7.5 Statistical inference and hypothesis testing

Statistical inference and hypothesis testing are crucial tools in epidemiology and biostatistics. These methods help researchers draw conclusions about populations from sample data, enabling evidence-based decisions in public health.

Hypothesis testing involves formulating null and alternative hypotheses, then using statistical tests to evaluate them. Understanding p-values, significance levels, and confidence intervals is key to interpreting results and assessing their practical importance in public health contexts.

Statistical Inference in Public Health

Fundamentals of Statistical Inference

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• Statistical inference draws conclusions about populations based on sample data, enabling evidence-based decisions in public health
• Central limit theorem states sampling distribution of the mean approaches normal distribution as sample size increases (regardless of population distribution)
• Sampling methods affect representativeness and generalizability of statistical inferences
  • Simple random sampling (every individual has equal chance of selection)
  • Stratified sampling (population divided into subgroups before sampling)
  • Cluster sampling (groups of individuals selected rather than individuals)
• Type I and Type II errors pose risks in statistical inference
  • Type I error rejects a true null hypothesis (false positive)
  • Type II error fails to reject a false null hypothesis (false negative)
• Statistical power measures probability of correctly rejecting a false null hypothesis
  • Influenced by sample size, effect size, and significance level
  • Higher power increases likelihood of detecting true effects

Advanced Concepts in Statistical Inference

• Bayesian inference incorporates prior knowledge and updates probabilities with new data
  • Contrasts with frequentist approaches that rely solely on observed data
  • Useful in situations with limited data or strong prior beliefs
• Sampling distribution represents all possible sample statistics from repeated sampling
  • Forms basis for inferential statistics and hypothesis testing
  • Shape affected by sample size and population parameters
• Confidence intervals provide range of plausible values for population parameters
  • 95% confidence interval most commonly reported in public health research
  • Wider intervals indicate less precise estimates
• Effect sizes quantify magnitude of differences or relationships between variables
  • Cohen's d for continuous outcomes (small: 0.2, medium: 0.5, large: 0.8)
  • Odds ratios for categorical outcomes (1 indicates no effect)

Hypothesis Testing with Statistical Methods

Formulating and Testing Hypotheses

• Null and alternative hypotheses form foundation of hypothesis testing
  • Null hypothesis typically represents no effect or difference
  • Alternative hypothesis represents researcher's expectation or claim
• Parametric tests assume normally distributed data and are used for continuous outcomes
  • t-tests compare means between two groups (independent or paired)
  • ANOVA compares means across multiple groups
• Non-parametric tests used when data violate assumptions of parametric tests or for ordinal outcomes
  • Mann-Whitney U test (alternative to independent t-test)
  • Kruskal-Wallis test (alternative to one-way ANOVA)
• Chi-square tests employed for categorical data to assess associations between variables
  • Used in epidemiological studies to compare observed and expected frequencies
  • Assumptions include independence of observations and adequate sample size

Advanced Statistical Methods

• Regression analyses model relationships between variables and predict outcomes
  • Linear regression for continuous dependent variables
  • Logistic regression for binary dependent variables
  • Multiple regression incorporates multiple independent variables
• Multiple comparison procedures adjust for increased Type I error risk in multiple hypothesis tests
  • Bonferroni correction divides significance level by number of tests
  • False Discovery Rate controls proportion of false positives among rejected hypotheses
• Meta-analysis combines results from multiple studies to increase statistical power
  • Provides overall effect size estimate across populations
  • Assesses heterogeneity between studies
• Bootstrapping resamples data to estimate sampling distribution and calculate confidence intervals
  • Useful when theoretical distributions are unknown or assumptions are violated
  • Provides robust estimates of standard errors and confidence intervals

Interpreting Statistical Significance

Understanding P-values and Significance Levels

• P-values represent probability of obtaining results as extreme as observed, assuming null hypothesis is true
  • Smaller p-values indicate stronger evidence against null hypothesis
  • Do not directly measure magnitude of effect or practical importance
• Significance level (α) sets threshold for rejecting null hypothesis
  • Commonly set at 0.05 in public health research
  • Represents acceptable Type I error rate
• Confidence intervals provide range of plausible values for population parameters
  • 95% confidence interval interpreted as range that would contain true parameter in 95% of repeated samples
  • Narrower intervals indicate more precise estimates
• Statistical versus practical significance distinguishes between chance results and meaningful implications
  • Statistically significant results may not always be practically important
  • Consider effect sizes and context when interpreting results

Advanced Interpretation Techniques

• Effect sizes quantify magnitude of differences or relationships between variables
  • Cohen's d for continuous outcomes (small: 0.2, medium: 0.5, large: 0.8)
  • Relative risk and odds ratios for categorical outcomes
• Power analysis determines sample size needed to detect meaningful effects
  • Considers desired power, effect size, and significance level
  • Helps researchers plan studies with adequate statistical power
• Sensitivity and specificity assess performance of diagnostic tests
  • Sensitivity measures true positive rate
  • Specificity measures true negative rate
• Receiver Operating Characteristic (ROC) curves evaluate trade-off between sensitivity and specificity
  • Area under curve (AUC) indicates overall test performance
  • Perfect test has AUC of 1, random guessing has AUC of 0.5

Evaluating Statistical Inference in Research

Critical Appraisal of Statistical Methods

• Publication bias skews available evidence in public health literature
  • Statistically significant results more likely to be published
  • Can lead to overestimation of effect sizes in meta-analyses
• P-hacking and data dredging compromise research integrity
  • Manipulating data or analyses to achieve statistically significant results
  • Can lead to false positive findings and irreproducible results
• Reproducibility crisis highlights importance of transparent reporting
  • Detailed description of statistical methods and results crucial
  • Pre-registration of study protocols helps prevent selective reporting
• Sensitivity analyses assess robustness of statistical inferences
  • Vary assumptions or data to test stability of results
  • Crucial for policy recommendations and decision-making

Ethical Considerations and Advanced Techniques

• Ethical considerations in statistical inference include potential harm from errors
  • Type I errors may lead to unnecessary interventions or resource allocation
  • Type II errors may result in missed opportunities for effective public health measures
• Bayesian decision theory incorporates prior knowledge and uncertainty into decision-making
  • Combines prior beliefs with new data to update probabilities
  • Useful in situations with limited data or strong prior information
• Propensity score matching reduces bias in observational studies
  • Balances covariates between treatment and control groups
  • Improves causal inference in non-randomized studies
• Multilevel modeling accounts for hierarchical structure in data
  • Analyzes nested data (individuals within communities within countries)
  • Allows for estimation of effects at different levels of analysis