BADM3400 Chapter 8: Regression Analysis

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Questions and Answers

What is the first step in identifying significant independent variables in a regression model?

Check for multicollinearity among variables.

Examine the p-values of the independent variables. (correct)

Remove variables with insignificant p-values.

Calculate the adjusted R² for the model.

What should be done after identifying an independent variable with the largest p-value that exceeds the significance level?

Leave the variable in the model for further analysis.

Remove that variable from the model and evaluate adjusted R². (correct)

Drop all independent variables with p-values above the significance level.

Increase the level of significance before removal.

Which of the following statements is TRUE about multicollinearity?

Multicollinearity reduces the number of variables needed in the model.

Multicollinearity always leads to improved model precision.

Multicollinearity implies strong correlations among independent variables. (correct)

Multicollinearity only affects the dependent variable.

What effect does significant multicollinearity have on regression coefficients?

It can cause the signs of coefficients to become misleading. Signup and view all the answers

What is the proper way to handle independent variables with high p-values in regression analysis?

Evaluate their significance and remove them one at a time. Signup and view all the answers

What tool is used in Excel to add a trend line to a data series?

Right click on data series and choose Add trend line Signup and view all the answers

Which statement correctly describes the R-squared (R²) value?

It indicates how well the trend line fits the data. Signup and view all the answers

What is a characteristic of higher order polynomials in trend analysis?

They can be difficult to interpret visually. Signup and view all the answers

In regression analysis, how many independent variables are involved in multiple regression?

Two or more independent variables Signup and view all the answers

What is the primary limitation of higher order polynomial trend lines?

They can significantly increase model complexity. Signup and view all the answers

Which mathematical function is commonly used in simple linear regression?

Linear function Signup and view all the answers

What does the term 'dependent variable' refer to in regression analysis?

It varies based on changes in the independent variable. Signup and view all the answers

Why is it recommended not to use polynomials beyond the third order?

They may provide poor insights visually. Signup and view all the answers

What assumption is checked by examining if successive observations in a dataset are not related?

Independence of Errors Signup and view all the answers

Which plot is primarily used to assess the assumption of homoscedasticity in regression analysis?

Residual plot Signup and view all the answers

What percentage of the variation in the dependent variable does an R-square value of 0.53 explain?

53% Signup and view all the answers

What does a residual histogram appearing slightly skewed imply about the normality of errors?

Normality does not hold, but it is not serious Signup and view all the answers

From which type of data can we generally assume that the independence of errors holds?

Cross-sectional data Signup and view all the answers

What is suggested as the best approach when adjusting the variables in a regression model?

Systematically evaluate the significance of each variable Signup and view all the answers

Which of the following is NOT an indicator of a good regression model?

Ensuring model complexity to explain variation Signup and view all the answers

What key characteristic of a regression model indicates that the errors are normally distributed?

Symmetrical shape of residual histogram Signup and view all the answers

What is the purpose of preparing a scatter chart before performing simple linear regression?

To confirm a linear relationship between variables. Signup and view all the answers

In the regression equation $Market value = a + b \times x$, what does x represent?

Square footage of the home. Signup and view all the answers

What is a characteristic feature of the best-fitting regression line in simple linear regression?

It minimizes the sum of the squared residuals. Signup and view all the answers

Which condition must be checked to satisfy the assumptions of linear regression?

Homoscedasticity ensures that variation about the regression line is random. Signup and view all the answers

What can be inferred if the residual plot appears random?

The linearity assumption is satisfied. Signup and view all the answers

What does hypothesis testing for regression coefficients help determine?

The significance of the relationship between the independent and dependent variables. Signup and view all the answers

Why is normality of errors an important assumption in regression analysis?

It impacts the reliability of hypothesis tests. Signup and view all the answers

What does a histogram of standard residuals reveal in regression analysis?

The distribution of errors in the model. Signup and view all the answers

What is the effect of multicollinearity in regression analysis?

It affects the estimation of coefficients and their statistical significance. Signup and view all the answers

Which of the following defines homoscedasticity in regression analysis?

The variance of residuals is constant across all levels of the independent variable. Signup and view all the answers

Study Notes

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.

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Description

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.