Explained variance

Definition

Explained variance is a statistical measure that reflects how much of the total variability in a dataset can be accounted for by a particular model or set of predictors. It quantifies the portion of variance that is attributed to the relationship between the independent and dependent variables, indicating how well the model explains the observed data.

5 Must Know Facts For Your Next Test

1. Explained variance helps assess the effectiveness of a regression model in capturing the relationship between variables.
2. Higher explained variance values suggest that the model is better at predicting outcomes, while lower values indicate a poor fit.
3. Explained variance can be computed as the ratio of explained variance to total variance, often represented mathematically as $R^2 = \frac{\text{Explained Variance}}{\text{Total Variance}}$.
4. It is crucial for model selection, as comparing explained variances of different models can guide the choice of the most appropriate model for the data.
5. Explained variance is sensitive to outliers, which can disproportionately affect both explained and total variance measures.

Review Questions

• How does explained variance inform us about the effectiveness of a predictive model?
  • Explained variance provides insight into how well a predictive model captures the relationship between independent and dependent variables. A higher explained variance indicates that a greater proportion of the total variability in the data can be attributed to the model, suggesting it is effective in making accurate predictions. Conversely, low explained variance signals that the model may not be fitting the data well and that there may be important predictors not included in the analysis.
• Discuss how explained variance is related to residual variance and what this relationship reveals about model performance.
  • Explained variance is directly related to residual variance in that they together make up total variance. Specifically, total variance is equal to explained variance plus residual variance. This relationship reveals that if a model has high explained variance, it corresponds to low residual variance, meaning there are fewer errors in prediction. This balance is crucial when evaluating model performance; a well-fitting model will minimize residuals while maximizing explained variance.
• Evaluate the implications of explained variance when comparing different statistical models for prediction accuracy.
  • When comparing different statistical models, explained variance serves as a critical metric for assessing prediction accuracy. A model with a higher explained variance relative to others demonstrates its superiority in capturing data relationships and making accurate predictions. However, it's essential to consider other factors such as overfitting, complexity, and generalizability. Evaluating explained variance alongside these considerations helps in selecting not just the most accurate but also the most robust model for future data.

Related terms

Total Variance: The overall variability present in the data, encompassing both explained and unexplained variance.

Coefficient of Determination (R²): A statistical measure that indicates the proportion of the variance in the dependent variable that can be predicted from the independent variables, often expressed as a value between 0 and 1.

Residual Variance: The portion of total variance that remains unexplained by the model, representing the errors or differences between observed and predicted values.