# Confidence interval

Senior Resident at Dr Renju Ravi
Jul. 7, 2015

### Confidence interval

1. CONFIDENCE INTERVAL Dr.RENJ U
2. OVERVIEW INTRODUCTION CONFIDENCE INTERVAL CONFIDENCE LEVEL CONFIDENCE LIMITS HOW TO SET? FACTORS – SET SIGNIFICANCE APPLICATIONS
3. INTRODUCTIO N 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 – Wider
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 Confidenc e Limit Width of confidenceinterval
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 0
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 n σ zME α/ 2 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. SIGNIFICANC E 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. QUESTION S 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 - Range 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 Smallest 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. EXAMPL E 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. EXAMPL E Study on a diagnostic test 100% sensitivity when the test is performed for 20 patients who have the disease. Test 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