Glossary

12.3 The Regression Equation

2 min read•june 25, 2024

helps us understand relationships between variables. It's like drawing a line through scattered dots to see patterns. We use this to predict outcomes based on input data.

The gives us a formula for that line. It shows how one variable changes when another does. This helps us make educated guesses about future trends or unknown values.

The Regression Equation

Least-squares regression line calculation

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  • fits data points best by minimizing between points and line
  • Equation of least-squares regression line: y^=b0+b1x\hat{y} = b_0 + b_1x
    • y^\hat{y}: predicted value of ()
    • b0b_0: of regression line
    • b1b_1: of regression line
    • xx: value of ()
  • Calculate slope (b1b_1) and y-intercept (b0b_0) using formulas:
    1. b1=∑i=1n(xi−xˉ)(yi−yˉ)∑i=1n(xi−xˉ)2b_1 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2}
    2. b0=yˉ−b1xˉb_0 = \bar{y} - b_1\bar{x}
    • xˉ\bar{x}: mean of
    • yˉ\bar{y}: mean of
    • xix_i and yiy_i: individual values of explanatory and response variables

Interpretation of regression slope

  • Slope (b1b_1) represents change in response variable (yy) for one-unit increase in explanatory variable (xx)
  • Interpretation depends on context of variables studied
    • Salary example: slope of 1,500 indicates employee's salary increases by $1,500 on average for each additional year of experience

Correlation and determination coefficients

  • (rr) measures strength and direction of linear relationship between explanatory and response variables
    • rr ranges from -1 to 1, values closer to -1 or 1 indicate stronger linear relationship
    • Positive rr: positive linear relationship
    • Negative rr: negative linear relationship
  • (r2r^2) represents proportion of variation in response variable explained by explanatory variable in regression model
    • r2r^2 ranges from 0 to 1, values closer to 1 indicate more variation explained by explanatory variable
    • Example: r2=0.75r^2 = 0.75 means 75% of variation in response variable explained by explanatory variable, 25% due to other factors or random variation

Linear Regression Analysis

  • is a statistical method used to model the relationship between variables
  • Scatterplots are used to visualize the relationship between two variables
  • The (regression line) is determined through
  • Regression analysis helps identify patterns and make predictions based on the relationship between variables

Key Terms to Review (22)

Coefficient of determination: The coefficient of determination, denoted as $R^2$, measures the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It ranges from 0 to 1, where a higher value indicates a better fit of the model.
Coefficient of Determination: The coefficient of determination, denoted as $R^2$, is a statistical measure that represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s) in a regression model. It is a key concept in understanding the strength and predictive power of a regression analysis.
Correlation Coefficient: The correlation coefficient is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, with -1 indicating a perfect negative correlation, 0 indicating no correlation, and 1 indicating a perfect positive correlation.
Dependent Variable: The dependent variable is the outcome or response variable in a study or experiment. It is the variable that is measured or observed to determine the effect of the independent variable. The dependent variable depends on or is influenced by the independent variable.
Explanatory variable: An explanatory variable is a type of independent variable used in experiments to explain variations in the response variable. It is manipulated by researchers to observe its effect on the dependent variable.
Explanatory Variable: An explanatory variable, also known as an independent variable, is a variable that is manipulated or controlled in a study to determine its effect on the dependent or response variable. It is the variable that is believed to influence or cause changes in the outcome or dependent variable.
Least-Squares Line: A least-squares line, also known as the line of best fit, is a straight line that minimizes the sum of the squared differences between observed values and the values predicted by the line. It is used in linear regression to model the relationship between two variables.
Least-Squares Regression Line: The least-squares regression line is a statistical model that represents the best-fitting straight line through a set of data points, minimizing the sum of the squared vertical distances between the data points and the line.
Line of Best Fit: The line of best fit, also known as the regression line, is a straight line that best represents the relationship between two variables in a scatter plot. It is used to make predictions and estimate the value of one variable based on the value of the other variable.
Linear regression: Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. It aims to predict the value of the dependent variable based on the values of the independent variables.
Linear Regression: Linear regression is a statistical technique used to model the relationship between a dependent variable and one or more independent variables. It aims to find the best-fitting straight line that describes the linear association between the variables.
Predictor Variable: A predictor variable, also known as an independent variable, is a variable that is used to predict or explain the outcome of a dependent variable in a regression analysis. It is the variable that is manipulated or controlled to observe its effect on the dependent variable.
Regression Analysis: Regression analysis is a statistical technique used to model the relationship between a dependent variable and one or more independent variables. It allows researchers to estimate the average change in the dependent variable associated with a one-unit change in the independent variable, while controlling for other factors.
Regression Equation: The regression equation is a mathematical model that describes the relationship between a dependent variable and one or more independent variables. It is used to predict the value of the dependent variable based on the values of the independent variables.
Residuals: Residuals, in the context of statistical analysis, refer to the differences between the observed values and the predicted values from a regression model. They represent the unexplained or unaccounted-for portion of the variability in the dependent variable, providing insights into the quality and fit of the regression model.
Response variable: A response variable is the outcome or dependent variable that researchers measure in an experiment to determine the effect of treatments. It is what changes as a result of variations in the independent variable.
Response Variable: The response variable, also known as the dependent variable, is the variable that is measured or observed in an experiment or study. It is the outcome or the characteristic of interest that may be influenced or predicted by the independent variable(s).
Scatterplot: A scatterplot is a type of data visualization that displays the relationship between two variables by plotting individual data points on a coordinate plane. It allows for the visual exploration of the strength and direction of the association between the variables.
Slope: Slope is a measure of the steepness or incline of a line, typically represented as the ratio of the change in the vertical direction (rise) to the change in the horizontal direction (run) between two points on the line. It serves as a key component in understanding linear relationships and is vital for forming predictions based on data trends.
Sum of Squared Errors (SSE): Sum of Squared Errors (SSE) measures the total deviation of observed values from the values predicted by a regression model. It is calculated by summing the squared differences between observed and predicted values.
Sum of Squared Vertical Distances: The sum of squared vertical distances is a measure used in linear regression analysis to determine the goodness of fit between a regression line and the observed data points. It quantifies the total deviation of the data points from the predicted values on the regression line.
Y-intercept: The y-intercept is the point at which a linear equation or regression line intersects the y-axis, representing the value of the dependent variable when the independent variable is zero. It is a crucial parameter in understanding the relationship between two variables and making predictions.
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