BUA212 Regression Analysis PDF

BUA212 REGRESSION ANALYSIS Miss Ogheneofejiro Edewor   What is Regression Analysis?  Types of Regression Analysis  Regression Model  Assumptions of Regression Analysis Outline  Steps in Conducting Regression Analysis  Business...

BUA212 REGRESSION ANALYSIS Miss Ogheneofejiro Edewor   What is Regression Analysis?  Types of Regression Analysis  Regression Model  Assumptions of Regression Analysis Outline  Steps in Conducting Regression Analysis  Business Applications of Regression Analysis   Regression analysis is a statistical method used to understand relationships between variables.  It involves identifying how the dependent variable What is (outcome) changes when one or more Regression independent variables (predictors) change.  It can be utilized to assess the strength of the Analysis? relationship between variables and for modeling the future relationship between them.  It measures cause and effect.   Simple Regression: Examines the relationship between one dependent variable and one independent variable. Example: The effect of advertising expenditure Types of (independent variable) on sales revenue (dependent variable). Regression  Multiple Regression: Analysis Explores the relationship between one dependent variable and two or more independent variables. Example: The impact of advertising expenditure, product price, and distribution channels on sales revenue.   Simple Regression: 𝑌=𝛽0+𝛽1𝑋+𝜀 Regression  Multiple 𝑌=𝛽0+𝛽1𝑋1+𝛽2𝑋2+... +𝛽𝑛𝑋𝑛+𝜀 Regression: Model   Dependent Variable (Y): The outcome we want to predict or explain.  Independent Variable(s) (X): Factors believed to Key influence the dependent variable. Components  Regression Coefficients (β): Values that quantify of a the relationship between the dependent and Regression independent variables. Model  Error Term (ε): The difference between the observed and predicted values of the dependent variable.   Linearity: The relationship between the dependent and independent variables is linear. Assumptions  Independence: Observations are independent of of each other. Regression  Homoscedasticity: Constant variance of errors. Analysis  Normality: The residuals (errors) are normally distributed.  1. Define the research problem and identify variables. 2. Collect and preprocess data. Steps in 3. Choose the appropriate regression model (simple or multiple). Conducting 4. Fit the regression model to the data. Regression 5. Evaluate model performance using statistical Analysis metrics such as R², adjusted R², and significance tests. 6. Interpret the results and apply them to business decisions.   R² (Coefficient of Determination)  Definition: R² indicates the proportion of variance in the dependent variable (Y) that is explained by the independent variable(s) (X).  Range: 0≤ R² ≤1 Interpreting  Interpretation:  A higher R² value (closer to 1) suggests that the model Regression explains a large portion of the variability in the Results dependent variable.  A low R² value indicates the model explains less variability, suggesting additional factors not included in the model may influence the dependent variable.  Example: If R² =0.85, it means 85% of the variation in sales revenue is explained by advertising expenditure.   Adjusted R²  Definition: Adjusted R² adjusts R² for the number of predictors in the model, penalizing for the inclusion of irrelevant variables. Interpreting  Interpretation: Regression  Useful in multiple regression to assess if adding more variables improves the model’s performance. Results  If Adjusted R² is significantly lower than R2R^2R2, it suggests overfitting or inclusion of irrelevant predictors.  Example: If R² =0.85 and Adjusted R² =0.80, the model performs well without excessive complexity.   Significance Tests (P-values)  Definition: Tests the null hypothesis that a regression coefficient is equal to zero (no effect).  Interpretation: Interpreting  P-value < 0.05: Statistically significant relationship Regression between the independent and dependent variable.  P-value Results ≥ 0.05: No statistically significant relationship.  Example: If the coefficient for "Advertising Expenditure" has a P-value of 0.02, advertising significantly influences sales.  Simple Regression Example  Scenario: A company wants to understand the impact of advertising expenditure (X) on sales revenue (Y). Example of  Regression Equation: Regression Y=β0+β1X+ϵ Equations  Given Values:  Intercept (β0): 500 (base sales revenue when no advertising is done).  Slope (β1): 2 (for every $1,000 spent on advertising, sales increase by $2,000).  Equation: Y=500+2X  Prediction Example: If advertising expenditure (X) is $3,000: Y=500+2(3,000)=500+6,000 = 6,500. The predicted sales revenue is $6,500. Multiple Regression Example  Scenario: A business wants to determine how advertising expenditure (X1​) and price discounts (X2​) affect sales revenue (Y).  Regression Equation: Y=β0+β1X1+β2X2+ϵ  Given Values:  Intercept (β0​): 300 (base sales when no advertising or discounts are applied).  Advertising coefficient (β1​): 1.5 (for every $1,000 spent on advertising, sales increase by $1,500).  Discount coefficient (β2​): 2 (for every 1% discount, sales increase by $2,000).  Equation: Y=300+1.5X1+2X2  Prediction Example: If advertising expenditure (X1​) is $4,000 and price discounts (X2​) are 5%: Y=300+1.5(4,000)+2(5) (recall that for every 1% discount, sales increase by $2,000) Y=300+6,000+10,000 =16,300 The predicted sales revenue is $16,300.   Forecasting Regression models help businesses predict future trends based on historical data. Example: Forecasting future sales based on past advertising efforts and market conditions. Business  Trend Analysis Applications Analyzing patterns decisions. over time to make informed of Regression Example: demand. Identifying seasonal trends in product Analysis  Resource Allocation Determining the impact of different factors on performance, helping allocate resources effectively. Example: Estimating the return on investment (ROI) of various marketing strategies.  Pricing Strategies Understanding how price changes affect demand for products or services. Example: Predicting customer demand based on varying pricing schemes.  Operational Efficiency Identifying factors that impact production costs or efficiency. Example: Assessing how machine downtime, labor hours, and material costs influence production output. ANY QUESTION?