Introduction
Regression analysis is a statistical method used to understand the relationship between a dependent variable and one or more independent variables.
In biostatistics, regression helps to model and analyze the relationships among various biological and medical variables, aiding in predictions and understanding the effect of multiple factors on health outcomes.
Properties of Regression
Linearity: Assumes a linear relationship between the dependent and independent variables.
Additivity: The combined effect of the independent variables on the dependent variable is additive.
Independence: Observations of the dependent variable are independent of each other.
Homoscedasticity: The variance of the residuals (errors) is constant across all levels of the independent variables.
Normality: The residuals should be approximately normally distributed.
Methods to Study Regression
1. Least Squares Method:
Minimizes the sum of the squared residuals to find the best-fitting line or curve.
2. Maximum Likelihood Estimation (MLE):
Finds parameter values that maximize the likelihood that the process described by the model produced the observed data.
3. Bayesian Methods:
Incorporate prior distributions with the data to update beliefs about the parameters.
4. Robust Regression:
Used when data contains outliers or violates the assumptions of traditional regression models.
Types of Regression
Simple Linear Regression: Models the relationship between a single independent variable and a dependent variable.
Multiple Linear Regression: Involves two or more independent variables predicting a single dependent variable.
Logistic Regression: Used when the dependent variable is binary (e.g., presence or absence of disease).
Poisson Regression: Applied to count data where the outcome variable represents the number of events.
Cox Proportional Hazards Regression: Used in survival analysis to examine the effect of variables on the time to an event.
Nonlinear Regression: Used when the relationship between the variables is nonlinear.
Polynomial Regression: Models the relationship as an nth-degree polynomial.
Ridge and Lasso Regression: Techniques that add regularization to prevent overfitting by penalizing large coefficients.
Regression Equations
1.. For X upon Y: X = a + bY
a is the intercept (the value of X when Y is zero).
b is the slope (the rate at which X changes for each unit change in Y).
2. For Y upon X: Y = a + bX
a is the intercept (the value of Y when X is zero).
b is the slope (the rate at which Y changes for each unit change in X).
Regression Line
A regression line is a straight line that best fits the data on a scatter plot, representing the relationship between two variables.
It is used in regression analysis to predict the value of a dependent variable based on the value of an independent variable.
1.. For X upon Y: X=a + bY
Models X as the dependent variable influenced by Y.
2. For Y upon X: Y = a + bX
Models Y as the dependent variable influenced by X.
Applications in Biostatistics
In biostatistics, the regression line helps in several ways:
1. Making Predictions:
By inputting a value of X (e.g., age, body mass index, or dosage of a drug), one can predict the expected value of Y (e.g., blood pressure, cholesterol level, or recovery time).
2. Measuring Relationships:
The slope and intercept provide insights into the nature of the relationship between variables.
For instance, a positive slope indicates a positive relationship, while a negative slope indicates a negative relationship.
3. Assessing Model Fit:
The closeness of data points to the regression line can be quantified by statistical measures such as R-squared or adjusted R-squared, indicating how well the independent variables explain the variance in the dependent variable.
Example Data:
Let's create an example dataset with the variables 'Hours Studied' (X) and 'Exam Score' (Y):
Regression Analysis:
To explore this dataset, let's calculate the regression equation and plot the data points along with the regression line. I'll generate a graph and display the regression equation:
Regression Line Equation:
The equation of the regression line is:
𝑌 = 42.00 + 7.29𝑋
Where:
𝑌 is the predicted exam score.
42.00 is the intercept, indicating that even with zero hours of study, the baseline score is 42.
7.29 is the slope, indicating that for each additional hour of study, the exam score increases by 7.29 points on average.
This example illustrates how regression can help in predicting relationships and outcomes, and how the regression line can be visualized and interpreted to understand these relationships better.
