Measures of Regression and Prediction Intervals: Variation, Coefficient of Determination, and Standard Error of Estimate

Correlation and Regression

Measures of Regression and Prediction Intervals

This section explores the quantitative assessment of how well a regression line fits a set of data. It covers the types of variation about a regression line, the coefficient of determination, and the standard error of estimate, all of which are essential for interpreting linear regression results in statistics.

Variation About a Regression Line

Types of Variation

Variation about a regression line is divided into three main types, each representing a different aspect of how data points deviate from the regression model:

• Total Variation: The sum of the squares of the differences between each observed y-value and the mean of y. It quantifies the overall spread of the data.

• Explained Variation: The sum of the squares of the differences between each predicted y-value (from the regression line) and the mean of y. It measures how much of the total variation is accounted for by the regression model.

• Unexplained Variation: The sum of the squares of the differences between each observed y-value and its corresponding predicted y-value. It represents the portion of variation not explained by the regression model.

For each ordered pair $(x_i, y_i)$ in a data set, the following deviations are calculated:

• Total deviation: $y_i - \overline{y}$

• Explained deviation: $\hat{y}_i - \overline{y}$

• Unexplained deviation: $y_i - \hat{y}_i$

Formulas:

• Total variation: $\sum (y_i - \overline{y})^2$

• Explained variation: $\sum (\hat{y}_i - \overline{y})^2$

• Unexplained variation: $\sum (y_i - \hat{y}_i)^2$

• Total variation = Explained variation + Unexplained variation

Coefficient of Determination

Definition and Interpretation

The coefficient of determination, denoted as $r^2$, is a statistical measure that quantifies the proportion of the total variation in the dependent variable that is explained by the regression model. It is calculated as:

$r^2 = \frac{\text{Explained variation}}{\text{Total variation}}$

• Interpretation: An $r^2$ value close to 1 indicates that most of the variation in the dependent variable is explained by the regression model, while a value close to 0 indicates little explanatory power.

• Example: In a study of children's vocabulary size and age, $r^2 \approx 0.964$ means that 96.4% of the variation in vocabulary size is explained by age, and 3.6% is unexplained.

Application Example:

• Given data for building heights and number of stories, $r^2 \approx 0.905$ indicates that 90.5% of the variation in stories is explained by height.

Standard Error of Estimate

Definition and Calculation

The standard error of estimate ($s_e$) measures the average distance that the observed values fall from the regression line. It is the standard deviation of the observed $y_i$ values about the predicted $\hat{y}_i$ values for a given $x_i$ value. The formula is:

$s_e = \sqrt{\frac{\sum (y_i - \hat{y}_i)^2}{n - 2}}$

• Interpretation: The smaller the standard error, the closer the data points are to the regression line, indicating a better fit.

• Calculation Steps:

  1. Make a table with columns for $x_i$, $y_i$, $\hat{y}_i$, $y_i - \hat{y}_i$, and $(y_i - \hat{y}_i)^2$.

  2. Use the regression equation to calculate predicted $y$ values.

  3. Sum the squared differences $(y_i - \hat{y}_i)^2$.

  4. Apply the formula to find $s_e$.

Example: For children's age and vocabulary size, $s_e = \sqrt{\frac{312625.8}{11-2}} \approx 186.38$. This means the average distance from the regression line is about 186 vocabulary size units.

Excel Application: For building heights and stories, the standard error of estimate is 2.138, meaning the average distance from the regression line is about 2.138 stories.

Summary Table: Variation and Regression Measures

Measure	Formula	Interpretation
Total Variation	$\sum (y_i - \overline{y})^2$	Overall spread of data
Explained Variation	$\sum (\hat{y}_i - \overline{y})^2$	Variation explained by regression
Unexplained Variation	$\sum (y_i - \hat{y}_i)^2$	Variation not explained by regression
Coefficient of Determination ($r^2$)	$\frac{\text{Explained variation}}{\text{Total variation}}$	Proportion of explained variation
Standard Error of Estimate ($s_e$)	$\sqrt{\frac{\sum (y_i - \hat{y}_i)^2}{n - 2}}$	Average distance from regression line

Practice Example: Coefficient of Determination

Sample Calculation

Given temperature ($x$) and number of ice cream cones sold per hour ($y$):

• Data: $x = 65, 70, 75, 80, 85, 90, 95, 100, 105$; $y = 8, 10, 11, 13, 12, 16, 19, 22, 23$

• Possible $r^2$ values: 0.946, 0.973, 0.923, 0.986

• Interpretation: The correct $r^2$ value indicates the proportion of variation in ice cream sales explained by temperature.

Additional info: These measures are fundamental for evaluating the effectiveness of linear regression models in statistics, and are directly relevant to Chapter 9: Correlation and Regression.