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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
Ill Non ill Total
Exposed a b L1
Non exposed c d L0
Total a+c b+d L1 + L0
Risk among exposed and unexposed
in a cohort study
Risk among exposed : R1 = a/L1
Risk among unexposed : R0 = c/L0Cumulative risk cohorts
Ill Non ill Total
Exposed 20 4 24
Non exposed 25 247 272
Total 45 251 296
Risk of anthrax among persons exposed and unexposed to
slaughtering cows, Sarkarpara, Murshidabad, West Bengal
Risk among exposed : R1 = a/L1= 83%
Risk among unexposed : R0 = c/L0 = 9%
Relative risk = 83% / 9% = 9.1Cumulative 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
€
RR=RR* e±1.96 ( )Var lnRR
€
Var(ln RR) =1a−
1a+ b
+1c−
1c+ d
Formula of the 95% confidence interval
• Confidence interval of relative risk
• Formula of the variance
Cumulative risk cohorts
Ill Non ill Total
Exposed 20 4 24
Non exposed 25 247 272
Total 45 251 296
Risk of anthrax among persons exposed and unexposed to
slaughtering cows, Sarkarpara, Murshidabad, West Bengal
€
Var(ln RR) =120
−1
20 + 4+
125
−1
25 + 272=0.0446
Cumulative risk cohorts
Ill Non ill Total
Exposed 20 4 24
Non exposed 25 247 272
Total 45 251 296
Risk of anthrax among persons exposed and unexposed to
slaughtering cows, Sarkarpara, Murshidabad, West Bengal
€
RRL =9.1* e−1.96 0.0446
=6.0
€
RRU =9.1* e+1.96 0.0446
=14
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
Events Person-years Rate
Exposed a PT1 Rate1
Non exposed c PT0 Rate0
Total a+c PT Rate
Calculation of a relative rate in a cohort study
Relative rate = Rate1/Rate0 = (a/PT1) / (c/PT0)
Incidence density cohorts
Events Person-years Rate
Exposed 32 12,000 Rate1
Non exposed 20 15,000 Rate0
Total a+c 27,000 Rate
Calculation of a relative rate in a cohort study
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
• Confidence interval of relative rate
• Formula of the variance
Formula of the 95% confidence interval
€
RR=RR* e±1.96 Var(lnRR)
€
Var(ln RR) =1a+1c
Incidence density cohorts
Events Person-years Rate
Exposed 32 12,000 Rate1
Non exposed 20 15,000 Rate0
Total a+c 27,000 Rate
Calculation of a relative rate in a cohort study
€
Var(ln RR) =132
+120
=0.0812
Incidence density cohorts
Events Person-years Rate
Exposed 32 12,000 Rate1
Non exposed 20 15,000 Rate0
Total a+c 27,000 Rate
Calculation of a relative rate in a cohort study
€
RRL =1.95 * e−1.96 0.0812
=1.12
€
RRU =1.95 * e+1.96 0.0812
=3.41
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
Cases Controls Total
Exposed a b N/A
Non exposed c d N/A
Total a+c b+d N/A
Odds ration in a case control study
OR = ad/bc
Case control studies
Cases Controls Total
Exposed 37 26 N/A
Non exposed 3 14 N/A
Total 40 40 N/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
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
• Confidence interval of odds ratio
• Formula of the variance
Formula of the 95% confidence interval
€
OR=OR* e±1.96 Var(lnOR)
€
Var(ln OR) =1a+1b+1c+1d
Case control studies
Cases Controls Total
Exposed 37 26 N/A
Non exposed 3 14 N/A
Total 40 40 N/A
Consumption of pump A water among cholera cases and controls, Barwai, Bhopal, Madhya Pradesh,
India, 2006
€
Var(ln RR) =137
+126
+13+
114
=0.457
Case control studies
Cases Controls Total
Exposed 37 26 N/A
Non exposed 3 14 N/A
Total 40 40 N/A
Consumption of pump A water among cholera cases and controls, Barwai, Bhopal, Madhya Pradesh,
India, 2006
€
ORL =6.6 * e−1.96 0.457
=1.76
€
ORU =6.6 * e+1.96 0.457
=25
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)
€
RR=RR1±1.96 χ 2
€
OR=OR1±1.96 χ 2
• 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