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

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

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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

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Key areas Cohort studies (cumulative incidence) Cohort studies (incidence density) Case control studies

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Cohort study for calculation of cumulative incidence Cumulative risk cohorts

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IllNon illTotal ExposedabL 1 Non exposedcd L 0 Totala+cb+d L 1 + L 0 Risk among exposed and unexposed in a cohort study Risk among exposed : R 1 = a/L 1 Risk among unexposed : R 0 = c/L 0 Cumulative risk cohorts

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IllNon illTotal Exposed20424 Non exposed25247 272 Total45251 296 Risk of anthrax among persons exposed and unexposed to slaughtering cows, Sarkarpara, Murshidabad, West Bengal Risk among exposed : R 1 = a/L 1 = 83% Risk among unexposed : R 0 = c/L 0 = 9% Relative risk = 83% / 9% = 9.1 Cumulative risk cohorts

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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

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

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IllNon illTotal Exposed20424 Non exposed25247 272 Total45251 296 Risk of anthrax among persons exposed and unexposed to slaughtering cows, Sarkarpara, Murshidabad, West Bengal Cumulative risk cohorts

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IllNon illTotal Exposed20424 Non exposed25247 272 Total45251 296 Risk of anthrax among persons exposed and unexposed to slaughtering cows, Sarkarpara, Murshidabad, West Bengal Lower limit: Upper limit: Cumulative risk cohorts

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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

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Cohort study for calculation of incidence density Incidence density cohorts

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EventsPerson-yearsRate ExposedaPT 1 Rate 1 Non exposedcPT 0 Rate 0 Totala+cPT Rate Calculation of a relative rate in a cohort study Relative rate = Rate 1 /Rate 0 = (a/PT 1 ) / (c/PT 0 ) Incidence density cohorts

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EventsPerson-yearsRate Exposed3212,000Rate 1 Non exposed2015,000Rate 0 Totala+c27,000 Rate Calculation of a relative rate in a cohort study Rate 1 = 32/ 12,000 = 2.6 per 1,000 PY Rate 0 = 20/ 15,000 = 1.33 per 1,000 PY Relative rate = 2.6/ 1.33 = 1.95 Incidence density cohorts

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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

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

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EventsPerson-yearsRate Exposed3212,000Rate 1 Non exposed2015,000Rate 0 Totala+c27,000 Rate Calculation of a relative rate in a cohort study Incidence density cohorts

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EventsPerson-yearsRate Exposed3212,000Rate 1 Non exposed2015,000Rate 0 Totala+c27,000 Rate Calculation of a relative rate in a cohort study Lower limit: Upper limit:

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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

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Case control study for calculation of odds ratios Case control studies

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Cases ControlsTotal ExposedabN/A Non exposedcdN/A Totala+cb+dN/A Odds ration in a case control study OR = ad/bc Case control studies

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Cases ControlsTotal Exposed3726N/A Non exposed314N/A Total4040N/A Consumption of pump A water among cholera cases and controls, Barwai, Bhopal, Madhya Pradesh, India, 2006 OR = ad/bc = (37x14)/ (3x26) = 6.6 Case control studies

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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

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

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Cases ControlsTotal Exposed3726N/A Non exposed314N/A Total4040N/A Consumption of pump A water among cholera cases and controls, Barwai, Bhopal, Madhya Pradesh, India, 2006 Case control studies

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Cases ControlsTotal Exposed3726N/A Non exposed314N/A Total4040N/A Consumption of pump A water among cholera cases and controls, Barwai, Bhopal, Madhya Pradesh, India, 2006 Lower limit: Upper limit: Case control studies

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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

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

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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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