🐛Biostatistics Unit 7 – Correlation and Linear Regression in Biology

Key Concepts

• Correlation measures the strength and direction of the linear relationship between two continuous variables
• Pearson's correlation coefficient (r) quantifies the linear association between two variables and ranges from -1 to +1
  • Values close to -1 indicate a strong negative linear relationship
  • Values close to +1 indicate a strong positive linear relationship
  • Values close to 0 indicate a weak or no linear relationship
• Spearman's rank correlation coefficient (ρ) assesses the monotonic relationship between two variables, which can be non-linear
• Linear regression models the relationship between a dependent variable (Y) and one or more independent variables (X) using a linear equation
• Least squares method estimates the best-fitting line by minimizing the sum of squared residuals (differences between observed and predicted values)
• Coefficient of determination (R^2) measures the proportion of variance in the dependent variable explained by the independent variable(s)

Data Collection and Preparation

• Collect data on the variables of interest ensuring a representative sample of the population
• Check for missing values, outliers, and data entry errors that may affect the analysis
• Assess the normality of the data using histograms, Q-Q plots, or statistical tests (Shapiro-Wilk test)
  • Non-normal data may require data transformations (log, square root) or non-parametric methods
• Examine the presence of outliers using box plots or Z-scores
  • Outliers can have a significant impact on correlation and regression results
• Ensure that the data is measured on a continuous scale for Pearson's correlation and linear regression
• Consider the sample size to ensure adequate statistical power and precision of estimates
• Organize data in a structured format (spreadsheet or data frame) for easy analysis

Types of Correlation

• Positive correlation indicates that as one variable increases, the other variable also tends to increase
  • Example: Height and weight in humans
• Negative correlation indicates that as one variable increases, the other variable tends to decrease
  • Example: Age and flexibility in adults
• Zero correlation suggests no linear relationship between the variables
  • Example: Shoe size and IQ
• Pearson's correlation coefficient (r) is used for linear relationships between two continuous variables
• Spearman's rank correlation coefficient (ρ) is used for monotonic relationships, which can be non-linear, or when data is ordinal
• Point-biserial correlation is used when one variable is continuous and the other is dichotomous (binary)
  • Example: Height and gender (male/female)

Interpreting Correlation Coefficients

• Correlation coefficients range from -1 to +1, with 0 indicating no linear relationship
• The sign (+/-) of the correlation coefficient indicates the direction of the relationship
• The magnitude of the correlation coefficient indicates the strength of the relationship
  • Values close to -1 or +1 suggest a strong relationship
  • Values close to 0 suggest a weak relationship
• Statistical significance (p-value) assesses whether the observed correlation is likely due to chance
  • A small p-value (typically < 0.05) indicates a statistically significant correlation
• Correlation does not imply causation; other factors may influence the relationship
• Consider the context and subject matter when interpreting the practical significance of a correlation

Introduction to Linear Regression

• Linear regression models the relationship between a dependent variable (Y) and one or more independent variables (X)
• Simple linear regression involves one independent variable, while multiple linear regression involves two or more independent variables
• The general equation for a simple linear regression is: $Y = β_0 + β_1X + ε$
  • $β_0$ is the y-intercept, $β_1$ is the slope, and $ε$ is the error term
• The least squares method estimates the regression coefficients by minimizing the sum of squared residuals
• The coefficient of determination (R^2) measures the proportion of variance in Y explained by X
  • R^2 ranges from 0 to 1, with higher values indicating a better fit
• Residual plots help assess the assumptions of linear regression (linearity, homoscedasticity, normality of residuals)

Assumptions and Limitations

• Linear regression assumes a linear relationship between the dependent and independent variables
  • Non-linear relationships may require data transformations or alternative models
• Homoscedasticity assumes that the variance of the residuals is constant across all levels of the independent variable(s)
  • Heteroscedasticity (non-constant variance) can affect the validity of the model
• Independence of observations assumes that the residuals are not correlated with each other
  • Autocorrelation can occur in time series or spatially correlated data
• Normality of residuals assumes that the residuals follow a normal distribution
  • Non-normal residuals can affect the validity of hypothesis tests and confidence intervals
• Outliers and influential points can have a significant impact on the regression results
  • Identify and carefully consider the treatment of outliers (removal or robust methods)
• Multicollinearity occurs when independent variables are highly correlated with each other
  • Multicollinearity can affect the interpretation of regression coefficients and model stability

Applications in Biology

• Correlation and regression are widely used in various fields of biology to explore relationships and make predictions
• Ecology: Relationship between species abundance and environmental factors (temperature, precipitation)
• Genetics: Association between genetic markers and quantitative traits (height, disease risk)
• Physiology: Relationship between body size and metabolic rate in animals
• Epidemiology: Identifying risk factors for diseases and predicting disease occurrence
• Conservation biology: Modeling species distribution and habitat suitability based on environmental variables
• Neuroscience: Examining the relationship between brain activity and behavioral or cognitive measures
• Pharmacology: Dose-response relationships and drug efficacy studies

Statistical Software and Tools

• Statistical software packages (R, Python, SAS, SPSS) provide functions for correlation and regression analysis
• R:

  cor()

  function for correlation,

  lm()

  function for linear regression
  • Packages like

    ggplot2

    and

    corrplot

    for data visualization
• Python:

  scipy.stats

  module for correlation,

  statsmodels

  and

  scikit-learn

  for regression
  • Libraries like

    matplotlib

    and

    seaborn

    for data visualization
• Spreadsheet software (Microsoft Excel, Google Sheets) can perform basic correlation and regression analysis
  • Add-ins or built-in functions (CORREL, PEARSON, SLOPE, INTERCEPT)
• Online tools and calculators are available for quick correlation and regression calculations
  • Example: VassarStats, MedCalc, GraphPad QuickCalcs
• Ensure data privacy and security when using online tools or sharing data with third-party software