BIVARIATE NORMAL TOLERANCE REGION PLOT

BIVARIATE NORMAL TOLERANCE REGION PLOT

Graphics Command

Generates a bivariate normal tolerance region plot.

A tolerance interval calculate a confidence interval that contains at least a fixed percentage (or proportion) of the data.

There are two probability values involved in the tolerance region:

1. The coverage probability is the fixed percentage of the data to be covered. We call this .

2. The confidence level. We call this .

That is, we can state with % confidence that at least % of the data fall within the given limits. For example, if = 0.95 and = 0.90, we say that we have a "95% confidence interval for 90% coverage".

The TOLERANCE LIMITS command is used to compute univariate normal tolerance intervals (it will also compute non-parametric tolerance interals). The BASIS TOLERANCE LIMITS is used to compute normal, lognormal, and Weibull tolerance limits.

The BIVARIATE NORMAL TOLERANCE REGION PLOT is used for the case where we have bivariate, normally distributed data. In this case, if X and Y denote our bivariate data, we define a region A such that

Pr(Pr[(X,Y) A] > } =

Hall and Sheldon state that with a properly chosen K the following ellipse is the smallest region that will contain of the probability mass:



When we have a sample of n data points, we replace the population means, standard deviations, and correlation with the sample values in the above formula.

The problem then becomes how to find the appropriate value of K.

Hall and Sheldon describe how to find K for several different cases. We focus on their case 4 where the population means and standard deviations and the population correlation between X and Y are unknown.

Hall and Sheldon developed a table (Table 3 in their paper) using Monte Carlo methods for = 0.75, 0.90, and 0.95 and = 0.50, 0.80, 0.90, and 0.95 for n = 10 to 50.

Dataplot will use the tabled values when appropriate. For n > 50 or for and values not available in the table, Dataplot uses an approximation given on pp. 325-327 of Krishnamoorthy.

We recommend using the tabled values for and for smaller values of n.

The BIVARIATE NORMAL TOLERANCE REGION PLOT generates a scatter plot of Y versus X and then overlays the tolerance region ellipse on this plot. Note that Dataplot allows multiple ellipses (corresponding to different values of ) to be overlaid on the plot.

You can specify the desired value of by entering the command

LET GAMMA = <value>

If you do not specify a value for , then 0.95 will be used.

To specify a single value of , then enter the command

LET DELTA = <value>

To specify multiple values of , see Syntax 3 below. If no values are specified, then 0.90 will be used.

BIVARIATE NORMAL TOLERANCE REGION PLOT <y1> <y2>
            <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax is used when there are no groups in the data and there is a single value of .

BIVARIATE NORMAL TOLERANCE REGION PLOT <y1> <y2> <lab>
            <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
            <lab>is a group-id variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax is used when there are multiple groups in the data (e.g., multiple labs or multiple methods) and there is a single value of .

BIVARIATE NORMAL TOLERANCE REGION PLOT <y1> <y2> <lab> <delta>
            <SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
            <y2> is the second response variable;
            <lab> is a group-id variable;
            <delta> is a variable containing the desired values of delta;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax is used when there are multiple groups in the data (e.g., multiple labs or multiple methods) and multiple values of are desired.

If you want multiple values of when there are no groups, create a group-id variable with all values equal. For example

LET LAB = 1 FOR I = 1 1 N

BIVARIATE NORMAL TOLERANCE REGION PLOT Y1 Y2
BIVARIATE NORMAL TOLERANCE REGION PLOT Y1 Y2 LAB
BIVARIATE NORMAL TOLERANCE REGION PLOT Y1 Y2 LAB DELTA
BIVARIATE NORMAL TOLERANCE REGION PLOT Y1 Y2 LAB ...
            SUBSET LAB > 2

You can use the CHARACTER and LINE commands to control the appearance of the plot. If there are NGROUP groups and NDELTA values of delta in the data, settings 1 through NDELTA control the appearance of the NDELTA ellipses and settings NDELTA + 1 through NDELTA + NGROUP control the appearance of the raw data for each group.

None

None

TOLERANCE LIMITS	= Generate normal or nonparameteric tolerance limits.
BASIS TOLERANCE LIMITS	= Generate normal, lognormal, and Weibull tolerance limits.
YOUDEN PLOT	= Generate a Youden plot.
PLOT	= Generates a data or function plot.
CHARACTERS	= Sets the type for plot characters.
LINES	= Sets the type for plot lines.

Hall and Sheldon (1979), "Improved Bivariate Normal Tolerance Regions with Some Applications", Journal of Quality Technology, Vol. 11, No. 1, pp. 13-19.

Krishnamoorthy (2006), Handbook of Statistical Distributions with Applications, Chapman & Hall/CRC, pp. 325-327.

Interlaboratory Analysis

2007/4

 
ORIENTATION SQUARE
DIMENSION 20 COLUMNS
.
SKIP 25
READ STG.DAT LABID F LABAVE FAVE
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TITLE OFFSET 2
TITLE CASE ASIS
LABEL CASE ASIS
TITLE Youden with Bivariate Normal Tolerance (95%-90%)
LET GAMMA = 0.95
LET DELTA = 0.90
X1LABEL Filter Average Value
Y1LABEL Lab Value for Filter
X2label Labs 1 to 9 with Bivariate Normal Contour
X3LABEL SED/ITL/NIST March 2007
.
TIC OFFSET UNITS DATA
XLIMITS 0 1200
XTIC OFFSET -25 100
YLIMITS 0 1500
YTIC OFFSET -350 100
.
CHARACTERS BLANK 1 2 3 4 5 6 7 8 9
LINES BLANK ALL
LINES SOLID
.
.  Base tolerance curve on all labs, but only plot first 9
.
BIVARIATE NORMAL TOLERANCE REGION PLOT LABAVE FAVE LABID