SED navigation bar go to SED home page go to Dataplot home page go to NIST home page SED Home Page SED Staff SED Projects SED Products and Publications Search SED Pages
Dataplot Vol 1 Vol 2

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

      \( LCL = \frac{\mbox{tanh}(Z - \Phi^{-1}(1 - \alpha/2))}{\sqrt{n - 3}} \)
      \( UCL = \frac{\mbox{tanh}(Z + \Phi^{-1}(1 - \alpha/2))}{\sqrt{n - 3}} \)

    with tanh and \( \Phi^{-1} \) 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

      \( \begin{array}{lcl} Z & = & \mbox{atanh}(r) \\ & = & \left( \log{\frac{1 + r}{1 - r}} \right) /2 \end{array} \)

    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: 12/11/2023
Please email comments on this WWW page to [email protected].