Confidence interval

1. CONFIDENCE INTERVAL Dr.RENJU

2. OVERVIEW INTRODUCTION CONFIDENCE INTERVAL CONFIDENCE LEVEL CONFIDENCE LIMITS HOW TO SET? FACTORS – SET SIGNIFICANCE APPLICATIONS

3. INTRODUCTION Statistical parameter Descriptive statistics : Describe what is there in our data Inferential statistics : Make inferences from our data to more general conditions

4. Inferential statistics Data taken from a sample is used to estimate a population parameter Hypothesis testing (P-values) Point estimation (Confidence intervals)

5. POINT ESTIMATE Estimate obtained from a sample Inference about the population Point estimate is only as good as the sample it represents Random samples from the population - Point estimates likely to vary

6. ISSUE ??? Variation in sample statistics

7. SOLUTION Estimating a population parameter with a confidence interval

8. CONFIDENCE INTERVAL A range of values so constructed that there is a specified probability of including the true value of a parameter within it

9. CONFIDENCE LEVEL Probability of including the true value of a parameter within a confidence interval Percentage

10. CONFIDENCE LIMITS Two extreme measurements within which an observation lies End points of the confidence interval Larger confidence – W ider interval

11. A point estimate is a single number A confidence interval contains a certain set of possible values of the parameter Point Estimate Lower Confidence Limit Upper Confidence Limit Width of confidence interval

12. HOW TO SET

13. CONCEPTS NORMAL DISTRIBUTION CURVE MEAN ( µ ) STANDARD DEVIATION (SD) RELATIVE DEVIATE (Z)

14. NORMAL DISTRIBUTION CURVE

15. Perfect symmetry Smooth Bell shaped Mean (µ) Median Mode SD( σ ) - 1 Area - 1

16. RELATIVE DEVIATE (Z) Distance of a value (X) from mean value (µ) in units of standard deviation (SD) Standard normal variate

17. Z = x – µ SD

18. CONFIDENCE LIMITS From µ - Z(SD) To µ + Z(SD)

19. CONFIDENCE INTERVAL

20. FACTORS – TO SET CI Size of sample Variability of population Precision of values

21. SAMPLE SIZE Central Limit Theorem “Irrespective of the shape of the underlying distribution, sample mean & proportions will approximate normal distributions if the sample size is sufficiently large” Large sample – Narrow CI

22. SKEWED DISTRIBUTION

23. VARIABILITY OF POPULATION

24. POPULATION STATISTICS Repeated samples Different means Standard normal curve Bell shape Smooth Symmetrical

25. POPULATION STATISTICS

26. Population mean (µ) Standard error - Sampling (SD/√n) Z = x – µ SD/√n Confidence limits From µ - Z(SE) To µ + Z(SE)

27. 95% 95% sample means are within 2 SD of population mean

28. Precision of values Greater precision Narrow confidence interval Larger sample size

29. Precision of values

30. SIGNIFICANCE

31. 95% Significance Observed value within 2 SD of true value

32. Confidence Interval and α Error Type I error Two groups Significant difference is detected Actual – No difference exists False Positive

33. Confidence level is usually set at 95% (1 –  ) = 0.95

34. Margin of Error x

35. Margin of error Reduce the SD ( σ ↓) Increase the sample size (n ↑) Narrow confidence level (1 – ) ↓

36. P value 95% CI corresponds to hypothesis testing with P <0.05

37. SIGNIFICANCE If CI encloses no effect, difference is non significant

38. P value – Statistical significance Confidence Interval – Clinical significance

39. APPLICATIONS CLINICAL TRIALS

40. Margin of error Increase the sample size Reduce confidence level Dynamic relation Confidence intervals and sample size

41. Example Series of 5 trials Equal duration Different sample sizes To determine whether a novel hypolipidaemic agent is better than placebo in preventing stroke

42. Smallest trial  8 patients Largest trial  2000 patients ½ of the patients in each trial – New drug All trials - Relative risk reduction by 50%

43. Questions In each individual trial, how confident can we be regarding the relative risk reduction Which trials would lead you to recommend the treatment unequivocally to your patients

44. More confident - Larger trials CI - R ange within which the true effect of test drug might plausibly lie in the given trial data

45. Greater precision Narrow confidence intervals Large sample size

46. THERAPEUTIC DECISIONS Recommend for or against therapy ?

47. Minimally Important Treatment Effect S mallest amount of benefit that would justify therapy Points

48. Uppermost point of the bell curve Observed effect Point estimate Observed effect

49. Tails of the bell curve Boundaries of the 95% confidence interval Observed effect

50. Trial 1

51. Trial 2

52. CI overlaps the smallest treatment benefit Not Definitive Need narrower Confidence interval Larger sample size

53. Trial 3

54. Trial 4

55. CI overlaps the smallest treatment benefit Not Definitive Need narrower Confidence interval Larger sample size

56. Confidence intervals for extreme proportions Proportions with numerator – 0 Proportions approaching - 1 Proportions with numerators very close to the corresponding denominators

57. Numerator - 0 Rule of 3 Proportion – 0/n Confidence level – 95% Upper boundary – 3/n

58. Example 20 people – Surgery None had serious complications Proportion 0/20 3/n – 3/20 15%

59. Proportions approaching - 1 Translate 100% into its complement

60. Example Study on a diagnostic test 100% sensitivity when the test is performed for 20 patients who have the disease. T est identified all 20 with the disease as positive – 100% No falsely negatives – 0%

61. 95% Confidence level Proportion of false negatives - 0 /20 3/n rule Upper boundary - 15% (3 /20 ) Sensitivity Lower boundary Subtract this from 100% 100 – 15 = 85%

62. Numerators very close to the denominators Rule Numerator X 1 5 2 7 3 9 4 10

63. 95% Confidence level Upper boundary –

64. CONCLUSION Confidence interval Confidence level Confidence limits 95% Observed value within 2 SD Population statistics

65. THANK YOU