# The strength of the association Measures of association FETP India.

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The strength of the association Measures of association FETP India

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

Key areas Cohort studies Case control studies Cross-sectional survey

IllNon-illTotal Exposedaba+b Non-exposedcdc+d Totala+cb+da+b+c+d Presentation of the data of an analytical study in a 2 x 2 table

IllNon-illTotal ExposedabL 1 Non-exposedcd L 0 Totala+cb+d L 1 + L 0 Presentation of the data of an analytical cohort study in a 2 x 2 table Cohorts

IllNon-illTotal ExposedabL 1 Non-exposedcd L 0 Totala+cb+d L 1 + L 0 Calculation of the risk for the whole population in a cohort study Cohorts R = (a+c)/(L 1 + L 0 )

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

Approaches that may be used to compare two risks in a cohort study Additive approach  Risk difference Multiplicative approach  Cumulative incidence Relative risk  Person-time analysis Relative rate  Alternative approach Disease odds ratio Cohorts

Calculation of a risk difference in a cohort study Risk difference = R 1 - R 0 = (a/L 1 ) - (c/L 0 ) IllNon-illTotal ExposedabL 1 Non-exposedcd L 0 Totala+cb+d L 1 + L 0 Cohorts

LeishmaniasisNon-illTotal Water bodies139487626 No water bodies11114125 Total150601751 Risk difference = (139/626) - (11/125) = 22% - 9% = 13% Risk of leishmaniasis according to water bodies within 25 metres of house, Chatrakhali, West Bengal, 2004-6 Cohorts

IllNon-illTotal ExposedabL 1 Non-exposedcd L 0 Totala+cb+d L 1 + L 0 Calculation of a relative risk in a cohort study Cohorts Relative risk = R 1 /R 0 = (a/L 1 ) / (c/L 0 )

LeishmaniasisNon-illTotal Water bodies139487626 No water bodies11114125 Total150601751 Relative risk = (139/626) / (11/125) = 22% / 9% = 2.5 Risk of leishmaniasis according to water bodies within 25 metres of house, Chatrakhali, West Bengal, 2004-6 Cohorts

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

The odds Probability of occurrence of an event ______________________________________ Probability of non-occurrence of this event Cohorts

Calculation of a disease odds ratio in a cohort study Odds of disease in exposed: = (a/L 1 )/(b/L 1 )= a/b Odds of disease in unexposed = (a/L 0 )/(b/L 0 ) = c/d Disease odds ratio (OR): (a/b) / (c/d) = ad/bc IllNon-illTotal ExposedabL 1 Non-exposedcd L 0 Totala+cb+d L 1 + L 0 Cohorts

Reasons to prefer relative risks to diseases odds ratios in cohort studies The relative risk corresponds to an intuitive notion The OR is less meaningful to most readers The OR is larger than the relative risk Cohorts

Calculation of an exposure-odds ratio in a cohort study Odds of exposure in ill: = (a/a+c)/(c/a+c)= a/c Odds of exposure in non-ill = (b/b+d)/(d/d+d 0 ) = b/d Exposure odds ratio (OR): (a/c) / (b/d) = ad/bc IllNon-illTotal ExposedabL 1 Non-exposedcd L 0 Totala+cb+d L 1 + L 0 Cohorts

Magical trick #1 In a cohort study, the ratio of the odds of disease among exposed and unexposed is equal to the ratio of the odds of exposure among ill and non-ill Cohorts

Calculation of a relative risk for a rare disease in a cohort study b # a+b; d # c+d Relative risk = a (c+d)/c(a+b) # ad/bc IllNon-illTotal Exposed a bL 1 Non-exposed c d L 0 Total a+c b+d L 1 + L 0 Cohorts

Magical trick #2 In a cohort study, when the disease is rare, the ratio of the odds of disease is almost equal to the relative risk Cohorts

Case ControlTotal Exposeda.f 1 b.f 2 N/A Non-exposedc.f 1 d.f 2 N/A TotalC 1.f 1 C 0.f 2 N/A Impossibility to calculate a relative risk in a case control study Cases are sampled from all cases (sampling fraction: f 1 ) Controls are sampled from all controls (sampling fraction: f 2 ) f 1 and f 2 are unknown, risks cannot be calculated Case control

Case ControlTotal Exposeda.f 1 b.f 2 N/A Non-exposedc.f 1 d.f 2 N/A TotalC 1.f 1 C 0.f 2 N/A Odds of exposure among cases= (a.f 1 )/(c.f 1 ) Calculation of the exposure-odds ratio in a case control study Odds of exposure among controls = (b.f 2 )/(d.f 2 ) Exposure odds ratio: ad/bc Case control

Magical trick #3 In a case control study, the exposure-odds ratio is equal to the disease-odds ratio of the corresponding cohort Case control

Magical trick #2 + #3 = #4 In a case control study, when the disease is rare, the exposure-odds ratio is almost equal to the relative risk in the corresponding cohort Case control

S. typhusControlsTotal Sleeping in work clothes661379 Changing clothes to sleep563389 Total12246168 Sleeping in work clothes and scrub typhus, Darjeeling, West Bengal, India, 2005-6 Case control Odds ratio = (66x33)/(56x13) =3.0

IllNon-illTotal Exposedab L 1 Non-exposedcd L 0 Totala+cb+da+b+c+d Presentation of the data of an analytical cross-sectional study in a 2 x 2 table Cross sectional studies Prevalence in exposed: P 1, Prevalence in unexposed: P 0 Prevalence ratio = P 1 /P 0 = (a/L 1 ) / (c/L 0 ) Formula identical to the relative risk, but could be calculated both ways

TrachomaNo trachomaTotal Dirty face54337391 Clean face5014591509 Total10417961900 Prevalence ratio = (54/391)/(50/1509) = 4.2 Prevalence of trachoma according to facial hygiene, Burkina Faso, 1991 Cross sectional studies

Limitations of causal inference in analytical cross-sectional studies Prevalent cases Exposure and outcome examined simultaneously Cross sectional studies

Take-home messages Calculate relative risks in cohort studies Calculate odd ratio in case control studies, it will approximate the relative risk if the disease is rare Calculate prevalence ratio in cross-sectional surveys and beware of chicken-and-egg causality issues

Association does not systematically mean causation: Potential explanations for an association ?Bias ?Chance ?Confounding factor ?Causation  After the first three have been ruled out

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