Glossary

12.2 The Regression Equation

3 min read•june 27, 2024

Regression equations help us understand relationships between variables. They use data to find the best-fitting line, showing how one thing changes when another does. This powerful tool lets us predict outcomes and analyze trends.

The equation has two key parts: and . The slope shows how much y changes when x increases by one. The y-intercept is where the line crosses the y-axis. Together, they paint a picture of the data's pattern.

The Regression Equation

Least-squares regression line calculation

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  • best fits data points minimizes sum of squared
    • Residuals represent vertical distances between data points and regression line (yi−y^iy_i - \hat{y}_i)
  • in form y^=b0+b1x\hat{y} = b_0 + b_1x
    • y^\hat{y} represents predicted value of y for given x
    • b0b_0 represents y-intercept, value of y when x is 0 (height at origin)
    • b1b_1 represents slope, change in y for one-unit increase in x (rise over run)
  • Calculate slope (b1b_1) using formula: 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}
    • xix_i and yiy_i represent individual data points (coordinates)
    • xˉ\bar{x} and yˉ\bar{y} represent means of x and y, respectively (averages)
  • Calculate y-intercept (b0b_0) using formula: b0=yˉ−b1xˉb_0 = \bar{y} - b_1\bar{x}
    • Substitute slope (b1b_1) and means (xˉ\bar{x} and yˉ\bar{y}) into equation

Interpretation of slope and y-intercept

  • Slope (b1b_1) represents change in (y) for one-unit increase in (x)
    • Positive slope indicates positive between x and y (direct)
    • Negative slope indicates negative linear relationship between x and y (inverse)
  • Y-intercept (b0b_0) represents value of response variable (y) when predictor variable (x) is 0
    • Y-intercept may not have meaningful interpretation if x cannot realistically be 0 ()
  • Interpret slope and y-intercept using context of data and units of variables
    • Slope units: units of y per unit of x (mph per year)
    • Y-intercept units: same as units of y (starting salary in dollars)

Strength of linear relationships

  • (r) measures strength and direction of linear relationship between two variables
    • r ranges from -1 to 1
      1. r = 1 indicates perfect positive linear relationship (straight line increasing)
      2. r = -1 indicates perfect negative linear relationship (straight line decreasing)
      3. r = 0 indicates no linear relationship (scattered points)
    • Stronger linear relationship as r approaches -1 or 1 (tighter clustering around line)
    • Calculate r using formula: r=∑i=1n(xi−xˉ)(yi−yˉ)∑i=1n(xi−xˉ)2∑i=1n(yi−yˉ)2r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2}\sqrt{\sum_{i=1}^{n}(y_i - \bar{y})^2}}
  • (r2r^2) represents proportion of variation in response variable (y) explained by predictor variable (x)
    • r2r^2 ranges from 0 to 1
      1. r2=1r^2 = 1 indicates all variation in y explained by x (perfect fit)
      2. r2=0r^2 = 0 indicates none of variation in y explained by x (no relationship)
    • Calculate r2r^2 by squaring correlation coefficient (r)
    • r2r^2 often expressed as percentage (50% of variation explained)
  • Visualize relationship between variables using a

Assessing model fit and reliability

  • Examine residuals for patterns or trends to assess model assumptions
    • Check for (constant variance of residuals across predictor values)
  • Identify potential that may influence regression results
  • measures average deviation of observed values from predicted values
  • Calculate confidence intervals for slope and intercept to assess precision of estimates

Key Terms to Review (17)

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 valuable tool for assessing the goodness of fit and the strength of the relationship between the variables in a regression analysis.
Confidence Interval: A confidence interval is a range of values that is likely to contain an unknown population parameter, such as a mean or proportion, with a specified level of confidence. It provides a way to quantify the uncertainty associated with estimating a population characteristic from a sample.
Correlation Coefficient: The correlation coefficient is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. It is a value that ranges from -1 to 1, with -1 indicating a perfect negative linear relationship, 0 indicating no linear relationship, and 1 indicating a perfect positive linear relationship.
Extrapolation: Extrapolation is the process of using a known set of data or information to estimate or predict values or outcomes beyond the original range of observation. It involves extending the known pattern or trend of a variable to make inferences about values that lie outside the original data set.
Homoscedasticity: Homoscedasticity is a statistical concept that refers to the assumption of equal variance or constant variance across different groups or observations within a dataset. It is a crucial assumption in various statistical analyses, including regression analysis and hypothesis testing.
Least-Squares Regression Line: The least-squares regression line is a statistical technique used to find the line of best fit for a set of data points, minimizing the sum of the squared vertical distances between the data points and the line. This line represents the linear relationship between two variables and is a crucial component in the topics of 'The Regression Equation' and 'Prediction'.
Linear Relationship: A linear relationship is a mathematical association between two variables where the change in one variable is proportional to the change in the other variable. This relationship can be represented by a straight line when the variables are plotted on a graph.
Outliers: Outliers are data points that lie an abnormal distance from other values in a dataset. They are observations that are markedly different from the rest of the data, often due to measurement errors, experimental conditions, or natural variability within the population.
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 a variable that is hypothesized to influence or determine the value of another variable.
R-squared: R-squared, also known as the coefficient of determination, 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 analysis. It is a key metric used to assess the goodness of fit of a regression model.
Regression Equation: The regression equation is a mathematical model that describes the relationship between a dependent variable and one or more independent variables. It allows for the prediction of the dependent variable's value based on the values of the independent variables.
Residuals: Residuals, in the context of regression analysis, refer to the differences between the observed values of the dependent variable and the predicted values based on the regression model. They represent the unexplained or unaccounted for variation in the data, providing valuable insights into the model's fit and the presence of any patterns or anomalies.
Response Variable: The response variable, also known as the dependent variable, is the variable of interest that is measured or observed in a study. It is the variable that is expected to change or respond based on changes in the independent or explanatory variable(s).
Scatter Plot: A scatter plot is a type of data visualization that displays the relationship between two numerical variables. It presents data points on a coordinate plane, with one variable plotted on the x-axis and the other on the y-axis, allowing for the identification of patterns, trends, and potential correlations between the variables.
Slope: Slope is a measure of the steepness or incline of a line or surface. It represents the rate of change between two variables, typically the dependent and independent variables in a linear relationship.
Standard Error of Estimate: The standard error of estimate is a measure of the average amount of error or variability in the predictions made from a regression equation. It quantifies the uncertainty in the predicted values by providing an estimate of the standard deviation of the residuals or the difference between the observed and predicted values.
Y-intercept: The y-intercept is the point where 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 behavior of linear relationships and making predictions.
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