Error Variance in Statistical Models

Definition

• Error variance, also known as residual variance, is the variance in a dependent variable that cannot be explained by the independent variable(s) in a statistical model.

Calculation

• Error variance is calculated as the average of the squared differences between observed and predicted values of the dependent variable.
• The formula for error variance is: σ² = Σ(ei²) / (n - k - 1), where:
  • σ² is the error variance
  • ei is the residual for each observation
  • n is the total number of observations
  • k is the number of independent variables

Interpretation

• A small error variance indicates that the model is a good fit to the data, as most of the variance in the dependent variable can be explained by the independent variable(s).
• A large error variance indicates that the model is a poor fit to the data, as most of the variance in the dependent variable remains unexplained.

Importance

• Error variance is an important concept in statistical modeling, as it helps to evaluate the goodness of fit of a model and identify potential sources of error.
• A low error variance is often desirable, as it indicates that the model is able to accurately predict the dependent variable.

Relationship with R-squared

• Error variance is related to R-squared (R²), which is a measure of the proportion of variance in the dependent variable that is explained by the independent variable(s).
• A high R² value indicates a low error variance, and vice versa.

Definition of Error Variance

• Error variance, also known as residual variance, is the unexplained variance in a dependent variable.
• It represents the variance that cannot be attributed to the independent variable(s) in a statistical model.

Calculating Error Variance

• Error variance is calculated as the average of the squared differences between observed and predicted values of the dependent variable.
• The formula for error variance is: σ² = Σ(ei²) / (n - k - 1), where:
  • σ² is the error variance
  • ei is the residual for each observation
  • n is the total number of observations
  • k is the number of independent variables

Interpreting Error Variance

• A small error variance indicates a good fit of the model to the data.
• A large error variance indicates a poor fit of the model to the data.
• Error variance helps to evaluate the goodness of fit of a model and identify potential sources of error.

Importance of Error Variance

• Error variance is a crucial concept in statistical modeling.
• A low error variance is desirable, as it indicates that the model accurately predicts the dependent variable.

Relationship with R-squared

• Error variance is related to R-squared (R²), which measures the proportion of variance in the dependent variable explained by the independent variable(s).
• A high R² value indicates a low error variance, and vice versa.