 Dataplot Vol 1 Vol 2

# SD PREDICTION LIMITS

Name:
SD PREDICTION LIMITS
Type:
Analysis Command
Purpose:
Generates a prediction interval for the standard deviation of one or more new observations given a previous sample.
Description:
Given a sample of n observations with standard deviation s, the two-sided prediction interval to contain the standard deviation of m new indpendent, identically distributed observations is

$$\mbox{lower prediction limit} = s \sqrt{\frac{1}{F_{(1-\alpha/2;n-1,m-1)}}}$$

$$\mbox{upper prediction limit} = s \sqrt{F_{(1-\alpha/2;m-1,n-1)}}$$

with F denoting the percent point function of the F distribution.

The one-sided lower prediction limit is

$$\mbox{lower prediction limit} = s \sqrt{\frac{1}{F_{(1-\alpha;n-1,m-1)}}}$$

The one-sided upper prediction limit is

$$\mbox{upper prediction limit} = s \sqrt{F_{(1-\alpha;m-1,n-1)}}$$

In this formula, the only value from the new observations is the sample size. That is, it can be applied before the new data is actually collected. The number of observations for the new sample is entered with the command

LET NNEW = <value>

If NNEW is not defined, then a value of 1 is used.

This prediction interval is based on the assumption that the underlying data is approximately normally distributed. The prediction interval for the standard deviation is highly sensitive to non-normality in the data. It is recommended that the original data be tested for normality before using these normal based intervals.

Syntax 1:
<LOWER/UPPER> <LOGNORMAL/BOXCOX> SD PREDICTION LIMITS <y>
<SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

If LOWER is specified, a one-sided lower prediction limit is returned. If UPPER is specified, a one-sided upper confidence limit is returned. If neither is specified, a two-sided limit is returned.

If the keyword LOGNORMAL is present, the log of the data will be taken, then the normal prediction limits will be computed, and then the computed normal lower and upper limits will be exponentiated to obtain the lognormal prediction limits.

Similarly, if the keyword BOXCOX is present, a Box-Cox transformation to normality will be applied to the data before computing the normal prediction limits. The computed lower and upper limits will then be transformed back to the original scale.

This syntax supports matrix arguments for the response variable.

Syntax 2:
MULTIPLE <LOWER/UPPER> <LOGNORMAL/BOXCOX>
SD PREDICTION LIMITS <y1> ... <yk>
<SUBSET/EXCEPT/FOR qualification>
where <y1> .... <yk> is a list of 1 to 30 response variables;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax will generate a prediction interval for each of the response variables. The word MULTIPLOT is optional. That is,

MULTIPLE SD PREDICTION LIMITS Y1 Y2 Y3

is equivalent to

SD PREDICTION LIMITS Y1 Y2 Y3

If LOWER is specified, a one-sided lower prediction limit is returned. If UPPER is specified, a one-sided upper prediction limit is returned. If neither is specified, a two-sided limit is returned.

If the keyword LOGNORMAL is present, the log of the data will be taken, then the normal prediction limits will be computed, and then the computed normal lower and upper limits will be exponentiated to obtain the lognormal prediction limits.

Similarly, if the keyword BOXCOX is present, a Box-Cox transformation to normality will be applied to the data before computing the normal prediction limits. The computed lower and upper limits will then be transformed back to the original scale.

This syntax supports matrix arguments for the response variables.

Syntax 3:
REPLICATED <LOWER/UPPER> <LOGNORMAL/BOXCOX>
SD PREDICTION LIMITS <y> <x1> ... <xk>
<SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
<x1> .... <xk> is a list of 1 to 6 group-id variables;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax performs a cross-tabulation of the <x1> ... <xk> and generates a prediction interval for each unique combination of the cross-tabulated values. For example, if X1 has 3 levels and X2 has 2 levels, six confidence intervals will be generated.

If LOWER is specified, a one-sided lower prediction limit is returned. If UPPER is specified, a one-sided upper prediction limit is returned. If neither is specified, a two-sided limit is returned.

If the keyword LOGNORMAL is present, the log of the data will be taken, then the normal prediction limits will be computed, and then the computed normal lower and upper limits will be exponentiated to obtain the lognormal prediction limits.

Similarly, if the keyword BOXCOX is present, a Box-Cox transformation to normality will be applied to the data before computing the normal prediction limits. The computed lower and upper limits will then be transformed back to the original scale.

This syntax does not support matrix arguments.

Examples:
SD PREDICTION LIMITS Y1
SD PREDICTION LIMITS Y1 SUBSET TAG > 2
MULTIPLE SD PREDICTION LIMITS Y1 TO Y5
REPLICATED SD PREDICTION LIMITS Y X
Note:
A table of prediction limits is printed for alpha levels of 50.0, 80.0, 90.0, 95.0, 99.0, and 99.9.
Note:
In addition to the STANDARD DEVIATION PREDICTION LIMIT command, the following commands can also be used:

LET ALPHA = 0.05
LET NNEW = <value>

LET A = LOWER STANDARD DEVIATION PREDICTION LIMIT Y
LET A = UPPPER STANDARD DEVIATION PREDICTION LIMIT Y
LET A = ONE SIDED LOWER STANDARD DEVIATION PREDICTION
LIMIT Y
LET A = ONE SIDED UPPER STANDARD DEVIATION PREDICTION
LIMIT Y

LET A = SUMMARY LOWER STANDARD DEVIATION PREDICTION
LIMIT YSD N
LET A = SUMMARY UPPPER STANDARD DEVIATION PREDICTION
LIMIT YSD N
LET A = SUMMARY ONE SIDED LOWER STANDARD DEVIATION
PREDICTION LIMIT YSD N
LET A = SUMMARY ONE SIDED UPPER STANDARD DEVIATION
PREDICTION LIMIT YSD N

The first 2 commands specify the significance level and the number of new observations. The next 4 commands are used when you have raw data. The last 4 commands are used when only summary data (standard deviation, sample size) is available.

In addition to the above LET command, built-in statistics are supported for about 20 different commands (enter HELP STATISTICS for details).

Default:
None
Synonyms:
STANDARD DEVIATION PREDICTION INTERVAL is a synonym for STANDARD DEVIATION PREDICTION LIMITS

SD PREDICTION LIMIT is a synonym for STANDARD DEVIATION PREDICTION LIMIT

Related Commands:
 SD CONFIDENCE LIMITS = Generate a confidence limit for the standard deviation. CONFIDENCE LIMITS = Generate a confidence limit for the mean. PREDICTION LIMITS = Generate prediction limits for the mean. PREDICTION BOUNDS = Generate prediction limits to cover all new observations. TOLERANCE LIMITS = Generate a tolerance limit.
Reference:
Hahn and Meeker (1991), "Statistical Intervals: A Guide for Practitioners," Wiley, pp. 61-62.
Applications:
Confirmatory Data Analysis
Implementation Date:
2013/04
2014/06: Support for LOGNORMAL and BOXCOX options
Program 1:

SKIP 25
SET WRITE DECIMALS 5
LET NNEW = 5
.
SD PREDICTION LIMITS Y
LOWER SD PREDICTION LIMITS Y
UPPER SD PREDICTION LIMITS Y

The following output is generated
            Two-Sided Prediction Limits for the SD

Response Variable: Y

Summary Statistics:
Number of Observations:                             195
Sample Mean:                                    9.26146
Sample Standard Deviation:                      0.02278
Number of New Observations:                           5

Two-Sided Prediction Limits for the SD
------------------------------------------
Confidence          Lower          Upper
Value (%)          Limit          Limit
------------------------------------------
50.0        0.01579        0.02655
80.0        0.01173        0.03201
90.0        0.00959        0.03543
95.0        0.00791        0.03848
99.0        0.00517        0.04466
99.9        0.00287        0.05215

One-Sided Lower Prediction Limits for the SD

Response Variable: Y

Summary Statistics:
Number of Observations:                             195
Sample Mean:                                    9.26146
Sample Standard Deviation:                      0.02278
Number of New Observations:                           5

One-Sided Lower Prediction Limits for the SD
---------------------------
Confidence          Lower
Value (%)          Limit
---------------------------
50.0        0.02091
80.0        0.01462
90.0        0.01173
95.0        0.00959
99.0        0.00619
99.9        0.00342

One-Sided Upper Prediction Limits for the SD

Response Variable: Y

Summary Statistics:
Number of Observations:                             195
Sample Mean:                                    9.26146
Sample Standard Deviation:                      0.02278
Number of New Observations:                           5

One-Sided Upper Prediction Limits for the SD
---------------------------
Confidence          Upper
Value (%)          Limit
---------------------------
50.0        0.02091
80.0        0.02802
90.0        0.03201
95.0        0.03543
99.0        0.04212
99.9        0.05002

Program 2:

SKIP 25
SET WRITE DECIMALS 5
LET NNEW = 3
.
REPLICATED SD PREDICTION LIMITS Y X

The following output is generated
             Two-Sided Prediction Limits for the SD

Response Variable: Y
Factor Variable 1: X                            1.00000

Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99800
Sample Standard Deviation:                      0.00434
Number of New Observations:                           3

Two-Sided Prediction Limits for the SD
------------------------------------------
Confidence          Lower          Upper
Value (%)          Limit          Limit
------------------------------------------
50.0        0.00236        0.00553
80.0        0.00141        0.00753
90.0        0.00098        0.00896
95.0        0.00069        0.01038
99.0        0.00030        0.01381
99.9        0.00009        0.01937

Two-Sided Prediction Limits for the SD

Response Variable: Y
Factor Variable 1: X                            2.00000

Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99910
Sample Standard Deviation:                      0.00521
Number of New Observations:                           3

Two-Sided Prediction Limits for the SD
------------------------------------------
Confidence          Lower          Upper
Value (%)          Limit          Limit
------------------------------------------
50.0        0.00284        0.00664
80.0        0.00170        0.00904
90.0        0.00118        0.01076
95.0        0.00083        0.01247
99.0        0.00036        0.01658
99.9        0.00011        0.02324

Two-Sided Prediction Limits for the SD

Response Variable: Y
Factor Variable 1: X                            3.00000

Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99540
Sample Standard Deviation:                      0.00397
Number of New Observations:                           3

Two-Sided Prediction Limits for the SD
------------------------------------------
Confidence          Lower          Upper
Value (%)          Limit          Limit
------------------------------------------
50.0        0.00216        0.00506
80.0        0.00129        0.00689
90.0        0.00090        0.00820
95.0        0.00063        0.00950
99.0        0.00028        0.01264
99.9        0.00008        0.01772

Two-Sided Prediction Limits for the SD

Response Variable: Y
Factor Variable 1: X                            4.00000

Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99820
Sample Standard Deviation:                      0.00385
Number of New Observations:                           3

Two-Sided Prediction Limits for the SD
------------------------------------------
Confidence          Lower          Upper
Value (%)          Limit          Limit
------------------------------------------
50.0        0.00209        0.00490
80.0        0.00125        0.00668
90.0        0.00087        0.00794
95.0        0.00061        0.00921
99.0        0.00027        0.01224
99.9        0.00008        0.01717

Two-Sided Prediction Limits for the SD

Response Variable: Y
Factor Variable 1: X                            5.00000

Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99190
Sample Standard Deviation:                      0.00757
Number of New Observations:                           3

Two-Sided Prediction Limits for the SD
------------------------------------------
Confidence          Lower          Upper
Value (%)          Limit          Limit
------------------------------------------
50.0        0.00413        0.00965
80.0        0.00247        0.01314
90.0        0.00172        0.01563
95.0        0.00120        0.01811
99.0        0.00053        0.02409
99.9        0.00016        0.03377

Two-Sided Prediction Limits for the SD

Response Variable: Y
Factor Variable 1: X                            6.00000

Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99879
Sample Standard Deviation:                      0.00988
Number of New Observations:                           3

Two-Sided Prediction Limits for the SD
------------------------------------------
Confidence          Lower          Upper
Value (%)          Limit          Limit
------------------------------------------
50.0        0.00538        0.01259
80.0        0.00322        0.01714
90.0        0.00224        0.02039
95.0        0.00157        0.02363
99.0        0.00070        0.03142
99.9        0.00022        0.04406

Two-Sided Prediction Limits for the SD

Response Variable: Y
Factor Variable 1: X                            7.00000

Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    1.00150
Sample Standard Deviation:                      0.00787
Number of New Observations:                           3

Two-Sided Prediction Limits for the SD
------------------------------------------
Confidence          Lower          Upper
Value (%)          Limit          Limit
------------------------------------------
50.0        0.00429        0.01003
80.0        0.00257        0.01365
90.0        0.00178        0.01625
95.0        0.00125        0.01883
99.0        0.00055        0.02504
99.9        0.00017        0.03511

Two-Sided Prediction Limits for the SD

Response Variable: Y
Factor Variable 1: X                            8.00000

Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    1.00039
Sample Standard Deviation:                      0.00362
Number of New Observations:                           3

Two-Sided Prediction Limits for the SD
------------------------------------------
Confidence          Lower          Upper
Value (%)          Limit          Limit
------------------------------------------
50.0        0.00197        0.00462
80.0        0.00118        0.00628
90.0        0.00082        0.00748
95.0        0.00057        0.00867
99.0        0.00025        0.01153
99.9        0.00008        0.01616

Two-Sided Prediction Limits for the SD

Response Variable: Y
Factor Variable 1: X                            9.00000

Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99829
Sample Standard Deviation:                      0.00413
Number of New Observations:                           3

Two-Sided Prediction Limits for the SD
------------------------------------------
Confidence          Lower          Upper
Value (%)          Limit          Limit
------------------------------------------
50.0        0.00225        0.00527
80.0        0.00135        0.00717
90.0        0.00093        0.00853
95.0        0.00065        0.00989
99.0        0.00029        0.01315
99.9        0.00009        0.01844

Two-Sided Prediction Limits for the SD

Response Variable: Y
Factor Variable 1: X                           10.00000

Summary Statistics:
Number of Observations:                              10
Sample Mean:                                    0.99479
Sample Standard Deviation:                      0.00532
Number of New Observations:                           3

Two-Sided Prediction Limits for the SD
------------------------------------------
Confidence          Lower          Upper
Value (%)          Limit          Limit
------------------------------------------
50.0        0.00290        0.00679
80.0        0.00173        0.00924
90.0        0.00121        0.01099
95.0        0.00084        0.01273
99.0        0.00037        0.01694
99.9        0.00011        0.02375

Program 3:

.  Following example from Hahn and Meeker's book.
.
let ymean = 50.10
let ysd   = 1.31
let n1    = 5
let nnew  = 3
let alpha = 0.05
.
set write decimals 5
let slow1 = summary lower sd prediction limits ysd n1
let supp1 = summary upper sd prediction limits ysd n1
let slow2 = summary one sided lower sd prediction limits ysd n1
let supp2 = summary one sided upper sd prediction limits ysd n1
print slow1 supp1 slow2 supp2

The following output is generated
 PARAMETERS AND CONSTANTS--

SLOW1   --        0.20910
SUPP1   --        4.27492
SLOW2   --        0.29860
SUPP2   --        3.45211


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Date created: 04/15/2013
Last updated: 12/14/2020