---
title: Part III Measures of association
author: Arindam Basu
tags: epidemiology, hlth301, hlth214, hlth403
---

# Measures of Association

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## We will learn
- Attributable risk
- Attributable risk percent
- Relative Risk
- Odds and Odds Ratio
- Population Attributable risk%

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## Attributable Risk
- Risk in exposed = Incidence among exposed
- Risk in non-exposed = Incidence among non-exposed
- Attributable Risk = Risk in exposed - Risk in non-exposed
- **Excess Risk** attributable to exposure
- Similarly **Excess Risk** attributable to interventions

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## Illustration
- Incidence of asthma among those exposed to air pollution: 19 per 1000 person-years
- Incidence of asthma among those NOT exposed to air pollution: 10 per 1000 person-years
- Attributable risk of asthma with air pollution: 9 per 1000 person-years

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## Why is attributable risk important?
- Helps the provider to plan interventions
- Excess risk helps us to understand how much actual difference exists
- 10% change for a rare disease and a common disease carry different meanings

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## Illustration of excess risk, low frequency
- Disease incidence 1 in 1000 person-years at baseline
- With exposure, say there is a 20% jump in the incidence
- With exposure, the disease would be 1.2 per 1000 person-years

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### Illustration of excess risk, high freq
- Disease risk 10 in 1000 person-years at baseline
- Exposure leads to 20% jump in the incidence
- Post exposure, disease risk 12 per 1000 person-years

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## Attributable risk percent
- What percent of asthma among those who are exposed is attributed to air pollution?
- 9/19 = 47.3% of all exposed cases are due to air pollution
- Proportion of disease among exposed attributed to an exposure

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## Relative Risk (RR)
- Risk in exposed / Risk in non-exposed
- Measure used for causal inference
- Rate Ratio (ratio of incidences)
- Odds Ratio (ratio of two odds)

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## Illustration of Relative Risk
- Risk of asthma with air pollution 19 per 1000 person-years
- Risk of asthma without air pollution 10 per 1000 person-years
- Risk Ratio is 19 / 10 or 1.9
- Those exposed to air pollution are 1.9 times
- Risk Ratio same as Rate Ratio (same thing)

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## Concept of Odds and Odds Ratio
- Odds is a ratio of two probabiities
- Probability that an event E will occur is p(E)
- Probability that event E will not occur is (1 - p(E))
- Odds = p(E) / (1 - p(E))

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## Illustration of Odds
- Say there is a 10% chance of rain tomorrow
- Rain is an **event**
- Probability of rain is p(Rain) = 10%
- Probability of no rain tomorrow is 90%
- We say p(NO Rain) = 90%
- Odds of Rain = 10: 90 (ten against 90)

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## Odds Ratio
- Ratio of two odds
- Odds of **Exposure** IF *Diseased*
- Written as Odds(**E**xposure | **D**isease)
- Odds of **Exposure** IF *Non Diseased*
- Written as Odds(**E**xposure | **No D**isease)
- **Odds Ratio = Odds(E | D) / Odds(E | No D)**

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## Illustration of Odds Ratio

| Smoking | Lung Cancer | No Cancer |
| ---------- | ----------- | --------- |
| Smoker | 200 | 40 |
| Non-smoker | 20 | 180 |
| Total | 220 | 220 |

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## Odds of Smoking for those with cancer
- Out of 220 **with cancer**,
- 200 were smokers
- Probability of smoking = 200/220
- Probability of NOT smoking = (1 - 200/220) = 20/ 220
- Odds of **smoking** = (200/220) / (1 - 200/220)
- **Odds of Smoking IF CANCER = 200 / 20**

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## Odds of Smoking for those with NO Cancer
- Out of 220 with NO Cancer,
- 40 were smokers
- Probability of smoking = 40 / 220
- Probability of NOT smoking = 180 / 220
- **Odds of Smoking IF NO Cancer = 40 / 180**

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## Odds Ratio of Smoking and Cancer
- Odds of **Smoking IF CANCER**
- Divided by
- Odds of **Smoking IF NO CANCER**
- (200 / 20 ) DIVIDED BY (40 / 180)
- **(200 * 180) / (40 * 20) = 45**
- Smokers are 45 times at risk of Cancer compared with Non-smokers

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## Meaning of RR and OR
- OR and RR are essentially same
- particularly for rare disease conditions
- If RR or OR > 1, indicates risk
- If RR or OR < 1, indicates protective effect

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## How to use RR and OR
- RR or OR helps to indicate strength of association
- Using prevalence of exposure,
- RR and OR can be used to estimate PAR%
- PAR% = Population attributable risk %
- How much of the disease can be reduced
- If the risk factor can be completely eliminated
- Assuming a cause and effect association exists

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## Theory of PAR%
- In a study, we can have
- Everyone in the exposed group to be exposed
- Everyone in the non-exposed group to be not exposed
- This does not happen in real life
- In real life only some in the population are exposed
- Prevalence of exposure can be plugged in a formula
- Also called Population Attributable Fraction

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## Formula of PAR%

- **((pE * (RR - 1)) / (1 + pE * (RR - 1))) * 100**
- Where pE = prevalence of exposure
- RR = Relative Risk (also could be OR)

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## Illustration with air pollution example
- In the society about 20% were exposed to high air pollution
- pE = 0.20
- RR = 1.9
- PAR% = ((0.20 * 0.90) / (1 + 0.20 * 0.90)) * 100
- **PAR% = (0.18 / 1.8) * 100 = 10%**
- If air pollution were to be completely eliminated, it would reduce asthma by 10%

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## Illustration with smoking and lung cancer
- In a community, about 20% people smoke
- pE = 0.20
- RR = 45
- PAR% = (0.20 * 44 / (1 + 0.20 * 44)) * 100
- **PAR% = 89.8%**
- If smoking could be eliminated, it would reduce lung cancer by 89.8% or roughly by 90%!

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## Summary
- We learned about AR, RR, OR, PAR%
- Attributable Risk is the risk difference
- Attributable risk % tells us how much of the risk is accounted for by the exposure
- RR is a relative risk estimate, it can be expressed in two ways:
- Rate Ratio or Odds Ratio
- PAR% tells us how much we would be able to reduce the health effect if we are able to eliminate the risk factor