BADM3400 Chapter 8: Regression Analysis

Introduction to Business Analytics

• Course: BADM3400
• Lecturer: Jason Chan, PhD
• Chapter: 8 - Trend lines and Regression Analysis

Modeling Relationships and Trends in Data

• Create charts to better understand data sets.
• Use scatter charts for cross-sectional data.
• Use line charts for time series data.

Common Mathematical Functions for Predictive Analytical Models

• Linear: y = a + bx
• Logarithmic: y = ln(x)
• Polynomial (2nd order): y = ax² + bx + c
• Polynomial (3rd order): y = ax³ + bx² + cx + d
• Power: y = axb
• Exponential: y = abx (e is often used for constant b)

Excel Trendline Tool

• Right-click on data series and choose "Add Trendline".
• Check boxes to display equation and R-squared value on chart.

R-Squared (R²)

• Measures the "fit" of the line to the data.
• Values range from 0 to 1.
• A value of 1.0 indicates a perfect fit (all data points on the line).
• Higher values indicate better fit.

Example 1: Modeling a Price-Demand Function

• Linear demand function: Sales = 20512 – 95116(price)
• Data shows a relationship between price and sales.

Example 2: Predicting Crude Oil Prices

• Line chart shows historical data.
• Excel's Trendline tool used to model different types of functions with crude oil prices.

Caution About Polynomials

• R² values increase with polynomial order.
• Higher-order polynomials are often less smooth and hard to interpret.
• Avoid orders beyond third-order.
• Visual inspection is crucial for evaluating fit

Regression Analysis

• Tool for mathematical and statistical models.
• Characterizes relationships between dependent and independent variables (ratio or categorical).
• All variables should be numerical.

Simple Linear Regression

• Finds a linear relationship between one independent variable (X) and one dependent variable (Y).
• First, prepare a scatter chart to verify data has a linear trend.
• Use alternative approaches if the data isn't linear.

Example 3: Home Market Value Data

• House size (square footage) is related to market value.
• Data (house age, square feet, market value) is usually presented in table format.
• Scatter plot of the data should show a linear trend.

Finding the best fitting regression line

• Market value = a + bx, where 'x' represents square feet
• Visual inspection of lines (A and B) is needed to identify the best fitting line

Example 4: Using Excel to Find the Best Regression Line

• Market value = -32,673 + $35.036 x (square feet)
• Estimated market value of a home with 2,200 square feet is $109,752.

Least-Squares Regression

• Simple linear regression model: Y = a + bX + ε
• Estmiate population parameters using sample data.

Residuals

• Observed errors in estimating the dependent variable.
• Residual = Actual Y value − Predicted Y value
• Standard residuals above ±2 or ±3 are potential outliers.

Least Squares Regression (continued)

• Best-fitting line minimizes the sum of squares of residuals.
• Excel functions: INTERCEPT (known_y's, known_x's) and SLOPE (known_y's, known_x's)

Example 5: Using Excel Functions to Find Least-Squares Coefficients

• Data for house age, square feet, and market value.
• Slope (b1): 35.036
• Intercept (b0): $32,673
• Estimation for a house with 1,750 sq.ft. = 93,986(93,986 (93,986(93,987 using a different excel function)

Simple Linear Regression with Excel

• Data > Data Analysis > Regression
• Input Y range and X range (include headers)
• Check Labels box

Home Market Value Regression Results

• Regression statistics table generated by Excel (using home size data set)

Regression Statistics

• Multiple R - sample correlation coefficient (-1 to +1)
• R Square - coefficient of determination (0 to 1)
• Adjusted R Square - adjusts R² for sample size
• Standard Error - variability between observed and predicted Y values

Formulae (continued)

• r formula
• b1 formula
• b0 formula

Example 6: Interpreting Regression Statistics for Simple Linear Regression

• 53% of variation in home market values can be explained by their size.

Regression as Analysis of Variance

• ANOVA F-test to see if variation in Y is due to X levels
• Null hypothesis (H0): population slope coefficient = 0
• Alternate hypothesis (H1): population slope coefficient ≠ 0
• Excel provides the p-value (Significance F).

Example 7: Interpreting Significance of Regression

• p-value = 3.798×10−8.
• Statistical significance of home size as predictor of market value.

Testing Hypotheses for Regression Coefficients

• t-test can be used as an alternative method.
• Excel provides p-values for slope and intercept tests.

Confidence Intervals for Regression Coefficients

• Confidence intervals (Lower 95% & Upper 95%) show the range of possible values.
• Use them to test hypotheses about regression coefficients.

Example 9: Interpreting Confidence Intervals for Regression Coefficients

• Illustrates confidence intervals for intercept and slope in a home market value example.
• Estimates for market value of a home with 1750 sq. ft (at the confidence interval extremes).

Residual Analysis and Regression Assumptions

• Residual = Actual Y − Predicted Y
• Standard residual = residual/standard deviation.
• Rule of thumb: standard residuals outside of ±2 or ±3 are potential outliers.
• Residual plot.

Example 10: Interpreting Residual Output

• Interpretation of the first observation's residual and its standard residual.

Checking Assumptions

• Linearity: examine scatterplot and residual plots.
• Normality: examine a histogram of residuals.
• Homoscedasticity. (constant spread): examine the residual plot
• Independence of errors: successive observations shouldn't be related.

Example 11: Checking Regression Assumptions for the Home Market Value Data

• Scatter plot, residual plot, histogram, and plot of residuals.
• Assess linearity, normality, homoscedasticity, and independence of errors. (from the provided home market data set)

Multiple Linear Regression

• Linear regression model with more than one independent variable.

Estimated Multiple Regression Equation

• Partial regression coefficients explain the change in the dependent variable.
• Changes in independent variables.

Excel Regression Tool

• Independent variables in successive columns.
• Key distinctions in Multiple R and R² calculation.
• ANOVA for significance of the entire model.

ANOVA for Multiple Regression

• Tests the significance of the entire model.
• (Uses an F-statistic).
• Hypotheses about individual regression coefficients

Example 12: Interpreting Regression Results for the Colleges and Universities Data

• Data on colleges/universities.
• Provides a regression equation for estimating graduation rates.

Example 13: Identifying the Best Regression Model

• Identifying the best regression model from a data set (Banking Data).
• Dropping the significant variable (Home value).
• Re-running the regression

Multicollinearity

• Strong correlations among independent variables.
• Hard to isolate individual effects of independent variables.
• Inflates p-values and makes interpretation challenging.

This quiz covers Chapter 8 of Introduction to Business Analytics, focusing on trend lines and regression analysis. It explores various mathematical functions used in predictive analytical models, as well as tools in Excel for creating trendlines and measuring R-squared values. Test your understanding of modeling relationships and trends in data through practical applications.