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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

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

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

    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

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

    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: 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
    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
        
Program 2:
     
    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
        
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: 02/10/2015

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