MW

GPH 380E Foundations of Biostatistics and Epidemiology - Notes

Absolute Risk

  • Absolute Risk (AR): The incidence of disease in a population. It is the probability that a person in a given population will develop the disease over a specified period.
  • Use: Useful in clinical decision-making or policy-making to understand burden, planning, and baseline levels.
  • Limitation: AR does not imply causation and provides no direct comparison between exposed and unexposed groups.
  • Example: 83% of people who ate egg salad got sick vs. 30% who didn’t eat it.
  • Absolute risk for exposed: AR_{exposed} = 0.83 = 83\%.

Comparing Risks

  • To determine if exposure is associated with disease, compare:
    • Risk Ratio (Relative Risk, RR)
    • Risk Difference (Excess Risk, i.e., Absolute Risk Increase)
  • Attack Rate (AR): Quantifies risk in the exposed or at-risk population:
    • Formula: AR = \frac{\text{Number of new cases}}{\text{Population at risk}} \times 100

Difference vs. Ratio: Example data (Table 12.3)

  • Community A:
    • Exposed risk = 40%
    • Unexposed risk = 10%
    • Relative Risk (RR) = 4.0
    • Excess risk (Risk Difference) = 30%
  • Community B:
    • Exposed risk = 90%
    • Unexposed risk = 60%
    • Relative Risk (RR) = 1.5
    • Excess risk (Risk Difference) = 30%

Difference in interpretation of Difference vs. Ratio

  • Risk Difference (Absolute Risk Increase):
    • Tells how many more people per 100 got the disease because of exposure.
    • In both communities, 30 extra people per 100 exposed individuals got sick compared to the unexposed.
  • Relative Risk (Strength of Association):
    • Tells how much more likely exposed individuals are to develop the disease relative to the unexposed baseline.
    • Community A: Exposure increases risk 4-fold (RR = 4.0).
    • Community B: Risk increases 1.5-fold (RR = 1.5).

What story do these numbers tell?

  • Community A:
    • Background risk among the unexposed is low (10%).
    • Jump to 40% among exposed is a dramatic relative increase; RR = 4.0 indicates a strong association.
  • Community B:
    • Baseline risk among the unexposed is already high (60%).
    • Rise to 90% among exposed is a smaller relative jump; RR = 1.5 suggests a weaker association relative to the baseline.

Key Takeaway

  • Risk Difference vs Relative Risk:
    • Risk Difference: absolute burden (how many more cases per population unit).
    • Relative Risk: strength of association (how much more likely the exposed group is to be diseased).
  • Context matters:
    • Public health planning often emphasizes Risk Difference to estimate burden.
    • Causal inference or scientific understanding often emphasizes Relative Risk to judge strength of association.

Interpreting Relative Risk (RR) (Table 12.4)

  • If RR = 1:
    • Risk in exposed = risk in unexposed; no association.
  • If RR > 1:
    • Risk in exposed greater than risk in unexposed; positive association; possibly causal.
  • If RR < 1:
    • Risk in exposed less than risk in unexposed; negative association; possibly protective.

Calculating Relative Risk in Cohort Studies (Table 12.5)

  • Study design: Cohort
  • Setup (2x2 table):
    • Exposed vs Not exposed
    • Disease yes vs Disease no
  • Entries: a, b, c, d where:
    • a = disease in exposed
    • b = no disease in exposed
    • c = disease in nonexposed
    • d = no disease in nonexposed
  • Incidence in exposed: Incidence_{exposed} = \frac{a}{a+b}
  • Incidence in nonexposed: Incidence_{unexposed} = \frac{c}{c+d}
  • Relative Risk: RR = \frac{\dfrac{a}{a+b}}{\dfrac{c}{c+d}}
  • Alternative representation: RR = \frac{\text{Incidence in exposed}}{\text{Incidence in unexposed}}

Another view of the 2x2 in words

  • In a cohort table: Exposed column shows incidence in exposed (a/(a+b)); Not exposed column shows incidence in unexposed (c/(c+d)); RR is the ratio of these two incidences.

Odds Ratio (OR)

  • What is the Odds Ratio (OR)?
    • OR is a measure of association between exposure and disease.
    • It compares the odds of exposure among cases to the odds of exposure among controls.
    • Can be used in both cohort and case-control studies.

Why use Odds Ratio (OR) instead of Relative Risk (RR)?

  • RR requires incidence data in exposed and unexposed groups, which is straightforward in cohort studies.
  • In case-control studies, we start with diseased (cases) and non-diseased (controls), so incidence rates are unknown.
  • Therefore, the odds ratio is the best available measure of association in case-control designs.

Calculating Odds Ratios (OR)

  • Standard 2×2 table (Exposed vs Not Exposed; Disease Yes vs No):
    • Entries: a (exposed with disease), b (exposed without disease), c (unexposed with disease), d (unexposed without disease)
  • OR formula: OR = \frac{a \times d}{b \times c}

Interpreting OR

  • OR = 1 (\rightarrow) No association
  • OR > 1 (\rightarrow) Positive association (potential risk factor)
  • OR < 1 (\rightarrow) Negative association (potential protective factor)

When is OR a good estimate of RR?

  • OR approximates RR well only when the disease is rare (low incidence).
  • This is known as the rarity assumption.
  • Conditions for the rarity assumption:
    • Cases represent all people with the disease in the population.
    • Controls represent all people without the disease in the population.
    • The disease is infrequent in the population.

Summary of key equations

  • Absolute Risk (AR) in a population: AR = \frac{\text{Number of new cases}}{\text{Population at risk}} \times 100
  • Risk Difference (Absolute Risk Increase): ARD = AR{\text{exposed}} - AR{\text{unexposed}}
  • Relative Risk (RR): RR = \frac{\dfrac{a}{a+b}}{\dfrac{c}{c+d}}
  • Attack Rate (AR) (definition): as above in context of incidence in a population at risk
  • Odds Ratio (OR): OR = \frac{a \times d}{b \times c}
  • Interpretation of RR: RR = 1 \Rightarrow \text{no association}; RR > 1 \Rightarrow \text{positive association}; RR < 1 \Rightarrow \text{negative association}
  • Interpretation of OR: OR = 1 \Rightarrow \text{no association}; OR > 1 \Rightarrow \text{positive association}; OR < 1 \Rightarrow \text{negative association}

Practical notes for exams

  • Distinguish between Absolute Risk (AR) and Relative Risk (RR): AR gives burden; RR gives strength of association.
  • Use Risk Difference when planning public health burden or evaluating the number of additional cases due to exposure.
  • Use RR to discuss how much more likely exposure leads to disease, especially when baseline risk is informative.
  • Use OR in case-control studies or whenever full incidence data are not available; remember OR approximates RR only when disease is rare.
  • Always consider the study design when choosing measures of association (cohort vs case-control).

Key Terms and Definitions

  • Absolute Risk (AR): The probability that an individual in a given population will develop a disease over a specified period. It indicates the incidence of disease.
  • Risk Ratio (Relative Risk, RR): A measure of the strength of association between an exposure and a disease, indicating how many times more likely exposed individuals are to develop the disease compared to unexposed individuals.
  • Risk Difference (Absolute Risk Increase, ARD): The absolute difference in risk between exposed and unexposed groups, indicating the number of additional cases of disease attributable to the exposure per unit of population.
  • Attack Rate (AR): A measure of the proportion of a population that develops a disease during an outbreak, often used to quantify risk in an exposed or at-risk group.
  • Odds Ratio (OR): A measure of association that compares the odds of exposure among cases to the odds of exposure among controls. It is often used in case-control studies.
  • Rarity Assumption: The condition under which the Odds Ratio (OR) provides a good approximation of the Relative Risk (RR), specifically when the disease incidence is low.
  • Cohort Study: An observational study design where groups of exposed and unexposed individuals are followed over time to compare disease incidence.
  • Case-Control Study: An observational study design where individuals with a disease (cases) are compared to individuals without the disease (controls) to examine past exposures.

Equations and Their Usage

  • Absolute Risk (AR):
    • Formula: AR = \frac{\text{Number of new cases}}{\text{Population at risk}} \times 100
    • Usage: To understand the overall burden of a disease in a population, for clinical decision-making, and public health planning.
  • Attack Rate (AR):
    • Formula: AR = \frac{\text{Number of new cases}}{\text{Population at risk}} \times 100
    • Usage: Quantifies risk in a specific exposed or at-risk population, particularly useful during outbreaks to determine the proportion of people who became ill.
  • Risk Difference (Absolute Risk Increase, ARD):
    • Formula: ARD = AR{\text{exposed}} - AR{\text{unexposed}}
    • Usage: To determine the public health burden, indicating how many extra cases per population unit are directly due to the exposure. Essential for prioritizing interventions.
  • Relative Risk (RR):
    • Formula: RR = \frac{\dfrac{a}{a+b}}{\dfrac{c}{c+d}} = \frac{\text{Incidence in exposed}}{\text{Incidence in unexposed}}
    • Usage: To assess the strength of association between an exposure and an outcome, indicating how much more likely the exposed group is to develop the disease. Crucial for causal inference.
  • Odds Ratio (OR):
    • Formula: OR = \frac{a \times d}{b \times c}
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