Stats class 8

Risks and Odds

Introduction to Risk Measurement

• Probability value as a simple risk measurement.

• Case Study: Polio and the Salk Vaccine.

  • Polio risk with Salk Vaccine: 0.000164 out of 200,745.

  • Need for a comparison group to evaluate vaccine effectiveness.

Comparing Rates

• Rates with Control Group:

  • Polio with Placebo: 0.000571 out of 201,229.

• Importance of comparing treatment with placebo to understand vaccine efficiency.

Absolute Risk Reduction (ARR)

• Definition: Absolute Risk Reduction calculates the difference in rates between the treatment and control groups.

  • Formula: ( ARR = P_{treatment} - P_{control} )

• Polio rates:

  • Salk Vaccine Rate: 0.000164

  • Placebo Rate: 0.000571

  • Result: ARR = 0.000407 (0.0407% fewer events with Salk vaccine).

  • Conclusion: 0.0407% fewer events of polio with the vaccine.

Relative Risk (RR)

• Definition: Comparative measurement of risk between treatment and control groups, expressed as a ratio.

  • Formula: ( RR = \frac{P_{treatment}}{P_{control}} )

• Interpretation:

  • RR = 1: No difference in risk.

  • RR > 1: Higher risk in treatment group.

  • RR < 1: Higher risk in control group.

• Polio Rates:

  • Treatment Group: 0.000164

  • Control Group: 0.000571

  • Result: RR = 0.287, indicating the Salk vaccine group has a significantly lower risk.

  • Reciprocal RR suggests placebo group is 3.48 times as likely to get polio.

Number Needed to Treat (NNT)

• Definition: Number of patients that need to be treated to prevent one adverse event.

  • Formula: ( NNT = \frac{1}{ARR} )

• Example:

  • With ARR = 0.000407, NNT = 2,458 (patients need to be vaccinated to prevent one polio case).

  • Salk vaccine deemed effective considering serious implications of polio.

Odds

Odds Definition

• Odds represent the likelihood of an event occurring.

• Formula: ( Odds = \frac{P(A)}{1 - P(A)} )

• Example:

  • Odds of polio given the Salk vaccine calculated as follows:

  • Odd for treated group and placebo.

Odds Ratio (OR)

• Definition: Compares odds of an event among treatment vs. control groups.

  • Formula: ( OR = \frac{Odds_{treatment}}{Odds_{control}} )

• Interpretation:

  • OR = 1: Odds are equal for both groups.

  • OR > 1: Higher odds for treatment group.

  • OR < 1: Higher odds for control group.

• Calculation Example:

  • Odds for the Salk vaccine: ( \frac{33}{200,712} )

  • Odds for placebo: ( \frac{115}{201,114} )

• Result: Odds of polio significantly reduced (by 71%) with vaccination.

Rates

Definition of Rates

• Rate: Frequency count of events relative to the population size exposed to risk, often multiplied by a constant for clarity.

• Types of Rates:

  • Mortality, Fertility, Morbidity rates defined accordingly.

Calculation of Mortality Rate

• Crude Mortality Rate Example:

  • Using US data (787,650 deaths from cardiovascular disease, population 312,799,495).

  • Calculation gives a crude mortality rate of 2.5 per 1000 population.

• Implications of Crude Rates:

• Comparison not always fair (e.g., Florida and Colorado).

• Importance of using specific or adjusted rates rather than crude rates for accurate health assessments.

Adjusted and Specific Rates

• Specific Rate: Calculated for particular groups or causes.

• Adjusted Rate: Takes into account significant factors to allow fair comparisons.

• Essential for accurate interpretations of health data.