Regression Analysis Notes

Regression Analysis

Definition of Regression Equation

• Represents the algebraic relationship between two sets of data (variables).
• Specifically, it represents the best overall algebraic relationship.
• Variables are typically labeled $x$ and $y$.

Independent and Dependent Variables

• x variable: Independent variable.
  • Often used to make predictions using the regression equation.
• y variable: Dependent variable.
  • Its value depends on the $x$ value.
• Alternative names:
  • Explanatory variable ($x$).
  • Response variable ($y$).
  • Caution: These names can be misleading, suggesting a causal connection when one may not exist.

Linear Regression Equation Form

• Rearrangement of the slope-intercept form ($y = mx + b$).
• Statisticians typically present it as: $y = b<em>0 + b</em>1x$, where:
  • $b_0$: y-intercept.
  • $b_1$: Coefficient of the $x$ variable (slope).
• The order (y-intercept first) allows for extensions to multiple independent variables.
• Regression process determines the values of $b<em>0$ and $b</em>1$.

Regression Line (Best Fit Line)

• A graph of the regression equation.
• Visual representation of the trend on a scatter plot.

Excel Implementation: Adding Regression Line and Equation

• Steps to add a trendline:
  • Select the scatter plot.
  • Go to "Chart Tools" > "Design" > "Add Chart Element" > "Trendline" > "Linear".
• The trendline provides a visual representation of the data's overall pattern.
• Points close to the line are accurately predicted, while those farther have larger prediction errors.
• A regression line represents a prediction -- for a given $x$ value, the line predicts a corresponding $y$ value.

Displaying the Regression Equation

• Steps to display equation:
  • Go to "Add Chart Element" > "Trendline" > "More Trendline Options".
  • Check "Display Equation on Chart".
• Excel displays the equation in slope-intercept form, with a default of four decimal places for coefficients.
• Equation allows for prediction of $y$ (dependent variable) given $x$ (independent variable).

Interpretation of the Equation

• The coefficient of the $x$ value in the equation represents the relationship between the check total and the tip.
• Example: A coefficient of 0.1973 suggests an average tip rate of approximately 19.7%.
• The y-intercept adjusts this percentage (e.g it looks like it's about 55¢ less than 19.7% based on the other value in the equation).

Alternative Method: Excel's Data Analysis Toolpack

• Activating the Toolpack:
  • Go to "File" > "Options" > "Add-ins".
  • In the "Manage" dropdown, select "Excel Add-ins" and click "Go".
  • Check "Analysis ToolPak" and click "OK".
• Accessing Regression:
  • Go to the "Data" tab > "Analysis" > "Data Analysis".
  • Select "Regression" from the list.
• Specifying Input Ranges:
  • Input Y Range: Select the range of the dependent variable (tips).
  • Input X Range: Select the range of the independent variable (check totals).
  • Check "Labels" if you include column headings in your selection. this indicates that the first row contains labels.
• Output Options:
  • Choose "New Worksheet Ply" to create a new tab for the output; give it a descriptive name.
  • Check "Line Fit Plots" to generate a scatter plot with the regression line.

Regression Output Analysis

• Correlation Coefficient: The Toolpack calculates Pearson's Correlation Coefficient, same as the CORREL function.
• Regression Equation: Provides the intercept and coefficients for the regression equation.
• Line Fit Plots : A scatter plot with data points (actual tips) and predicted tips. Notice the axes are labeled for checks and tips.

Residual Output Table

• Observation: Numbers each check-tip pair.
• Predicted tip: The tip value predicted by the regression line for a given check total.
• Residuals: The difference between the actual tip and the predicted tip (Actual - Predicted). A negative residual means the prediction was too high; a positive residual means the prediction was too low.
• $Residuals = Observed - Predicted$