1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.3. Graphical Techniques: Alphabetic
1.3.3.33. 6Plot

Purpose:
Graphical Model Validation

The 6plot is a collection of 6 specific graphical techniques
whose purpose is to assess the validity of a Y versus X fit.
The fit can be a linear fit, a nonlinear fit, a LOWESS
(locally weighted least squares) fit, a spline fit, or any
other fit utilizing a single independent variable.
The 6 plots are:
 Scatter plot of the response
and predicted values versus the independent
variable;
 Scatter plot of the residuals
versus the independent variable;
 Scatter plot of the residuals
versus the predicted values;
 Lag plot of the residuals;
 Histogram of the residuals;
 Normal probability plot of the
residuals.

Sample Plot

This 6plot of
the PONTIUS.DAT data set,
which followed a linear fit, shows that the linear model is not
adequate. It suggests that a quadratic model would be a better
model.

Definition:
6 Component Plots

The 6plot consists of the following:
 Response and predicted values
 Vertical axis: Response variable, predicted values
 Horizontal axis: Independent variable
 Residuals versus independent variable
 Vertical axis: Residuals
 Horizontal axis: Independent variable
 Residuals versus predicted values
 Vertical axis: Residuals
 Horizontal axis: Predicted values
 Lag plot of residuals
 Vertical axis: RES(I)
 Horizontal axis: RES(I1)
 Histogram of residuals
 Vertical axis: Counts
 Horizontal axis: Residual values
 Normal probability plot of residuals
 Vertical axis: Ordered residuals
 Horizontal axis: Theoretical values from a
normal N(0,1) distribution for ordered residuals

Questions

The 6plot can be used to answer the following questions:
 Are the residuals approximately normally distributed
with a fixed location and scale?
 Are there outliers?
 Is the fit adequate?
 Do the residuals suggest a better fit?

Importance:
Validating Model

A model involving a response variable and a single independent
variable has the form:
\[ Y_{i} = f(X_{i}) + E_{i} \]
where Y is the response variable, X is the independent
variable, f is the linear or nonlinear fit function, and
E is the random component. For a good model, the
error component should behave like:
 random drawings (i.e., independent);
 from a fixed distribution;
 with fixed location; and
 with fixed variation.
In addition, for fitting models it is usually further assumed
that the fixed distribution is normal and the fixed location
is zero. For a good model the fixed variation
should be as small as possible. A necessary component
of fitting models is to verify these assumptions for the
error component and to assess whether the variation for
the error component is sufficiently small. The histogram,
lag plot, and normal probability plot are used to
verify the fixed distribution, location, and variation
assumptions on the error component. The plot
of the response variable and the predicted values versus
the independent variable is used to assess whether the
variation is sufficiently small. The plots of the
residuals versus the independent variable and the predicted
values is used to assess the independence assumption.
Assessing the validity and quality of the fit in terms of
the above assumptions is an absolutely vital part of
the modelfitting process. No fit should be considered
complete without an adequate model validation step.

Related Techniques

Linear Least
Squares
NonLinear Least
Squares
Scatter Plot
Run Sequence Plot
Lag Plot
Normal Probability Plot
Histogram

Case Study

The 6plot is used in the
Alaska pipeline
data case study.

Software

It should be feasible to write a macro for the 6plot in any
general purpose statistical software program that supports
the capability for multiple plots per page and supports the
underlying plot techniques.
