Odds Ratios, Risk Assessment, and Power in Statistical Analysis

Lecture 12: Odds Ratios

How to Assess Small Risks?

This lecture introduces the concept of odds ratios and their application in assessing small risks, particularly in epidemiological and biomedical research. Understanding how to quantify and interpret small risks is essential for making informed decisions in health and biological sciences.

• Odds Ratio (OR): A measure of association between exposure and outcome, commonly used in case-control studies.

• Risk Ratio (RR): Also known as relative risk, compares the probability of an event occurring in two groups.

• Power of a Test: The probability that a statistical test will correctly reject a false null hypothesis.

Statistical Power and Hypothesis Testing

Assessing Power Using Sampling Distributions

Statistical power is the likelihood that a test will detect an effect when there is one. It depends on sample size, effect size, and significance level (alpha). Power analysis is crucial for designing experiments and interpreting results.

• Sampling Distribution: The distribution of a statistic (e.g., sample proportion) over repeated samples.

• Null Hypothesis (H0): Assumes no effect or no difference.

• Alternative Hypothesis (HA): Assumes an effect or difference exists.

• Power Calculation: The area under the alternative distribution beyond the critical value (where H0 is rejected).

Example: In a study with 70 pieces of toast, the true proportion is 62%. The power is the probability of rejecting H0 if the sample proportion is 0.62.

Assumptions for Chi-Square Tests

Conditions for Validity

Chi-square tests are used to assess associations between categorical variables. Certain assumptions must be met for valid results.

• Data must be counts: Observations are frequencies, not percentages or measurements.

• Randomization: Data should be randomly sampled.

• Independence: Each sampling unit is counted only once (not paired data).

• Cell Frequencies:

  • For 2x2 tables: All cells should have at least 5 counts.

  • For larger tables: All cells should have at least 1 count, and no more than 20% of cells should have fewer than 5 counts.

• Paired Data: Use McNemar's test instead of chi-square.

Epidemiology: Probability, Risk, and Odds

Key Definitions and Formulas

Understanding probability, risk, and odds is fundamental in epidemiological studies.

• Probability: Proportion of cases in a set of independent events. Range: [0, 1].

• Risk: Probability that an event (e.g., disease) will occur over a specific period. Calculated from a random sample.

• Odds: Ratio of probability of success to probability of failure. Range: [0, ∞].

Formulas:

• Risk: $\text{Risk} = \frac{\#\text{Successes}}{\text{Total}} = p$

• Odds: $\text{Odds} = \frac{p}{1-p}$

Relative Risk vs. Odds Ratio

Comparing Measures of Association

Relative risk and odds ratio are both used to quantify the relationship between exposure and outcome, but they have different interpretations and applications.

• Relative Risk (RR): Compares risk in exposed vs. unexposed groups. $RR = \frac{p_{\text{exposed}}}{p_{\text{unexposed}}}$

• Odds Ratio (OR): Compares odds in exposed vs. unexposed groups. $OR = \frac{\text{Odds}_{\text{exposed}}}{\text{Odds}_{\text{unexposed}}}$

• Hazard Ratio (HR): Used in survival analysis; similar interpretation to RR.

Advantages and Disadvantages:

• RR is intuitive but cannot be calculated for case-control studies.

• OR can be calculated for all study designs and can include additional variables, but its interpretation is less intuitive.

• When the event is rare ($p$ small), OR approximates RR.

Worked Example: WNV Mortality Risk

Calculating Risk, RR, and OR

Given data on age groups and WNV mortality, calculate risk, relative risk, and odds ratio.

Age Group	No	Yes	Total
0 – 44	217	2	219
65 – 100	163	19	182

• Risk (Seniors): $\frac{19}{182} = 0.104$

• Risk (Young): $\frac{2}{219} = 0.0091$

• Relative Risk (RR): $RR = \frac{19/182}{2/219} = 11.4$

• Odds (Seniors): $\frac{19}{163} = 0.1166$

• Odds (Young): $\frac{2}{217} = 0.0092$

• Odds Ratio (OR): $OR = \frac{19/163}{2/217} = 12.64$

Additional info: For rare events, OR ≈ RR; for common events, they differ more.

Effect Size and Power

Defining Small, Medium, and Large Effects

Effect size quantifies the magnitude of a difference or association. Cohen's criteria are commonly used to classify effect sizes.

Test	Small	Medium	Large
t-Test (Cohen's d)	0.2	0.5	0.8
Chi-square (Cramer's V)	0.1	0.3	0.5
Odds Ratio (OR)	1.5	2.3	5.0

• Power: Typically, a study should have at least 80% power to detect a meaningful effect.

• Sample Size: Larger samples increase power; required sample size depends on effect size and alpha.

Summary Table: Chi-Square Test Assumptions

Assumption	Requirement
Data type	Counts (not percentages)
Randomization	Random sample
Independence	Each unit counted once
Cell frequencies (2x2)	All cells ≥ 5
Cell frequencies (larger table)	All cells ≥ 1; <20% cells <5

Key Formulas

• Risk: $\text{Risk} = \frac{\#\text{Successes}}{\text{Total}}$

• Odds: $\text{Odds} = \frac{p}{1-p}$

• Relative Risk: $RR = \frac{p_{\text{exposed}}}{p_{\text{unexposed}}}$

• Odds Ratio: $OR = \frac{\text{Odds}_{\text{exposed}}}{\text{Odds}_{\text{unexposed}}}$

Applications and Interpretation

When to Use RR vs. OR

• RR: Preferred in cohort studies and randomized trials.

• OR: Used in case-control studies and logistic regression.

• Interpretation: RR > 1 or OR > 1 indicates increased risk/odds; RR < 1 or OR < 1 indicates decreased risk/odds.

Example: In the WNV study, seniors have a much higher risk and odds of mortality compared to younger patients.

Summary

• Odds ratios and risk ratios are essential tools for quantifying associations in epidemiology.

• Statistical power and effect size are critical for designing studies and interpreting results.

• Chi-square tests require specific assumptions for valid inference.

• Correct interpretation of RR and OR depends on study design and event frequency.