🐛Biostatistics Unit 5 – Biological Hypothesis Testing & Inference

Key Concepts

• Null hypothesis ($H_0$) states there is no significant difference between specified populations, any observed difference is due to sampling or experimental error
• Alternative hypothesis ($H_A$) states there is a significant difference between specified populations, directly contradicting the null hypothesis
• P-value probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct
• Alpha ($\alpha$) level, also known as the significance level, is the probability threshold below which the null hypothesis is rejected (commonly set at 0.05)
• Type I error (false positive) occurs when the null hypothesis is rejected when it is actually true
  • Denoted by $\alpha$, the significance level
• Type II error (false negative) occurs when the null hypothesis is not rejected when it is actually false
  • Denoted by $\beta$, related to statistical power
• Statistical power probability of correctly rejecting a false null hypothesis, depends on sample size, effect size, and significance level

Types of Hypotheses in Biology

• One-tailed (directional) hypothesis specifies the direction of the expected difference between populations (e.g., group A has a higher mean than group B)
• Two-tailed (non-directional) hypothesis states that there is a difference between populations, but does not specify the direction of the difference
• Simple hypothesis specifies a single value for a population parameter (e.g., the mean weight of a certain species is 50 grams)
• Composite hypothesis specifies a range of values for a population parameter (e.g., the mean weight of a certain species is greater than 50 grams)
• Null hypothesis of no difference states that there is no significant difference between the populations being compared
  • Used as a starting point for statistical tests
• Alternative hypothesis of difference states that there is a significant difference between the populations being compared
  • Can be one-tailed or two-tailed
• Null hypothesis of no association states that there is no significant relationship between two variables (e.g., no correlation between body size and lifespan)

Steps in Biological Hypothesis Testing

• State the null and alternative hypotheses based on the research question and available data
• Choose an appropriate statistical test based on the type of data, sample size, and assumptions
  • Common tests include t-tests, ANOVA, chi-square, and correlation
• Set the significance level ($\alpha$) before conducting the test (usually 0.05)
• Collect data through experiments or observations, ensuring proper sampling techniques and experimental design
• Calculate the test statistic using the chosen statistical test and the collected data
• Determine the p-value associated with the test statistic, which represents the probability of obtaining the observed results if the null hypothesis is true
• Compare the p-value to the significance level ($\alpha$)
  • If p-value < $\alpha$, reject the null hypothesis in favor of the alternative hypothesis
  • If p-value ≥ $\alpha$, fail to reject the null hypothesis (insufficient evidence to support the alternative hypothesis)
• Interpret the results in the context of the original research question and consider the biological significance of the findings

Statistical Tests for Biological Data

• t-tests compare means between two groups (independent samples) or within a single group (paired samples)
  • Assumptions: normality, equal variances, and independence
• Analysis of Variance (ANOVA) compares means among three or more groups
  • One-way ANOVA for one independent variable, two-way ANOVA for two independent variables
  • Assumptions: normality, equal variances, and independence
• Chi-square test compares observed and expected frequencies of categorical variables
  • Goodness-of-fit test for a single variable, test of independence for two variables
  • Assumptions: large sample size, independence, and expected frequencies ≥ 5
• Correlation tests measure the strength and direction of the linear relationship between two continuous variables
  • Pearson correlation for normally distributed data, Spearman rank correlation for non-normal data
  • Assumptions: linearity, no outliers, and homoscedasticity
• Regression analysis models the relationship between a dependent variable and one or more independent variables
  • Linear regression for continuous variables, logistic regression for binary outcomes
  • Assumptions: linearity, independence, normality of residuals, and homoscedasticity
• Non-parametric tests (e.g., Mann-Whitney U, Kruskal-Wallis, Wilcoxon signed-rank) used when assumptions of parametric tests are violated
  • Less powerful than parametric tests but more robust to violations of assumptions

Interpreting Results and P-values

• P-value represents the probability of obtaining the observed results (or more extreme) if the null hypothesis is true
• A small p-value (typically < 0.05) indicates strong evidence against the null hypothesis, suggesting that the alternative hypothesis may be true
• A large p-value (≥ 0.05) indicates weak evidence against the null hypothesis, suggesting that the null hypothesis cannot be rejected based on the available data
• Statistical significance does not necessarily imply biological or practical significance
  • Consider the effect size and the context of the research question
• Confidence intervals provide a range of plausible values for a population parameter based on the sample data
  • Narrower intervals indicate more precise estimates
• Effect size measures the magnitude of the difference or relationship between variables
  • Cohen's d for t-tests, eta-squared for ANOVA, odds ratio for logistic regression
• Results should be interpreted cautiously, considering limitations of the study design, sample size, and potential confounding variables

Common Pitfalls and Misconceptions

• Multiple testing problem: conducting many statistical tests increases the likelihood of obtaining a significant result by chance (Type I error)
  • Use corrections such as Bonferroni or false discovery rate (FDR) to adjust p-values
• Confusing statistical significance with practical or biological significance
  • A statistically significant result may not be meaningful in the context of the research question
• Overinterpreting non-significant results as evidence of no effect (absence of evidence is not evidence of absence)
  • Consider the statistical power and the potential for Type II errors
• Assuming that a significant correlation implies causation
  • Correlation does not prove causation; consider potential confounding variables and the need for experimental manipulation
• Failing to check assumptions of statistical tests, leading to invalid or misleading results
  • Assess normality, equal variances, independence, and other assumptions before conducting tests
• Overfitting models by including too many predictors relative to the sample size
  • Use model selection techniques (e.g., AIC, BIC) and cross-validation to avoid overfitting
• Relying solely on p-values for decision-making without considering the context and the limitations of the study
  • Use a combination of p-values, effect sizes, confidence intervals, and biological knowledge to interpret results

Real-world Applications in Biology

• Comparing the effectiveness of different treatments or interventions in clinical trials (e.g., drug efficacy, surgical techniques)
• Assessing the impact of environmental factors on species abundance, diversity, or behavior (e.g., climate change, habitat fragmentation)
• Identifying genetic variants associated with diseases or traits using genome-wide association studies (GWAS)
• Evaluating the performance of diagnostic tests or biomarkers for detecting diseases or conditions (e.g., sensitivity, specificity)
• Investigating the relationship between diet, exercise, or other lifestyle factors and health outcomes (e.g., obesity, cardiovascular disease)
• Comparing the growth rates, survival, or reproductive success of different populations or species in ecological studies
• Assessing the effectiveness of conservation strategies for protecting endangered species or habitats
• Analyzing the expression levels of genes in different tissues, developmental stages, or experimental conditions using RNA-seq or microarray data

Advanced Topics and Future Directions

• Bayesian hypothesis testing incorporates prior knowledge and updates the probability of hypotheses based on observed data
  • Provides a more flexible and intuitive approach compared to frequentist methods
• Non-parametric bootstrapping resamples the observed data to estimate the sampling distribution of a statistic and construct confidence intervals
  • Useful when the assumptions of parametric tests are violated or the distribution is unknown
• Permutation tests generate a null distribution by randomly shuffling the observed data and calculating the test statistic for each permutation
  • Provides exact p-values and is useful when the assumptions of parametric tests are violated
• Mixed-effects models account for both fixed and random effects in the data, allowing for the analysis of hierarchical or clustered data structures
  • Useful for repeated measures, longitudinal studies, or multi-level data
• Machine learning techniques (e.g., random forests, support vector machines) can be used for classification, regression, or clustering of biological data
  • Provides a data-driven approach for identifying patterns and making predictions
• Integrating multiple data types (e.g., genomics, transcriptomics, proteomics) to gain a more comprehensive understanding of biological systems
  • Requires advanced statistical methods and bioinformatics tools for data integration and interpretation
• Developing new statistical methods and software tools to handle the increasing complexity and volume of biological data
  • Addressing challenges such as high-dimensionality, sparsity, and non-normality of data
• Promoting reproducibility and transparency in biological research by sharing data, code, and detailed methods
  • Using platforms such as GitHub, Jupyter notebooks, and open-access journals to facilitate collaboration and replication of results