Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Probability and Statistics → Statistics ↓


Regression Analysis


Regression analysis is a powerful statistical method that allows you to examine the relationship between two or more variables of interest. While there are many types of regression analysis, at its core, it helps to understand how the specific value of the dependent variable (or "criterion variable") changes when any one of the independent variables is altered, while the other independent variables are held constant.

Understanding the basics

To begin regression analysis, it is essential to understand the types of variables involved:

  • Dependent variable: This is the main factor you are trying to understand or predict. It is dependent on one or more independent variables.
  • Independent variable(s): These are the variables that you suspect have an effect on your dependent variable.

Simple linear regression

Simple linear regression is a method that helps you understand the relationship between two continuous variables: one independent (X) and one dependent (Y). We reach this by fitting a linear equation to the observed data. The equation is as follows:

Y = a + bX + ε
  • Y is the dependent variable we are trying to predict.
  • X is the independent variable that we are using to make a prediction.
  • a is the intercept of the line (the expected mean value of Y when X = 0).
  • b is the slope of the line (the change in Y for a one unit change in X).
  • ε is the error term (the difference between the actual and predicted Y values).
Visual Example of Simple Linear Regression

Interpretation of coefficients

In the equation Y = a + bX + ε, the coefficients a and b provide important information about the relationship between X and Y:

  • Intercept (a): It is the expected value of Y when the value of X is zero. It is the point where the regression line crosses the Y-axis.
  • Slope (b): It tells us the default change in the dependent variable (Y) for every one-unit change in X. A positive slope indicates a direct relationship, while a negative slope indicates an inverse relationship.

Multiple regression analysis

Multiple regression involves more than one independent variable and helps to understand how multiple factors affect the dependent variable. The equation for multiple regression is:

Y = a + b1X1 + b2X2 + ... + bnXn + ε
  • Y is the dependent variable.
  • X1, X2, ..., Xn are independent variables.
  • a is the intercept.
  • b1, b2, ..., bn are the coefficients corresponding to each independent variable.
  • ε is the error term.

Example

Suppose we want to predict the scores of students based on their study hours (X1) and sleep hours (X2). A possible regression equation could be:

Score = 10 + 5*(StudyHours) + 3*(SleepHours) + ε

Here, 5 is the coefficient that tells us that for every additional hour of study, the score can increase by 5 points, assuming sleep hours remain constant. Similarly, for every additional hour of sleep, the score can increase by 3 points, keeping study hours constant.

Assumptions of regression analysis

For regression analysis to be valid, several assumptions must be satisfied:

  • Linearity: The relationship between the independent and dependent variables must be linear.
  • Independence: The observations must be independent of each other.
  • Homoskedasticity: The variance of the errors should be the same at all levels of the indented variable.
  • Normal distribution of errors: The residuals should be approximately normally distributed.

Regression in behavior

In practice, regression analysis is used for forecasting and predictions in many areas. Here are some examples:

  • Economics: Predicting consumer spending based on factors such as income, interest rates, and inflation.
  • Medicine: Studying the effect of certain behaviors or exposures on health outcomes, such as heart disease.
  • Marketing: Forecasting sales based on advertising expenditure, seasonal factors, etc.
  • Real estate: Determining the value of a property based on characteristics such as square footage, number of rooms, and location.

Case study: home prices

Let us consider a case study of forecasting house prices based on various factors, such as:

  • Size of the house (in sq ft)
  • Number of bedrooms
  • Place
  • Age of the house

The potential regression equation can be structured as follows:

Price = a + b1*(Size) + b2*(Bedrooms) + b3*(Location) + b4*(Age) + ε

Each of these predictors has a corresponding coefficient that estimates its specific effect on the house price.

Conclusion

Regression analysis is a versatile tool that, when applied correctly, can uncover meaningful relationships between variables. By understanding these relationships, you can make informed decisions based on data rather than speculation. Whether forecasting future trends or analyzing existing patterns, regression provides a framework for understanding complex data.


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Regression Analysis

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