Residuals Analysis is a critical statistical technique used in the Improve Phase of Lean Six Sigma to validate regression models and ensure the accuracy of predictions. Residuals are the differences between observed values and predicted values from a regression model. Analyzing these residuals help…Residuals Analysis is a critical statistical technique used in the Improve Phase of Lean Six Sigma to validate regression models and ensure the accuracy of predictions. Residuals are the differences between observed values and predicted values from a regression model. Analyzing these residuals helps practitioners determine whether their model is appropriate and reliable for process improvement decisions.
There are four key assumptions that must be checked through residuals analysis: normality, independence, constant variance (homoscedasticity), and randomness. First, residuals should follow a normal distribution, which can be verified using a normal probability plot or histogram. If residuals appear normally distributed, the model is considered valid for making statistical inferences.
Second, residuals should be independent of each other, meaning one residual should not predict another. This is particularly important when data is collected over time. A pattern in residuals plotted against time order suggests autocorrelation, indicating the model may be missing important time-related factors.
Third, residuals should exhibit constant variance across all levels of predicted values. When plotted against fitted values, residuals should scatter randomly in a horizontal band. A funnel or cone shape indicates heteroscedasticity, suggesting the model performs differently at various prediction levels.
Fourth, residuals should appear random when plotted against fitted values and predictor variables. Any systematic patterns such as curves or trends indicate the model is not capturing all relationships in the data, and additional terms or transformations may be needed.
Practitioners typically use four-in-one residual plots to efficiently assess all these assumptions simultaneously. When violations are detected, corrective actions include transforming variables, adding polynomial terms, or considering alternative modeling approaches. Proper residuals analysis ensures that improvement recommendations are based on sound statistical foundations, leading to more effective and sustainable process improvements in Lean Six Sigma projects.
Residuals Analysis in Six Sigma Green Belt - Improve Phase
What is Residuals Analysis?
Residuals analysis is a statistical technique used to evaluate the validity of a regression model by examining the differences between observed values and predicted values. A residual is simply the difference between an actual data point and the value predicted by your regression model (Residual = Observed Value - Predicted Value).
Why is Residuals Analysis Important?
In the Improve Phase of DMAIC, residuals analysis serves several critical purposes:
• Model Validation: It confirms whether your regression model accurately represents the relationship between variables • Assumption Checking: It verifies that the assumptions underlying regression analysis are met • Error Detection: It helps identify outliers, influential points, and systematic patterns that may indicate model problems • Prediction Reliability: It ensures that predictions made from your model are trustworthy
How Does Residuals Analysis Work?
When performing residuals analysis, you examine four key assumptions:
1. Normality: Residuals should follow a normal distribution. This is checked using a normal probability plot or histogram of residuals.
2. Independence: Residuals should be independent of each other with no patterns when plotted against observation order.
3. Constant Variance (Homoscedasticity): The spread of residuals should remain constant across all fitted values. When plotted against fitted values, residuals should show a random scatter with consistent width.
4. Linearity: There should be no curved patterns in the residuals versus fitted values plot, indicating the linear model is appropriate.
Key Residual Plots to Understand:
• Residuals vs. Fitted Values: Should show random scatter around zero with constant spread • Normal Probability Plot: Points should fall approximately along a straight line • Residuals vs. Order: Should show no time-based patterns • Histogram of Residuals: Should appear approximately bell-shaped
Common Problems Identified Through Residuals Analysis:
• Funnel Shape: Indicates non-constant variance (heteroscedasticity) • Curved Pattern: Suggests a non-linear relationship exists • Clusters or Trends: May indicate missing variables or autocorrelation • Outliers: Large residuals may represent unusual observations requiring investigation
Exam Tips: Answering Questions on Residuals Analysis
Tip 1: Remember the acronym LINE - Linearity, Independence, Normality, Equal variance. These are the four assumptions you check with residuals.
Tip 2: When shown a residual plot, first look for random scatter. Any recognizable pattern suggests a problem with the model.
Tip 3: Know the difference between standardized and unstandardized residuals. Standardized residuals greater than 2 or 3 in absolute value typically indicate potential outliers.
Tip 4: Questions often present residual plots and ask you to identify violations. Practice recognizing funnel shapes (variance issues), curves (linearity issues), and trends (independence issues).
Tip 5: Understand that a good residual plot shows points randomly scattered around zero with no discernible pattern - this indicates your model is appropriate.
Tip 6: Be prepared to recommend corrective actions: transformations for non-linearity or non-constant variance, adding variables for unexplained patterns, or investigating outliers.
Tip 7: Remember that residuals analysis is performed AFTER fitting a regression model, not before. It is a diagnostic tool for model adequacy.
Tip 8: On calculation questions, always use the formula: Residual = Observed - Predicted. A positive residual means the model underpredicted; a negative residual means it overpredicted.