Dataplot Vol 1 Vol 2

# BOX COX HOMOSCEDASTICITY PLOT

Name:
BOX-COX HOMOSCEDASTICITY PLOT
Type:
Graphics Command
Purpose:
Generates a Box-Cox homoscedasticity plot.
Description:
Many statistical procedures (e.g., regression) make assumptions of constant variance relative to the value of an independent variable. For example, in regression it is assumed that the variance of the residuals does not depend on the value of the independent variable. This assumption is generally referred to as homogeneous variances or as homoscedasticity.

A Box-Cox homoscedasticity plot is a graphical technique for determining the Box-Cox transformation that yields the most constant variance of one variable relative to the values of a second variable.

The Box-Cox family is essentially the power-transformation family (adjusted to include log transformations). The form of the family is:

$$T_{y} = \frac{y^{\lambda}}{\lambda}$$

There are various methods for measuring constant variance. The particular method Dataplot uses is to divide the first variable into groups with the same value for the second value. For a given value of lambda, the standard deviation is computed for each group. The statistic used is the ratio of the minimum standard deviation to the maximum standard deviation (this ratio will always be between 0 and 1). The plot then consists of this statistic on the vertical axis versus the lambda parameter on the horizontal axis. The lambda corresponding to the highest ratio is the appropriate transformation to use to provide the most constant variance.

This command only applies if there is replication in the second response variable.

Syntax 1:
BOX-COX HOMOSCEDASTICITY PLOT <y> <x>
<SUBSET/EXCEPT/FOR qualification>
where <y> is the first response variable;
<x> is the second response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Syntax 2:
REPLICATED BOX-COX HOMOSCEDASTICITY PLOT <y> <x> <tag1>
<SUBSET/EXCEPT/FOR qualification>
where <y> is the first response variable;
<x> is the second response variable;
<tag1> is a group-id variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

A Box-Cox homoscedasticity plot will be generated for each distinct value of the group-id variable. These plots will be overlaid on the same plot.

Syntax 3:
REPLICATED BOX-COX HOMOSCEDASTICITY PLOT <y> <x> <tag1> <tag2>
<SUBSET/EXCEPT/FOR qualification>
where <y> is the first response variable;
<x> is the second response variable;
<tag1> is a group-id variable;
<tag2> is a group-id variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional.

The two group-id variables are cross-tabulated and a Box-Cox homoscedasticity plot will be generated for each distinct combination of values for the group-id variables. These plots will be overlaid on the same plot.

Examples:
BOX COX HOMOSCED PLOT Y X
REPLICATED BOX COX HOMOSCED PLOT Y X TAG
Note:
The number of observations in the two response variables (and any group-id variables) must be equal.
Note:
The TO syntax is supported for the BOX COX HOMOSCEDASTICITY command.
Default:
None
Synonyms:
BOX-COX HOMOGENITY PLOT
Related Commands:
 BOX-COX NORMALITY PLOT = Generates a Box-Cox normality plot. BOX-COX LINEARITY PLOT = Generates a Box-Cox linearity plot. LINES = Sets type for plot lines. CHARACTER = Sets type for plot lines. PROBABILITY PLOT = Generates a probability plot. PLOT = Generates a data or function plot.
Reference:
Box and Cox (1964), "An Analysis of of transformations," Journal of the Royal Statistical Society, Series B 26 (2): 211–252.
Applications:
Exploratory Data Analysis
Implementation Date:
93/7
2010/5: Support for REPLICATION option
Program:

SKIP 25
.
MULTIPLOT CORNER COORDINATES 0 0 100 100
MULTIPLOT SCALE FACTOR 2
MULTIPLOT 2 2
FIT Y X2
LINE SOLID BLANK
CHARACTER BLANK X
TITLE LINEAR FIT OF RAW DATA
PLOT PRED Y VS X2
.
TITLE BOX-COX HOMOSCEDASTICITY PLOT
X1LABEL LAMBDA
Y1LABEL CORRELATION COEFFICIENT
BOX-COX HOMOSCEDASTICTY PLOT Y X2
.
LET YTEMP = MAXIMUM YPLOT
RETAIN XPLOT SUBSET YPLOT = YTEMP
LET LAMBDA = XPLOT(1)
LET Y2 = (Y**LAMBDA - 1)/LAMBDA
FIT Y2 X2
TITLE LINEAR FIT OF TRANSFORMED DATA
X1LABEL ; Y1LABEL
PLOT PRED Y2 VS X2
END OF MULTIPLOT


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Date created: 11/30/2010
Last updated: 10/13/2015