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# How strong is strong

How strong is strong. Confidence intervals for measures of associations FETP India. Competency to be gained from this lecture. Calculate the confidence intervals of the measure of association that corresponds to a study design . Confidence intervals for measures of associations.

## How strong is strong

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1. How strong is strong Confidence intervals for measures of associations FETP India

2. Competency to be gained from this lecture Calculate the confidence intervals of the measure of association that corresponds to a study design

3. Confidence intervals for measures of associations • Exact method (Requires computer) • Provides largest confidence intervals • Maximizes the beta error • Semi-exact (Taylor’s series) • Based on the variance of the Log of the measure of association • Described in this lecture • Test-based (Miettinen) • Based on the Chi-square • Provides the most narrow confidence interval • Maximizes the power

4. Key areas • Cohort studies (cumulative incidence) • Cohort studies (incidence density) • Case control studies

5. Cohort study for calculation of cumulative incidence Cumulative risk cohorts

6. Risk among exposed and unexposed in a cohort study Ill Non ill Total Exposed a b L1 Non exposed c d L0 Total a+c b+d L1 + L0 Risk among exposed : R1 = a/L1 Risk among unexposed : R0 = c/L0 Cumulative risk cohorts

7. Risk of anthrax among persons exposed and unexposed to slaughtering cows, Sarkarpara, Murshidabad, West Bengal Ill Non ill Total Exposed 20 4 24 Non exposed 25 247 272 Total 45 251 296 Risk among exposed : R1 = a/L1= 83% Risk among unexposed : R0 = c/L0 = 9% Relative risk = 83% / 9% = 9.1 Cumulative risk cohorts

8. Interpretation of the relative risk • Those exposed to cow slaughtering have a risk of illness that is 9.1 greater than those who were not exposed • The sample size is limited • Could this association be an effect of chance alone? • Calculation of the confidence interval Cumulative risk cohorts

9. Formula of the 95% confidence interval • Confidence interval of relative risk • Formula of the variance Cumulative risk cohorts

10. Risk of anthrax among persons exposed and unexposed to slaughtering cows, Sarkarpara, Murshidabad, West Bengal Ill Non ill Total Exposed 20 4 24 Non exposed 25 247 272 Total 45 251 296 Cumulative risk cohorts

11. Risk of anthrax among persons exposed and unexposed to slaughtering cows, Sarkarpara, Murshidabad, West Bengal Ill Non ill Total Exposed 20 4 24 Non exposed 25 247 272 Total 45 251 296 Lower limit: Upper limit: Cumulative risk cohorts

12. Interpretation of the 95% confidence interval of the relative risk • Those exposed to cow slaughtering have a risk of illness that is 9.1 greater than those who were not exposed • While this estimate is based upon a sample, there is a 95% probability that the real relative risk lies between 6.0 and 14 • The same formula applies for analytical cross sectional studies Cumulative risk cohorts

13. Cohort study for calculation of incidence density Incidence density cohorts

14. Calculation of a relative rate in a cohort study Events Person-years Rate Exposed a PT1 Rate1 Non exposed c PT0 Rate0 Total a+c PT Rate Relative rate = Rate1/Rate0 = (a/PT1) / (c/PT0) Incidence density cohorts

15. Calculation of a relative rate in a cohort study Events Person-years Rate Exposed 32 12,000 Rate1 Non exposed 20 15,000 Rate0 Total a+c 27,000 Rate Rate1 = 32/ 12,000 = 2.6 per 1,000 PY Rate0 = 20/ 15,000 = 1.33 per 1,000 PY Relative rate = 2.6/ 1.33 = 1.95 Incidence density cohorts

16. Interpretation of the relative rate • Those exposed have a rate of illness that is 1.95 greater than those who were not exposed • The sample size is limited • Could this association be an effect of chance alone? • Calculation of the confidence interval Incidence density cohorts

17. Formula of the 95% confidence interval • Confidence interval of relative rate • Formula of the variance Incidence density cohorts

18. Calculation of a relative rate in a cohort study Events Person-years Rate Exposed 32 12,000 Rate1 Non exposed 20 15,000 Rate0 Total a+c 27,000 Rate Incidence density cohorts

19. Calculation of a relative rate in a cohort study Events Person-years Rate Exposed 32 12,000 Rate1 Non exposed 20 15,000 Rate0 Total a+c 27,000 Rate Lower limit: Upper limit:

20. Interpretation of the 95% confidence interval of the relative rate • Those exposed have a rate of illness that is 1.95 greater than those who were not exposed • While this estimate is based upon a sample, there is a 95% probability that the real relative rate lies between 1.1 and 3.4 Incidence density cohorts

21. Case control study for calculation of odds ratios Case control studies

22. Odds ration in a case control study Cases Controls Total Exposed a b N/A Non exposed c d N/A Total a+c b+d N/A OR = ad/bc Case control studies

23. Consumption of pump A water among cholera cases and controls, Barwai, Bhopal, Madhya Pradesh, India, 2006 Cases Controls Total Exposed 37 26 N/A Non exposed 3 14 N/A Total 40 40 N/A OR = ad/bc = (37x14)/ (3x26) = 6.6 Case control studies

24. Interpretation of the odds ratio • The odds of exposure to the pump water is 6.6 higher for cholera cases than for controls • Since the disease is rare, we infer that the risk of cholera is 6.6 higher for those who drank water from the pump • The sample size is limited • Could this association be an effect of chance alone? • Calculation of the confidence interval Case control studies

25. Formula of the 95% confidence interval • Confidence interval of odds ratio • Formula of the variance Case control studies

26. Consumption of pump A water among cholera cases and controls, Barwai, Bhopal, Madhya Pradesh, India, 2006 Cases Controls Total Exposed 37 26 N/A Non exposed 3 14 N/A Total 40 40 N/A Case control studies

27. Consumption of pump A water among cholera cases and controls, Barwai, Bhopal, Madhya Pradesh, India, 2006 Cases Controls Total Exposed 37 26 N/A Non exposed 3 14 N/A Total 40 40 N/A Lower limit: Upper limit: Case control studies

28. Interpretation of the 95% confidence interval of the odds ratio • Those exposed to the water from pump A have a risk of illness that is 6.6 greater than those who were not exposed • While this estimate is based upon a sample, there is a 95% probability that the real relative risk lies between 1.76 and 25 Case control studies

29. Formula for the test-based 95% confidence intervals (For larger studies) • Cohort study • Case control study

30. Take home messages • The 95% confidence interval generates a range of measures of association within which 95% of the values will fall • Three main methods are available, with an increasing degree of risk of alpha error (and decreasing degree of risk of beta error) • Statistical calculators (e.g., Epi-Info Statcalc) allow these calculations from a 2x2 table

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