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PREDICTION LIMITS

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

      \( \bar{x} \pm t_{(\alpha/2,n-1)} s \sqrt{\frac{1}{n} + \frac{1}{m}} \)

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

    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. However, this prediction interval is fairly robust against non-normality unless either the original sample size or the new sample is small or the departure from normality is severe (in particular, the data is not too skewed). Note that this includes the case of a prediction interval for a single future observation.

Syntax 1:
    <LOWER/UPPER> <LOGNORMAL/BOXCOX> 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 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 variable.

Syntax 2:
    MULTIPLE <LOWER/UPPER> <LOGNORMAL/BOXCOX>
                            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.

    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>
                            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 prediction 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:
    PREDICTION LIMITS Y1
    PREDICTION LIMITS Y1 SUBSET TAG > 2
    MULTIPLE PREDICTION LIMITS Y1 TO Y5
    REPLICATED 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 PREDICTION LIMITS command, the following commands can also be used:

      LET ALPHA = 0.05
      LET NNEW = 3

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

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

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

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

Note:
    Prediction limits can also be generated for the case where the interval will contain ALL of the new points. Enter HELP PREDICTION BOUNDS for details.
Default:
    None
Synonyms:
    PREDICTION INTERVALS is a synonum for PREDICTION LIMITS
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
Program 1:
     
    SKIP 25
    READ ZARR13.DAT Y
    SET WRITE DECIMALS 5
    LET NNEW = 5
    .
    PREDICTION LIMITS Y
    LOWER PREDICTION LIMITS Y
    UPPER PREDICTION LIMITS Y
        
    The following output is generated
                Two-Sided Prediction Limits for the Mean
     
    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 Mean
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        9.25448        9.26843
            80.0        9.24818        9.27473
            90.0        9.24440        9.27851
            95.0        9.24110        9.28181
            99.0        9.23461        9.28831
            99.9        9.22697        9.29594
     
     
                One-Sided Lower Prediction Limits for the Mean
     
    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 Mean
    ---------------------------
      Confidence          Lower
       Value (%)          Limit
    ---------------------------
            50.0        9.26146
            80.0        9.25275
            90.0        9.24818
            95.0        9.24440
            99.0        9.23724
            99.9        9.22912
     
     
                One-Sided Upper Prediction Limits for the Mean
     
    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 Mean
    ---------------------------
      Confidence          Upper
       Value (%)          Limit
    ---------------------------
            50.0        9.26146
            80.0        9.27016
            90.0        9.27473
            95.0        9.27851
            99.0        9.28567
            99.9        9.29379
        
Program 2:
     
    SKIP 25
    READ GEAR.DAT Y X
    SET WRITE DECIMALS 5
    LET NNEW = 3
    REPLICATED PREDICTION LIMITS Y X
        
    The following output is generated
                Two-Sided Prediction Limits for the Mean
     
    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 Mean
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.99598        1.00001
            80.0        0.99404        1.00195
            90.0        0.99275        1.00324
            95.0        0.99152        1.00447
            99.0        0.98870        1.00729
            99.9        0.98432        1.01167
     
     
                Two-Sided Prediction Limits for the Mean
     
    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 Mean
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.99668        1.00151
            80.0        0.99435        1.00384
            90.0        0.99280        1.00539
            95.0        0.99133        1.00686
            99.0        0.98794        1.01025
            99.9        0.98268        1.01551
     
     
                Two-Sided Prediction Limits for the Mean
     
    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 Mean
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.99355        0.99724
            80.0        0.99177        0.99902
            90.0        0.99060        1.00019
            95.0        0.98947        1.00132
            99.0        0.98689        1.00390
            99.9        0.98288        1.00791
     
     
                Two-Sided Prediction Limits for the Mean
     
    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 Mean
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.99641        0.99998
            80.0        0.99469        1.00170
            90.0        0.99355        1.00284
            95.0        0.99246        1.00393
            99.0        0.98995        1.00644
            99.9        0.98607        1.01032
     
     
                Two-Sided Prediction Limits for the Mean
     
    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 Mean
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.98839        0.99540
            80.0        0.98500        0.99879
            90.0        0.98275        1.00104
            95.0        0.98061        1.00318
            99.0        0.97568        1.00811
            99.9        0.96805        1.01574
     
     
                Two-Sided Prediction Limits for the Mean
     
    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 Mean
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.99422        1.00337
            80.0        0.98979        1.00780
            90.0        0.98687        1.01072
            95.0        0.98407        1.01352
            99.0        0.97765        1.01994
            99.9        0.96769        1.02990
     
     
                Two-Sided Prediction Limits for the Mean
     
    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 Mean
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.99785        1.00514
            80.0        0.99432        1.00867
            90.0        0.99199        1.01100
            95.0        0.98976        1.01323
            99.0        0.98464        1.01835
            99.9        0.97671        1.02628
     
     
                Two-Sided Prediction Limits for the Mean
     
    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 Mean
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.99872        1.00207
            80.0        0.99709        1.00370
            90.0        0.99602        1.00477
            95.0        0.99499        1.00580
            99.0        0.99264        1.00815
            99.9        0.98898        1.01181
     
     
                Two-Sided Prediction Limits for the Mean
     
    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 Mean
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.99638        1.00021
            80.0        0.99453        1.00206
            90.0        0.99330        1.00329
            95.0        0.99213        1.00446
            99.0        0.98944        1.00715
            99.9        0.98528        1.01131
     
     
                Two-Sided Prediction Limits for the Mean
     
    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 Mean
    ------------------------------------------
      Confidence          Lower          Upper
       Value (%)          Limit          Limit
    ------------------------------------------
            50.0        0.99233        0.99726
            80.0        0.98994        0.99965
            90.0        0.98836        1.00123
            95.0        0.98686        1.00273
            99.0        0.98339        1.00620
            99.9        0.97803        1.01156
        
Program 3:
     
    .  Following example from Hahn and Meeker's book.
    .
    let ymean = 50.10
    let ysd   = 1.31
    let n1    = 5
    let nnew  = 3
    .
    set write decimals 5
    let slow1 = summary lower prediction limits ymean ysd n1
    let supp1 = summary upper prediction limits ymean ysd n1
    let slow2 = summary one sided lower prediction limits ymean ysd n1
    let supp2 = summary one sided upper prediction limits ymean ysd n1
    print slow1 supp1 slow2 supp2
        
    The following output is generated
     PARAMETERS AND CONSTANTS--
    
        SLOW1   --       47.44381
        SUPP1   --       52.75619
        SLOW2   --       48.06049
        SUPP2   --       52.13951
        

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Date created: 04/15/2013
Last updated: 02/10/2015

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