Statistical Methods and Probability in Biomedical Research

Introduction

This study guide reviews key statistical concepts and tests commonly used in biomedical and health sciences research. Understanding these principles is essential for interpreting data, evaluating research findings, and applying statistical reasoning in scientific studies.

Probability and Its Rules

Definition of Probability

• Probability is the likelihood or chance that a specific event will occur in a given trial.

• Mathematically, the probability of event E, denoted as P(E), satisfies:

• P(E) = 0 means the event will never happen; P(E) = 1 means the event will always happen.

• The greater the value of P(E), the more likely the event is to occur.

Why Probability is Needed

• Uncertainty: Biological systems are complex and variable; observations and experiments are subject to error.

• Sampling Variation: Research is typically conducted on samples, not entire populations, introducing variability.

The Complement Rule

• If two events, E1 and E2, are complements, then

• Example: If the probability a student is female is 0.92, then the probability a student is male is 1 - 0.92 = 0.08.

Additive Rules

• First Additive Rule (Mutually Exclusive Events): If E1 and E2 cannot both occur,

• Second Additive Rule (General Case): For any two events,

Multiplicative Rule

• For independent events, the probability that both occur is the product of their probabilities:

• Example: Probability of getting heads on two consecutive coin tosses:

Joint, Marginal, and Conditional Probabilities

• Joint Probability: Probability of two events occurring together (e.g., positive test and disease present).

• Marginal Probability: Probability of a single event, regardless of other events.

• Conditional Probability: Probability of one event given that another event has occurred.

Conditional probability formula:

• Sensitivity: Probability of a positive test given disease is present.

• Specificity: Probability of a negative test given disease is absent.

Statistical Tests Overview

Common Statistical Tests and Their Assumptions

Types of Correlation Coefficients

• Pearson: Both variables are continuous.

• Spearman: One or both variables are ordinal, or data are skewed/non-linear.

• Point-biserial: One variable is continuous, the other is dichotomous.

• Phi: Both variables are dichotomous.

Statistical Significance and Errors

The p Value

• p stands for probability.

• It is the probability of observing a test statistic as extreme as, or more extreme than, the observed value if the null hypothesis is true.

• Commonly used to determine statistical significance in hypothesis testing.

• Low p values (e.g., < 0.05) suggest that the observed result is unlikely under the null hypothesis.

Type I and Type II Errors

• Type I Error (α): Rejecting the null hypothesis when it is actually true (false positive).

• Type II Error (β): Failing to reject the null hypothesis when it is actually false (false negative).

• Lowering α reduces Type I error but increases the risk of Type II error.

• Larger sample sizes reduce the risk of Type II error by increasing statistical power.

Alpha Level (α)

• The threshold p value for rejecting the null hypothesis (commonly set at 0.05).

• Smaller α values make it harder to reject the null hypothesis, reducing Type I error but increasing Type II error.

• Very small p values (e.g., < 0.001) indicate strong evidence against the null hypothesis.

Applied Examples of Statistical Tests

Example 1: One-Sample t-Test

• Used to compare the mean systolic blood pressure of a sample of 13-year-olds to the national average (113 mmHg).

• Hypotheses: H0: Sample mean = 113; HA: Sample mean ≠ 113.

• Assumption: Normality of the sample distribution.

Example 2: Spearman Correlation

• Examines association between mother's educational status (ordinal) and gross annual income (continuous or ordinal).

• Spearman's correlation is used due to ordinal data.

• Hypotheses: H0: No association; HA: Association exists.

Example 3: Correlation with Ordinal and Dichotomous Variables

• Investigates association between religious attendance (ordinal) and lifetime marijuana use (dichotomous).

• Appropriate correlation method depends on variable types (often Spearman or point-biserial).

Example 4: Dichotomized Variable Correlation

• Religious attendance dichotomized (never vs. attends) and correlated with marijuana use (dichotomous).

• Pearson or Phi coefficient may be used.

Example 5: Independent Samples t-Test

• Compares systolic blood pressure between adolescent smokers and non-smokers.

• Assumptions: Normality, homogeneity of variance.

Example 6: Independent Samples t-Test

• Compares diastolic blood pressure between male and female children.

• Assumptions: Normality, homogeneity of variance.

Example 7: One-Way ANOVA

• Compares mean height across five cohorts (groups) of children.

• Assumptions: Independence, normality, homogeneity of variance.

Example 8: Correlation Matrix

• Examines associations among several continuous health variables (e.g., diastolic BP, heart rate, BMI, cholesterol).

• Pearson correlation is appropriate for continuous, normally distributed variables.

Example 9: Paired Samples t-Test

• Compares means of homework grades at two time points for the same students.

• Assumptions: Normality of difference scores.

Example 10: Correlation Matrix with Mixed Variable Types

• Analyzes relationships among dichotomous, ordinal, and continuous variables (e.g., marijuana use, ethnicity, education, stigma, religiosity).

• Choose correlation type based on variable measurement levels.

Example 11: One-Way ANOVA

• Compares age across three political affiliation groups (Democrat, Republican, Independent).

• Assumptions: Independence, normality, homogeneity of variance.

Example 12: Simple Linear Regression

• Predicts midterm grades from homework 2 grades.

• Regression equation:

• Interpretation involves identifying independent variable (IV), dependent variable (DV), R2 (variance explained), and slope.

Summary Table: Statistical Test Selection

Key Takeaways

• Choose statistical tests based on research question, variable types, and data structure.

• Understand and check assumptions before interpreting results.

• Interpret p values, effect sizes, and confidence intervals in context.

• Be aware of the risks of Type I and Type II errors and how sample size and alpha level affect them.

Additional info: These statistical concepts are foundational for interpreting research in microbiology, epidemiology, and biomedical sciences, where hypothesis testing, probability, and data analysis are essential for drawing valid scientific conclusions.