SD PREDICTION LIMITS

SD PREDICTION LIMITS

Analysis Command

Generates a prediction interval for the standard deviation of one or more new observations given a previous sample.

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.

<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.

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.

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.

SD PREDICTION LIMITS Y1
SD PREDICTION LIMITS Y1 SUBSET TAG > 2
MULTIPLE SD PREDICTION LIMITS Y1 TO Y5
REPLICATED SD PREDICTION LIMITS Y X

A table of prediction limits is printed for alpha levels of 50.0, 80.0, 90.0, 95.0, 99.0, and 99.9.

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).

None

STANDARD DEVIATION PREDICTION INTERVAL is a synonym for STANDARD DEVIATION PREDICTION LIMITS

SD PREDICTION LIMIT is a synonym for STANDARD DEVIATION PREDICTION LIMIT

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.

Hahn and Meeker (1991), "Statistical Intervals: A Guide for Practitioners," Wiley, pp. 61-62.

Confirmatory Data Analysis

2013/04
2014/06: Support for LOGNORMAL and BOXCOX options

 
SKIP 25
READ ZARR13.DAT Y
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
    

 
SKIP 25
READ GEAR.DAT Y X
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
    

 
.  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