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## How strong is strong

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**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**• 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**Key areas**• Cohort studies (cumulative incidence) • Cohort studies (incidence density) • Case control studies**Cohort study for calculation of cumulative incidence**Cumulative risk cohorts**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**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**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**Formula of the 95% confidence interval**• Confidence interval of relative risk • Formula of the variance Cumulative risk cohorts**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**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**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**Cohort study for calculation of incidence density**Incidence density cohorts**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**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**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**Formula of the 95% confidence interval**• Confidence interval of relative rate • Formula of the variance Incidence density cohorts**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**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:**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**Case control study for calculation of odds ratios**Case control studies**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**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**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**Formula of the 95% confidence interval**• Confidence interval of odds ratio • Formula of the variance Case control studies**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**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**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**Formula for the test-based 95% confidence intervals (For**larger studies) • Cohort study • Case control study**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