Sample Size. Vorasith Sornsrivichai, MD., FETP Epidemiology Unit, Faculty of Medicine Prince of Songkla University

Transcription

1 Sample Size Vorasith Sornsrivichai, MD., FETP Epidemiology Unit, Faculty of Medicine Prince of Songkla University

2 All nature is but art, unknown to thee; All chance, direction, which thou canst not see; All discord, harmony not understood; All partial evil, universal good; And spite of pride, in erring reason's spite, One truth is clear, Whatever is, is right ~ Alexander Pope ~

3 How Much Is Enough? Is sample size of 30 subjects enough? If I sampling 0% of population will it be OK? Can I just use all 4 patients I have? 3

4 Objectives To learn how to calculate the sample size needed to obtain a specified precision for an estimate of a parameter To learn how to calculate the sample size needed to provide a specified power for a comparative study 4

5 Outline of Presentation Review of basic principle Determination of sample size Sample size calculation 5

6 Source: 6

7 Two Types Of Study Objective Estimation: Approximation of some parameters (magnitude or difference or ratio) Critical feature is the precision of the estimation. e.g. A public health officer seeks to estimate the proportion of children in the district receiving vaccinations. Hypothesis testing: Examination of proposed assumption Critical feature is the power of the study e.g. Is drug B more effective than drug A? 7

8 Determinants of The Sample Size Effect size Level of significance Power of the test Variation of the outcome 8

9 Other Determinants of The Sample Size Research questions and objective of the study Defining the population and the population size Type of outcome e.g. dichotomous, continuous Outcome measurement e.g. single, repeated measurements Sampling technique e.g. cluster sampling Type of statistical methods Type of analysis e.g. subgroup analysis Non-responses or lost to follow-up 9

10 Effect Size RR, OR, RD, etc. The higher the effect size, the lower the sample size needed 0

11 To err is human (, to forgive divine) ~ Alexander Pope~

12 Errors Study Truth Results H o is not true H 0 is true β Reject H o Power α Type I error Fail to reject H 0 β Type II error α Confidence

13 Significance False detection of difference/association by chance or Type I error α Statistical significance VS Epidemiological & Clinical significance 3

14 Power of the test β Power (- ) is the probability of rejecting H o when H o is not true H a H 0 Study number Power = 9/0 *00 = 90% 4

15 Knowledge is an unending adventure at the edge of uncertainty. ~ Jacob Bronowski ~

16 Uncertainty Variability in the population: not all samples would give exactly the same finding, i.e., there is uncertainty in making an inference However, the uncertainty can usually be quantified Uncertainty can be reduced by using a sufficiently large sample 6

17 Population Sample n = n = 5 n = 0

18 Central Limit Theorem If samples are drawn from a non-normally distributed parent population, the frequency distribution of the population of sample means approaches the normal distribution as the sample size increases. Population Sample n = n = 5 n = 0

19 Sampling Distributions As the sample size increases: the sample means tend to be distributed normally the width of the distribution decreases As the number of samples increases: the mean of the distribution of sample means tends to the mean of the population The above is also true for sample estimates of population proportion as long as the proportion is not too close to 0 or 9

20 Standard Normal Distribution X-3SE X-SE X-SE X X+SE X+SE X+3SE 0

21 Estimation Big n Small n Narrow SE Wide SE Distribution of estimate of the means from many samples

22 Estimation (large sample) Range of population values of X bar compatible with our study value d d Study value of X bar d = precision Sampling distributions from populations with various values of X bar

23 Estimation (small sample) Range of population values of X bar compatible with our study value d d Study value of X bar d = precision Sampling distributions from populations with various values of X bar 3

24 Population μ~50 cm. σ~ 5 cm. Population μ~50 cm. σ~ 0 cm. x Estimate of mean height α = 0.05 d=3 cm. x n= n=45 d d X Distribution of means of hypothetical samples

25 Population N= Population N= σ σ σ σ Data Sample A n=00 μ SD A ~ σ Data Sample B n=5 μ SD B ~ σ Estimation Uncertainty in measure sample A X A SE A =SD A / 00 Estimation Uncertainty in measure sample B X B SE B =SD B / 5 X X 5

26 Sample Size Calculation

27 Sample Size Calculation Available tables Nomogram Manual calculation Software: EpiInfo, STATA, R, OpenEpi 7

28 Available Table e.g. sample size to estimate P within d absolute percentage points with 99% confidence 8

29 Nomogram 9

30 OpenEpi Open Source Epidemiologic Statistics for Public Health

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49 Considerations The appropriate sample size may not be the same for all objectives in a study. Therefore calculate for all objectives then decide All sample size calculations considered here and in most computer programs assume simple random sampling Other sampling method e.g. cluster sampling may require adjustments 49

50 Considerations Calculated sample size is the minimum sample needed Add more (~0 30%) for non-response and lost to follow up E.g. suppose 0% of subjects in the study are expected to refuse to participate or to drop out before the study ends. The total number of n/(-0.) eligible subjects would have to be approached in the first instance 50

51 Inappropriate Sample Size Too SMALL wide CI unable to detect a real effect may miss important association Too BIG waste of reource (effort, time, money) even very small effects become statistical significant may be unethical 5

52 Although our intellect always longs for clarity and certainty, our nature often finds uncertainty fascinating. ~ Karl von Clausewitz ~

53 Sample Size Calculation One sample Estimating: proportion, mean Hypothesis testing: proportion, mean Two sample Estimating: difference between two proportions, two means Hypothesis testing: difference between two proportions, two means 53

54 Sample size calculations for estimation are based on : d = Z α / SE d /α In each case, we just put in the appropriate expression for standard error e.g. SD/ n 54

55 Estimating A Population Mean d = Z α / SE d = Z σ α/ n SE = σ n n = Z α/ σ d 55

56 Example (Estimating a population mean) An estimate is desired of the average retail price of 0 tablets of a tranquilizer. It is required to be within 0 % of the true average price with 95 %CI. The SD in price was estimated as 85 %. How many pharmacies should be randomly selected? n = Z α / d σ n = (.96) (0.85) /(0.) ~78 56

57 Estimating A Population Proportion d = Z α / SE SE = p( n p) n = Z α/ p( d p) 57

58 Example (Estimating a population proportion) A district public health officer seeks to estimate the proportion of children in the district receiving appropriate childhood vaccinations. How many children must be studied if the resulting estimate is to fall within 0 % of the true proportion with 95% CI. n = Z p( - p )/d α / n = (.96) (0.5)/(0.) =

59 Parameter Estimation The sample selected will be largest when P = 0.5 When one has no idea what the level of P is in the population, choosing 0.5 for P will always provide enough observations. P P(-P)

60 α Hypothesis Testing mean of B - mean of A Δ * Δ is minimum effect worth detecting β * = ( Δ Z Δ= ( Z a/ a/ SE SE 0 0 ) + ( Z ) = ( Z β β SE) SE) 60

61 Basic Equations Underlying Sample Size d Z = α / SE Δ = Δ = Z α / SE( Z SE If SE 0 = SE = SE then α/ Z β Z ) β SE a b Most sample size calculations for estimation and hypothesis testing are based on these equations. 6

62 How to Choose Δ Δ should be the minimum difference of clinical significance, or the minimum difference worth detecting. Previously reported differences may not be suitable for your study. It may be useful to consider the standardized effect size (ε = Δ/σ) when the outcome is a continuous variable. 6

63 Estimating The Difference Between Two Means d = Z α / n = SE ( + r ) If n n Z d SE α/ = SE = σ + n = σ σ n r + then n r n 63

64 Example 3 (Estimating the difference between two means) Nutritionists wish to estimate the difference in caloric intake at lunch between children in a school offering hot lunches and children in a school which does not. From other studies, they estimate that the SD of caloric intake among schoolchildren is 75 calories, and they wish to make their estimate to within 0 calories of the true difference with 95% confidence. (Equal numbers in each group) n [ + / r] Z = d n = * (.96) * 75 / 0 = ~ 09 (Note that r = n /n ) α / σ 64

65 65 Estimating the difference between two proportions r n p p n p p SE ) ( ) ( + = ) ( ) ( d r p p p p Z n + = α/ SE Z d = α / + = n p r p p p Z d ) ( ) ( α

66 Example 4 (Estimating the difference between two proportions) It is desired to estimate a risk difference in two industrial groups. How large a sample should be selected in each group for the estimate to be within 5 percentage points of the true difference with 95% confidence. It was observed that P = 0.4, P = 0.3. (Equal numbers in each group) Z / [ p ( p ) p ( p n α + = d ) / r] n =.96 [(0.40)(0.60) + (0.3)(0.68)]/(0.05) ~

67 Testing the hypothesis of a difference between two means ( ) Z Δ = SE + Z α / β Δ = n + n r SE = σ ( ) Z Z - + α β n σ + n r Δ = + nr r ( Z Z ) -α + β σ n = ( + r ) ( Z + ) α/ Z β Δ σ 67

68 Example 5 (Testing the hypothesis of a difference between two means) A study is being designed to measure the effect, on systolic blood pressure, of lowering sodium in the diet. From a pilot study it is observed that the SD of SBP in a community with high sodium diet is mm Hg, while that in a group with low sodium diet is 0.3 mm Hg. If alpha is 0.05 and beta is 0.0, how large a sample from each community should be selected in order to detect a mm Hg difference in blood pressure between the communities? (Equal group size and use pooled variance) n = ( + / r)( Z α / + Z β ) Δ n = [ ] (5.05) / = ~ 657 σ 68

69 69 Testing the hypothesis of a difference between two proportions) Ho true: Ho not true: 0 SE Z SE Z + = Δ β α / / ] [ / ) ( ) ( ) / )( ( Δ = r p p p p Z r p p Z n β α n r p p SE + = ) ( 0 ( ) ( ) r rp p p + + = rn p p n p p SE ) ( ) ( + = a n r p p p p Z n r p p Z ) ( ) ( ) )( ( = Δ β

70 Example 6 (Testing the hypothesis of a difference between two proportions) A case-control study is to be conducted with a case:control ratio of :. Exposure to the potential risk factor of interest among controls is expected to be 0%. How many cases and controls will be needed to detect an odds ratio of at least.0, at a significance level of 0.05 with a power of 80 percent? (Let n=number of cases, and n = number of controls) n = [ Z α / p( p)( + / r) + Z β p( p) + p ( p ) / r ] Δ p = ( p + rp ) /( + r) n [ ( 0.43)( + / ) ( 0.33) + 0.0( 0.0) / ] = n = 3 and n = * 3 = 64 ( ) 70

71 7 Power Determination Power = - =-function(a)=-p(z ) Continuous data (n t =n c ) (n t n c ) c t c t t c c t c n n n n Z A n Z A / ) ( / / / σ μ μ σ μ μ α α + = = β β

72 Exercise To compare a new antihypertensive drug with the standard treatment (n=50 in each group). The difference in BP treated with these two drugs was 4 mmhg. The variance was 40 mmhg. The significant level was 0.5. The researcher found no difference in these two drugs. Do you agree with this conclusion? A= 4.96 = (40 ) /50 Power = -function(a) =-f( ) = =0.835 Power = 83.5% 7

73 73 Power Determination Discrete data and proportion n t =n c n t n c c t t c c c t c t c c t t c c t c c n P Q n P Q P P n PQ n PQ Z A n P Q P Q P P n PQ Z A / ) / / / / ) / + + = + = α α P Q P P P c t = + = ) ( /

74 Exercise To compare between kinds of anti UV cream, A and B. Seventy five of 00 patients treated with cream A whereas 65 out of 00 patients treated with cream B improved. The researcher concluded that these two kinds of cream were not different at 5% of level of significance. Do you agree with this conclusion? A =.645 (.70 )(.30 ) / [(0.65)(0.35) + (0.75)(0.5)] /00 = 0.06 Power = - f(0.06) = = 0.54 ~54% 74

75 Interpretations Of Negative Findings - Power Calculations For a hypothesis-testing study which fails to reject the null hypothesis, it is useful to conduct a posthoc power calculation. We can use a rearrangement of the relevant sample-size equation. This should be done using the clinically relevant difference for the Δ of the equation (not the difference found in the study). 75

76 Power depends on: β the size of difference the treatment makes the rates of events among control patients the alpha level in use β the number of patients in the trial Nonrejection region Rejection region 76