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BIVARIATE NORMAL TOLERANCE REGION PLOTName:
There are two probability values involved in the tolerance region:
That is, we can state with \( \gamma \)% confidence that at least (\ \delta; \)% of the data fall within the given limits. For example, if \( \gamma \) = 0.95 and \( \delta \) = 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
Hall and Sheldon state that with a properly chosen K the following ellipse is the smallest region that will contain δ of the probability mass: \( \frac{1}{1 - \rho^2} \left[ \left( \frac{x - \mu_x}{\sigma_x} \right) ^2 - 2 \rho \left( \frac{x - \mu_x}{\sigma_x} \right) \left( \frac{y - \mu_y}{\sigma_y} \right) + \left( \frac{y - \mu_y}{\sigma_y} \right) ^2 \right] = K \) 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 \( \gamma \) = 0.75, 0.90, and 0.95 and \( \delta \) = 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 \( \gamma \) and \( \delta \) values not available in the table, Dataplot uses an approximation given on pp. 325-327 of Krishnamoorthy. We recommend using the tabled values for \( \gamma \) and \( \delta \) 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 \( \delta \)) to be overlaid on the plot. You can specify the desired value of \( \gamma \) by entering the command
If you do not specify a value for \( \gamma \), then 0.95 will be used. To specify a single value of \( \delta \), then enter the command
To specify multiple values of \( \delta \), see Syntax 3 below. If no \( \delta \) values are specified, then 0.90 will be used. Alternatively, a bivariate normal confidence region plot can be generated. In this case, we are generating a joint confidence interval for the means of the two datasets. For this alternative, the ellipse has a similar form to the tolerance ellipse. However, the K is replaced with Hotelling's T2:
with F denoting the percent point function of the F distribution. The bivariate normal confidence regions are discussed in the ISO 13528 standard based on the method given by Jackson.
<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 \( \delta \).
<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 \( \delta \).
<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 \( \delta \) are desired. If you want multiple values of \( \delta \) when there are no groups, create a group-id variable with all values equal. For example
<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 \( \delta \) (here, \( \delta \) denotes confidence levels rather than coverage levels).
<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 \( \delta \) (here \( \delta \) denotes confidence levels rather than coverage levels).
<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 \( \delta \) (here \( \delta \) denotes confidence levels rather than coverage levels) are desired. If you want multiple values of \( \delta \) when there are no groups, create a group-id variable with all values equal. For example
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
BIVARIATE NORMAL CONFIDENCE REGION PLOT Y1 Y2
Krishnamoorthy (2006), Handbook of Statistical Distributions with Applications, Chapman & Hall/CRC, pp. 325-327. ISO 13528 (2005), "Statistical Methods for Use in Proficiency Testing by Interlaboratory Comparisons," ISO 13528:2005(E). Jackson (1956), "Quality Control Methods for Two Related Variables," Industrial Quality Control, 7, pp. 2-6.
ORIENTATION SQUARE DIMENSION 20 COLUMNS . SKIP 25 READ STG.DAT LABID F LABAVE FAVE . 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 ![]()
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Date created: 01/07/2008 |