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
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
.
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