Dataplot Vol 1 Auxiliary Chapter

# DEX CONTOUR PLOT

Name:
DEX CONTOUR PLOT
Type:
Graphics Command
Purpose:
Generates a linear dex contour plot.
Description:
A contour plot is a graphical technique for representing a 3-dimensional z = f(x,y) surface by plotting constant-z "slices" (contours) on a 2-dimensional format.

The dex contour plot is a specialized contour plot used in the design of experiments.. It is used in the case where you have two independent factors, each with two levels. The low level is coded as "-1" and the high level is coded as "+1". In addition, you can optionally have center points (center points are coded as "0" and represent the value of the independent variable halfway between the low level and high level of that variable). These types of designs are common for full and fractional factorial designs. The dex contour plot is typically generated for the two most important factors in the full or fractional factorial design. The determination of exactly which are the two most important factors is determined prior to generating the dex contour plot (e.g., through the use of dex mean plots).

The typical application of the dex contour plot is in determining settings that will maximize (or minimize) the response variable. It can also be helpful in determining settings that result in the response variable hitting a pre-determined target value.

The following are the primary steps in the construction of the dex contour plot.

1. The x and y axes of the plot represent the values of the first and second factor (independent) variable.

2. The 4 vertex points are drawn. The vertex points are (-1,-1), (-1,1), (1,1), (1,-1). At each vertex point, the average of all the response values at that vertex point is printed.

3. Similarly, if there are center points, a point is drawn at (0,0) and the average of the response values at the center points is printed.

4. The linear dex contour plot assumes the model:

Y = MU + 0.5*(B1*U1 + B2*U2 + B12*U1*U2)

where MU is the overall mean of the response variables. The values of B0, B1, B12, and MU are estimated from the vertex points using a Yates analysis (the Yates analysis utilizes the special structure of the 2-level full and fractional factorial designs to simplify the computation of these parameters). Note that Dataplot does not in fact generate a full Yates analysis at this point, it simply utilizes the Yates algorithm for determing the estimates for these specific parameters.

In order to generate a single contour line, we need a value for Y, say Y0. Next, we solve for U2 in terms of U1 and, after doing the algebra, we have the equation:

U2 = (2*(Y0 - MU) - B1*U1)/(B2 + B12*U1)

We generate a sequence of points for U1 in the range -2 to 2 and compute the corresponding values of U2. These points constitute a single contour line corresponding to Y = Y0.

The user specifies the target values for which contour lines will be generated.

5. Dataplot generates a curvature test. This is printed as part of the alpanumeric output, not on the graph itself.

The curvature test is a t-test based on the response values at the vertex points and the center points. Note that this test is only applied if center points are in fact present. Also, the center points are NOT utilized in the estimation of B1, B2, and B12. They are only used as part of the curvature test.

The curvature test is used in determining if the linear model is an fact an adequate model for the data. If the null hypothese is rejected, the linear model is not adequate. In that case, we would probably consider the use of a quadratic, as oppossed to a linear, model.

Dataplot does not generate a dex contour plot for the quadratic case directly. However, the built-in macro, DEXCONTQ.DP, can be used. Enter LIST DEXCONTQ.DP for details on how to run this macro. This macro uses the Dataplot CONTOUR PLOT and FIT commands to generate the quadratic dex contour plot.

Syntax:
DEX CONTOUR PLOT <y> <x1> <x2> <ycont> <SUBSET/EXCEPT/FOR qualification>
where <y> is the response (= dependent) variable;
<x1> is the first factor (= independent) variable;
<x2> is the second factor (= independent) variable;
<ycont> is the variable containing the desired contour levels;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
LET YCONT = SEQUENCE 30 5 90
DEX CONTOUR PLOT Y X1 X2 YCONT
Note:
The appearance of the plot is controlled by the settings of the LINE and CHARACTER command. In addition, you can use the attribute settings for LINE and CHARACTER (e.g., color, thickness) to further control the appearance.

Specifically, if the design has center points, then the line and character settings are controlled by:

 trace 1 labels for the center point and the points at (-1,-1), (-1,1), (1,1), (1,-1). The character setting is automatically set to a special type called ZVAL and line is set to blank. Although the character type is fixed, you can still set the attributes. For example, you may want to use CHARACTER OFFSET to position the text. trace 2 the center point. The line setting is typically blank and the character type is commonly set to CIRCLE. trace 3 line connecting (-1,-1), (1,-1), (1,1), (-1,1). The line type is typically set to SOLID and the character type is typically set to CIRCLE. trace 4+ the contour lines start with trace 4. There is one trace for each value of YCONT. This allows you to set the attributes for each contour line individually. For example, you may want to draw a thickened or colored line at the maximum or minimum contour level. By default, the contour levels are drawn from the maximum to the minimum value.

If the design does not have center points, then the line and character settings are controlled by:

 trace 1 labels for the vertex points at (-1,-1), (-1,1), (1,1), (1,-1). The character setting is automatically set to a special type called ZVAL and line is set to blank. Although the character type is fixed, you can still set the attributes. For example, you may want to use CHARACTER OFFSET to position the text. trace 2 line connecting (-1,-1), (1,-1), (1,1), (-1,1). The line type is typically set to SOLID and the character type is typically set to CIRCLE. trace 3+ the contour lines start with trace 3. There is one trace for each value of YCONT. This allows you to set the attributes for each contour line individually. For example, you may want to draw a thickened or colored line at the maximum or minimum contour level. By default, the contour levels are drawn from the maximum to the minimum value.

Note:
By default, the contour levels are drawn from the maximum value to the minimum value.

If you want to draw the contour levels from the minimum to the maximum, then enter the command:

SET DEX CONTOUR PLOT MINIMUM

To reset the default, enter

SET DEX CONTOUR PLOT MAXIMUM

Typically, it is common to highlight the maximum contour line when you are trying to maximize the response and to highlight the minimum contour line when you are trying to minimize the response. The SET DEX CONTOUR PLOT command can help simplify this by ensuring that the first contour line drawn is the one you highlight.

Note:
In cases where there are more than two important factors, you may want to generate the dex contour plot for several different pairs. All the pairwise combinations for 3 or more factors can be generated via a matrix plot. Enter HELP SCATTER PLOT MATRIX for details. The Program 2 example below gives an example of this.
Note:
The DEX CONTOUR PLOT command assumes a linear model for the fit. You can use the macro DEXCONTQ.DP to generate a dex contour plot assuming a quadratic model. Enter LIST DEXCONTQ.DP for details.
Default:
None
Synonyms:
None
Related Commands:
 DEX MEAN PLOT = Generate a dex mean plot. YATES = Perform a Yates analysis. CONTOUR PLOT = Generates a contour plot.
Reference:
"Statistics for Experimenters", Box, Hunter, and Hunter, John Wiley, 1978, pp. 296-300.
Applications:
Design of Experiments
Implementation Date:
2000/7
Program 1:
SKIP 25
LET YCONT = SEQUENCE 50 2 70
CHARACTER ZVAL CIRCLE CIRCLE
CHARACTER FILL OFF ON ON
LINE BLANK BLANK BLANK
LINE THICKNESS 0.1 0.1 0.1 0.3
DEX CONTOUR PLOT Y X1 X2

Program 2:
skip 25
read boxclean.dat y x1 x2 x3 x4
let ycont = sequence 10 1 20
.
multiplot corner coordinates 10 10 90 90
multiplot scale factor 2.5
xtic offset 5 5
ytic offset 5 5
tic offset units screen
x1label displacement 4
.
line bl so
char zval circ
char fill off on
line thickness 0.1 0.1 0.3
set matrix plot type dex contour plot
set matrix plot tag on
matrix plot y x1 x2 x3 x4 ycont
.
move 50 97
justification center
text dex contour plot matrix for boxclean.dat

Date created: 6/5/2001
Last updated: 4/4/2003