t ak e n f ro m a p o pu l a t i o n , t h e f i n d i n g ar e g e n e ral i z e d t o t h e p op u la t i o n .   O p t i mu m s a m pl e s i z e d e t e r m i na t i o n is r e qu i r e d f o l lo w i n g r e a s on s : f o r t h e 4. T o 5. T o 6. T o all o w f o r a p pro p r i a t e a n al y s i s pr o v i d e t h e d es i re d l eve l o f a cc u ra c y all o w v a l i d i t y o f s i g n i f i c a n c e t est . 7

5. HOW L AR G E A S AM PL E D O I N E E D ?  I f the s am p l e i s t oo s m a l l : 2. E v e n a w e l l c on d u c t e d s t u d y m a y f a i l to a n sw e r i t s r e s e a r c h q u e s t i on 3. I t 4. I t m a y m a y f a i l to d e t e c t i m p o r ta n t e ffe c t o r a s so ci a t i ons a s so ci a t e t h i s e f fe c t o r asso c i a t i on i m p r e c i s el y 8

6. C O N V E R S E L Y I f the s am p l e si ze i s too l a r ge :  2. T h e s t u d y wil l b e d i f f i c u l t and c o s t l y 3. T i m e c o n s t r a i nt 4. A v a i l a b l e c as e s e . g r a r e d is e a s e . 5. L oss o f a cc u r a c y . H e n c e , opt i m u m samp l e s i ze m u s t c omm e n c e m e nt of a s t ud y . b e d e t er m i n e d b e f o r e 9

7. IMPORTANT POINTS            R a n dom e r r or Sy s t e ma t i c e r r o r T y pe I (a) err or T y pe I I ( b ) e r r or Po we r (1 - b ) E ff e c t si z e ( b i as) P r e c isi o n A c cu r a c y (r e li a bi l i t y ) (V a l i d i t y ) N u l l h y poth e si s D e si g n e f f e c t A l t e r na t i v e h y p o th e s i s 1

8.  R a n d o m e r r o r : e rr or t h a t o cc ur b y c ha n c e . S ou r c e s a r e samp l e v a ri a bi l i t y , s u b je ct t o s u b je ct d if f e r e n c e s & m e asur e m e n t e r r o rs . I t c an b e r e duce d b y a ve r a g i n g , i n c re ase samp l e s i z e , re p e a t i n g the e x p eri m e nt. S y s t e m a t ic e rr o r : d e vi at i o n s not d u e to c hance a l on e . Sev er al f a c t o rs , e . g pa t ie nt s e l e c t i on c ri t e ri a may con t r i b u t e . I t c an b e r e du c e d b y g o od s t u d y d e s i g n a n d c o n du c t of the e x p eri m e n t. P r e c i sio n : the d e g re e t o w h i c h a v a ri a b l e has the s ame v a l ue w h e n m e asur e d s ev er al t i m e s. I t i s a f un c t i on of r a ndom e rr o r . A c c u r ac y : the d e g r e e t o w h i c h a v a ri a bl e actua ll y re p r e s e nt t he t r ue v a l u e . I t i s f u n c t i on of s y s te m a t i c e rr o r .    1 1

9. 1 2

10.  N u l l h yp o t h e s i s : I t st a t e t h a t t he r e is n o d i ffe r e n c e a m on g g r o up s o r n o a ss o c i a t i o n b et w ee n t h e pr e d i ct o r & t h e o u t c o m e v ar i abl e . Th is h y po t h es is n ee d t o b e t es t e d .  A l t e r n a t i v e h y p o t h e s i s : I t c o n t rad i c t t h e n ul l h y po t h es i s . I f t h e al te rna t i v e h y p o t he s is c a n no t b e tes t e d d i r ect l y , it is a cce p te d b y e x c lu s i o n i f t h e tes t o f s i g n i f i c an c e r e j ect s t h e n u l l t w o h y p o t h es i s . Th e r e t a i l e d ( tw o - s i d e d ) ar e t w o t y p es ; o n e t a i l ( o n e - s i d e d ) o r 1 3

11.  T y pe I( a ) e r r o r : I t o cc ur s if a n i n v est i g a t o r r e j ec t s a n u l l h y po t h es is t h a t i s a c t u a ll y t ru e in t h e p o p u la t i o n . T h e pr o ba b i l i t y o f m ak i n g ( a ) e r r o r is c all e d a s l eve l o f s i g n i f i c a n c e & c o n s i d e re d a s . 5 ( 5 % ) . I t i s s p e c i f i e d a s i n s a m pl e s i z e c o m p u t i n g . Z a i s a v al u e f ro m st and a r d Z a n o rma l d i s t r i b u t i o n ≡ a . S a m p l e s i z e i s i n ve rs e l y pr o p o r t i ona l t o ty p e I e rror . T y pe II ( β ) e rr o r : it o cc u r s i f t h e i n ve s t i g a t o r f a i l s t o r e j ec t  a n u l l h y p o t h es i s t ha t is ac t u a l l y f a ls e in t h e p op u la t i o n . I t s p ec i f ied i n t e r m s o f Z b in s am p l e s i z e c o m p u t i n g . Z b i s a is v a l u e fr o m s t a n dar d no r ma l d i s t r i b u t i o n ≡ β 1 4

12.  P o w e r ( 1 - β ) : T h i s i s t he p r o b a bi l i ty that the t e st w i l l c o rr e c t l y i d e nt i f y a s i g n i f ic a n t d i f f e r e n c e , e f fe ct o r a ss o ci a t ion i n the s a m p l e s h o u l d one e x i st i n the popu l a t i on. S amp l e si z e i s d ire ct l y p r o p o r t i o n al to the po we r o f the s t u d y . T h e l a r ge r the s a m p l e si z e , the s t udy wi l l h a v e g r e at e r po w e r to d e t e c t si g n i f i can c e d i ff er e n c e , e f fe c t or a s so ci a t i o n .  E f f ec t s i z e : is a me a s u r e of t h e s t r e n g th of t h e r el a tio n s h ip be t w ee n t w o v a r i a bl e s in a p o pu la t i on. I t i s t he m a g n i tude of t he e f f e ct u n d e r t he a l t e r na t i v e h y p oth e s is . T he b i gg e r t he s i ze of the e ff e c t i n the popu l at i on, t he e asi e r i t wil l b e t o f i nd. 1 5

13.  Desi g n e f f e ct : Ge o g r a ph ic c l u s te r i n g i s g e n e ral l y u s e d t o ma k e t h e s t u d y e as i e r & c h e a p e r t o p e r f o rm . Th e effec t o n t h e s a m p l e s i z e d e p e nd s o n th e n um b e r o f cl u s te r s & th e v a ri a n c e b etw e e n & w i th i n t h e cl u s t e r . I n p r a c t i ce , t h i s is d ete r m i n e d f r o m pr e v i ou s st ud i e s an d i s e xpr esse d a s a c o n s t an t c all e d ‘ d e sig n e f fe c t ’ o f t e n b e t wee n 1 . & 2 . . Th e s am p l e s i z e f o r s i mp l e ra n do m s a mp l e s ar e m ul t i pl i e d b y t h e d es i g n ef f ec t t o ob t a in t h e s a mp l e s i z e f o r t h e c l u s te r s a m pl e . 1 6

14. P OW E R AN A LYSIS  W h e n t h e es t i m a t e d s am p l e s i z e c a n no t b e i n c lu d e d i n a s t u d y , p o s t - h o c p o w e r a n al y s i s s h o ul d b e c a r r i e d o u t . Th e pr o b a b i l i t y o f c or rec t l y r e j ec t i n g t h e nu l l h y p o t he s is i s e qua l t o 1 – β , w h i c h is c all e d p o w e r . Th e p o w e r o f a tes t  r efe r s t o i t s ab i l i t y t o t h e p o w e r o f a t es t is lo o k i n g f o r , g i v e n i t s d etec t w ha t it i s l oo k i n g f or . o u r pr o b a b i l i t y o f f i n d i n g w h a t w e ar e s i z e .   p o s t- ho c p o w e r a n al y s is i s d o n e a fte r a s t u d y h a s b ee n c arr i e d o u t t o h e l p t o e x pla in t h e r es ul t s if a s t u d y w h i c h di d n o t f i n d a n y sig n i f i c a n t e f fe c t s . 1 8

15. AT W H AT S T AGE C AN SA MP LE SI Z E ADD R E S S E D ? B E I t c a n b e a d d re s s e d at t w o s t a g es : 2. C a l c u l a t e t he o p t i mum sa m p l e si z e r e q u ir e d du ri n g the p l ann i n g s t a g e , w h il e d e si g n i ng the s t ud y , u si ng a pp r op r i ate ap p r oa c h & i n f o r mat i o n on s ome pa r am e t er s. 3. O r th r o u g h po s t - hoc po w e r an a l y s i s at the s t a g e o f i n t er p r e ta t i on of the r e s u l t. 1 9

16. A P P R OA C H F OR E S TI M A TI N G SA M P L E SI Z E / P OWER A N ALYSIS  T h er e a r e m a ny d i ff er e nt a pp r oa c h e s f o r c a l c u l at i ng the s amp l e si z e f or d if fe re nt s t udy d esi g n s . S u c h as c a s e c o n t ro l de s i g n , c o h or t de s i g n , c ro s s s e c t i o n a l s t u d i e s , c li n ic a l t r i a l s , d i a g n o s t i c t es t s t u d i e s e tc. Wi t h i n e a c h s t udy d e s i g n th e r e c o u l d b e mo r e s u b - d e si g ns and the s a m p l e si ze c a l c u l a t i on wi l l v a r y a c co r d i n gly .   T h er e f o r e , o ne m u s t u s e the c o rr e c t a pp r oach f o r c omp u t i ng s a m p l e si ze a p p r op ri a t e t o t h e s t udy d e s i g n & i t s su b t y p e . t he

17. P A RAM E T E RS D e p e nd i n g upon the app r o a c h c h o se n f or c a l c u l a t i n g the s a m p l e si z e , one a l s o as; H y p o th e s i s P r e c isi o n T y pe I err or T y pe I I e rr or P o w e r E ff e c t si ze D e si g n ef f e c t n e e d s t o s p e c i f y s o me ad d i t i o n al pa r am e t er s s uch        2 4

18. P R O C E D UR E F OR C A L C UL A TI N G SA M P L E SI Z E. T h e r e a r e f o u r p r o c e du r e s s i z e : 1 . U se of f o r m u l ae 2 . R e ady m a d e t a b l e 3 . N om o g r a m s 4 . C o m p ut e r s o f t w a r e t hat cou l d b e us e d f o r c a l c u l at i ng s a m p l e 2 5

19. U S E OF F O R M U L A E F OR S A M P L E S I Z E C A LC U L A T ION & PO W ER AN A L Y S IS  T h er e a r e m a ny f o r mu l ae f or ca l cu l a t i ng s amp l e s i ze & p o we r i n d if fe re nt si t uat i o ns f o r d i ff ere nt s t udy d es ig n s . T h e a pp r op ri a t e s amp l e s i ze f o r popu l a t i on - b as e d s t u dy d e t e r m i n e d l a r ge l y b y 3 f acto r s  i s 1 . T h e 2 . T h e 3 . T h e e s t i m a t e d p re v a le n c e o f the d e sir e d l e v e l o f con f i d e n c e . v a ri a b l e of i n t ere s t . a cc e p t a b l e ma r g i n o f e rr o r . 2 6

20. Sample size techniques for analytic studies and experiments 1. State the null hypothesis and either a one- or two-sided alternative hypothesis. 2. Select the appropriate statistical test based on the type of predictor variable and outcome variable in those hypotheses. 3. Choose a reasonable effect size (and variability, if necessary). 4. Set α and β . (Specify a two-sided α unless the alternative hypothesis is clearly one-sided.) 5. Use the appropriate table or formula to estimate the sample size.

21. Possible outcomes for tests of hypotheses null hypothesis is true & was rejected (type I error) α null hypothesis is false & was rejected (correct conclusion) null hypothesis is true & was accepted (correct conclusion) null hypothesis is false & was accepted (type II error) β H0 is true H0 is false reject H0 accept H0

22. Statistical Power  Power is the probability that the null hypothesis is rejected, if a specific alternative hypothesis is true. ß represents Type II error, the probability of not rejecting the null hypothesis when the given alternative is true. 1-β = power.  The power of a study should be minimally 80% and often, studies are designed to have 90-95% power to detect a particular clinical effect.

23. What factors affect power? α , (usually 90% or 95 %) ; Confidence describes the test’s ability to minimize type -I errors (false positives ) β , Power: (1 minus type II error): usually 80% or 90% Probability that you don’t fail to reject no impact, that you find impact Power describes the test’s ability to minimize type -II errors (false negatives ) As power increases, the chances to say “no impact” when in reality there is positive impact, declines Power analysis can be used to calculate the minimum sample size required to accept the outcome of a statistical test with a particular level of confidence effect size, variability n

24. Calculating sample size Problem: The research question is whether elderly smokers have a greater incidence of skin cancer than non-smokers. Are view of previous literature suggests that the 5-year incidence of skin cancer is about 0.20 in elderly non-smokers . At α ( two-sided) = 0.05 and power = 0.80, how many smokers and non-smokers will need to be studied to determine whether the 5-year skin cancer incidence is at least 0.30 in smokers ?

25. RISK FACTORS OF DIABETIC NEPHROPATHY AMONG TYPE 2 DIABETES PATIENTS AT UNIVERSITY TEACHING HOSPITAL OF KIGALI (CHUK), RWANDA Two population proportion formula ……using Epi Info Version 7 statistical package software. The following assumptions will be used: 95% confidence interval, 80% power, considering rare case occurrence the ratio of 1 case to 3 controls will be used, Odds Ratio of 6 will be considered from other study that the OR of non-adherence to blood glucose measurement among diabetic nephropathy compare to those free of diabetic nephropathy, probability of not adhered to blood glucose control among diabetic nephropathy is 2.38% and probability of adhered to blood glucose control among patients free of diabetic nephropathy is 12.31% ( Hintsa et al., 2017) and 10% for the non-response rate. After considering all these, the sample size calculated is 308 diabetic patients (77 cases and 231 controls).

26. risk of HIV INFECTION AMONG MEN AGED 50 TO 75 YEARS USING ERECTILE DYSFUNCTION DRUGS ATtending at KENYATTA NATIONAL HOSPITAL

27. Example: Calculating sample size when using the t test Problem : The research question is whether there is a difference in the efficacy of salbutamol and ipratropium bromide for the treatment of asthma. The investigator plans a randomized trial of the effect of these drugs on FEV1 (forced expiratory volume in 1 second) after 2 weeks of treatment. A previous study has reported that the mean FEV1 in persons with treated asthma was 2.0 liters , with a standard deviation of 1.0 liter . The investigator would like to be able to detect a difference of 10% or more in mean FEV1 between the two treatment groups. How many patients are required in each group (salbutamol and ipratropium) at α (two-sided) = 0.05 and power = 0.80? n = ( Z α /2 +Z β ) 2  *2*σ 2  / d 2 ,

28. How to know the power for a specified sample size using soft wares OpenEpi Menu …. EpiInfo https ://www.stat.ubc.ca/~ rollin/stats/ssize/b2.html