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Dataplot Vol 1 Auxillary Chapter

CORRELATION CONFIDENCE LIMITS

Name:
    CORRELATION CONFIDENCE LIMITS
Type:
    Analysis Command
Purpose:
    Generates a confidence interval for the correlation between two response variables based on Fisher's normal approximation.
Description:
    The confidence interval for the correlation, r, between two response variables based on Fisher's normal approximation is computed as

      LCL = TANH(Z - NORPPF(1 - alpha/2)/SQRT(n-3))
      UCL = TANH(Z + NORPPF(1 - alpha/2)/SQRT(n-3))

    with tanh and NORPPF denoting the hyperbolic tangent function and the percent point function of the normal distribution, respectively, and with n and alpha denoting the sample size and the confidence level, respectively, and where

      Z = ATANH(r) = LOG[(1+r)/(1-r)]/2

    with atanh denoting the inverse hyperbolic tangent function.

Syntax 1:
    CORRELATION CONFIDENCE LIMITS <y1> <y2>
                            <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.
Syntax 2:
    CORRELATION CONFIDENCE 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 is used to generate correlation confidence limits for all the pairwise combinations of <y1> to <yk>

Examples:
    CORRELATION CONFIDENCE LIMITS Y1 Y2
    CORRELATION CONFIDENCE LIMITS Y1 TO Y10
    CORRELATION CONFIDENCE LIMITS Y1 SUBSET TAG > 2
Note:
    A table of confidence intervals is printed for confidence limits of 50.0, 75.0, 80.0, 90.0, 95.0, 99.0, and 99.9.
Default:
    None
Synonyms:
    None
Related Commands:
    CORRELATION = Compute the correlation between two variables.
    CORRELATIION MATRIX = Compute the correlation matrix of a set of response variables.
    PARTIAL CORRELATION = Compute the partial correlation between two variables given the effect of a third variable.
Reference:
    Peavy, Bremer, Varner, Hogben (1986), "OMNITAB 80: An Interpretive System for Statistical and Numerical Data Analysis," NBS Special Publication 701.
Applications:
    Linear Regression
Implementation Date:
    2012/06
Program 1:
     
    SKIP 25
    READ BERGER1.DAT Y X
    CORRELATION CONFIDENCE LIMITS Y X
        
    The following output is generated.
                Confidence Limits for the Correlation Coefficient
                     (Based on Fisher Normal Approximation)
     
    Response Variable 1: Y
    Response Variable 2: X
     
     
    Summary Statistics for Variable 1:
    Number of Observations:                             107
    Sample Mean:                                    33.5794
    Sample Standard Deviation:                      18.6000
     
    Summary Statistics for Variable 2:
    Number of Observations:                             107
    Sample Mean:                                    39.0990
    Sample Standard Deviation:                      24.0563
     
    Correlation Coefficient (r):                     0.9455
    atanh(r):                                        1.7883
     
     
     
    --------------------------------------------------
      Confidence  Normal          Lower          Upper
       Value (%)   Value          Limit          Limit
    --------------------------------------------------
          50.000   0.674         0.9381         0.9521
          75.000   1.150         0.9322         0.9563
          80.000   1.281         0.9305         0.9574
          90.000   1.644         0.9256         0.9602
          95.000   1.959         0.9210         0.9626
          99.000   2.575         0.9113         0.9668
          99.900   3.290         0.8987         0.9710
     
        
Program 2:
     
    .  This data is from page 202 of
    .
    .  Peavy, Bremer, Varner, Hogben (1986), "OMNITAB 80:
    .  An Interpretive System for Statistical and Numerical
    .  Data Analysis," NBS Special Publication 701.
    .
    .  Original source of the data is from
    .  Draper and Smith (1981), "Applied Regression Analysis",
    .  Wiley, p. 373.
    .
    dimension 40 columns
    .
    read matrix m
    42.2  11.2  31.9  167.1
    48.6  10.6  13.2  174.4
    42.6  10.6  28.7  160.8
    39.0  10.4  26.1  162.0
    34.7   9.3  30.1  140.8
    44.5  10.8   8.5  174.6
    39.1  10.7  24.3  163.7
    40.1  10.0  18.6  174.5
    45.9  12.0  20.4  185.7
    end of data
    .
    set write decimals 4
    correlation confidence limits m1 m2 m3 m4
        
    The following output is generated.
                Confidence Limits for the Correlation Coefficient
                     (Based on Fisher Normal Approximation)
     
    Response Variable 1: M1
    Response Variable 2: M2
     
     
    Summary Statistics for Variable 1:
    Number of Observations:                               9
    Sample Mean:                                    41.8555
    Sample Standard Deviation:                       4.1764
     
    Summary Statistics for Variable 2:
    Number of Observations:                               9
    Sample Mean:                                    10.6222
    Sample Standard Deviation:                       0.7462
     
    Correlation Coefficient (r):                     0.6837
    atanh(r):                                        0.8361
     
     
     
    --------------------------------------------------
      Confidence  Normal          Lower          Upper
       Value (%)   Value          Limit          Limit
    --------------------------------------------------
          50.000   0.674         0.5085         0.8045
          75.000   1.150         0.3509         0.8631
          80.000   1.281         0.3030         0.8762
          90.000   1.644         0.1631         0.9065
          95.000   1.959         0.0359         0.9269
          99.000   2.575        -0.2121         0.9551
          99.900   3.290        -0.4677         0.9747
     
     
                Confidence Limits for the Correlation Coefficient
                     (Based on Fisher Normal Approximation)
     
    Response Variable 1: M1
    Response Variable 2: M3
     
     
    Summary Statistics for Variable 1:
    Number of Observations:                               9
    Sample Mean:                                    41.8555
    Sample Standard Deviation:                       4.1764
     
    Summary Statistics for Variable 2:
    Number of Observations:                               9
    Sample Mean:                                    22.4222
    Sample Standard Deviation:                       7.9279
     
    Correlation Coefficient (r):                    -0.6159
    atanh(r):                                       -0.7184
     
     
     
    --------------------------------------------------
      Confidence  Normal          Lower          Upper
       Value (%)   Value          Limit          Limit
    --------------------------------------------------
          50.000   0.674        -0.7589        -0.4162
          75.000   1.150        -0.8299        -0.2438
          80.000   1.281        -0.8459        -0.1928
          90.000   1.644        -0.8831        -0.0469
          95.000   1.959        -0.9084         0.0814
          99.000   2.575        -0.9436         0.3212
          99.900   3.290        -0.9681         0.5545
     
     
                Confidence Limits for the Correlation Coefficient
                     (Based on Fisher Normal Approximation)
     
    Response Variable 1: M1
    Response Variable 2: M4
     
     
    Summary Statistics for Variable 1:
    Number of Observations:                               9
    Sample Mean:                                    41.8555
    Sample Standard Deviation:                       4.1764
     
    Summary Statistics for Variable 2:
    Number of Observations:                               9
    Sample Mean:                                   167.0666
    Sample Standard Deviation:                      12.6451
     
    Correlation Coefficient (r):                     0.8017
    atanh(r):                                        1.1034
     
     
     
    --------------------------------------------------
      Confidence  Normal          Lower          Upper
       Value (%)   Value          Limit          Limit
    --------------------------------------------------
          50.000   0.674         0.6794         0.8806
          75.000   1.150         0.5607         0.9175
          80.000   1.281         0.5228         0.9255
          90.000   1.644         0.4069         0.9441
          95.000   1.959         0.2943         0.9565
          99.000   2.575         0.0518         0.9734
          99.900   3.290        -0.2353         0.9851
     
     
                Confidence Limits for the Correlation Coefficient
                     (Based on Fisher Normal Approximation)
     
    Response Variable 1: M2
    Response Variable 2: M3
     
     
    Summary Statistics for Variable 1:
    Number of Observations:                               9
    Sample Mean:                                    10.6222
    Sample Standard Deviation:                       0.7462
     
    Summary Statistics for Variable 2:
    Number of Observations:                               9
    Sample Mean:                                    22.4222
    Sample Standard Deviation:                       7.9279
     
    Correlation Coefficient (r):                    -0.1724
    atanh(r):                                       -0.1742
     
     
     
    --------------------------------------------------
      Confidence  Normal          Lower          Upper
       Value (%)   Value          Limit          Limit
    --------------------------------------------------
          50.000   0.674        -0.4215         0.1007
          75.000   1.150        -0.5675         0.2870
          80.000   1.281        -0.6027         0.3354
          90.000   1.644        -0.6888         0.4599
          95.000   1.959        -0.7506         0.5552
          99.000   2.575        -0.8413         0.7050
          99.900   3.290        -0.9082         0.8239
     
     
                Confidence Limits for the Correlation Coefficient
                     (Based on Fisher Normal Approximation)
     
    Response Variable 1: M2
    Response Variable 2: M4
     
     
    Summary Statistics for Variable 1:
    Number of Observations:                               9
    Sample Mean:                                    10.6222
    Sample Standard Deviation:                       0.7462
     
    Summary Statistics for Variable 2:
    Number of Observations:                               9
    Sample Mean:                                   167.0666
    Sample Standard Deviation:                      12.6451
     
    Correlation Coefficient (r):                     0.7679
    atanh(r):                                        1.0153
     
     
     
    --------------------------------------------------
      Confidence  Normal          Lower          Upper
       Value (%)   Value          Limit          Limit
    --------------------------------------------------
          50.000   0.674         0.6291         0.8593
          75.000   1.150         0.4972         0.9023
          80.000   1.281         0.4558         0.9118
          90.000   1.644         0.3308         0.9337
          95.000   1.959         0.2119         0.9483
          99.000   2.575        -0.0362         0.9684
          99.900   3.290        -0.3167         0.9822
     
     
                Confidence Limits for the Correlation Coefficient
                     (Based on Fisher Normal Approximation)
     
    Response Variable 1: M3
    Response Variable 2: M4
     
     
    Summary Statistics for Variable 1:
    Number of Observations:                               9
    Sample Mean:                                    22.4222
    Sample Standard Deviation:                       7.9279
     
    Summary Statistics for Variable 2:
    Number of Observations:                               9
    Sample Mean:                                   167.0666
    Sample Standard Deviation:                      12.6451
     
    Correlation Coefficient (r):                    -0.6287
    atanh(r):                                       -0.7393
     
     
     
    --------------------------------------------------
      Confidence  Normal          Lower          Upper
       Value (%)   Value          Limit          Limit
    --------------------------------------------------
          50.000   0.674        -0.7676        -0.4333
          75.000   1.150        -0.8363        -0.2633
          80.000   1.281        -0.8517        -0.2128
          90.000   1.644        -0.8876        -0.0677
          95.000   1.959        -0.9120         0.0607
          99.000   2.575        -0.9458         0.3024
          99.900   3.290        -0.9694         0.5398
        

Date created: 01/17/2013
Last updated: 01/17/2013
Please email comments on this WWW page to alan.heckert@nist.gov.