INTRACLASS CORRELATION

INTRACLASS CORRELATION (LET)
CORRELATION RATIO (LET)

Let Subcommand

Given a response variable and a group-id variable, compute the correlation ratio or the intraclass correlation coefficient.

The formula for the intraclass correlation coefficient is
\( \begin{array}{lcl} \eta^2 & = & \frac{\sum_{i=1}^{p}{N_{i} (\bar{Y}_{i} - \bar{Y})^{2}}} {\sum_{i=1}^{p}{\sum_{j=1}^{N_{i}} {(Y_{ij} - \bar{Y})^{2}}}} \\ & = & \frac{\sigma_{\bar{Y}}^{2}} {\sigma_{Y}^{2}} \end{array} \)

The intraclass correlation is the ratio of the weighted variance of the group means divided by the variance of all samples.

The intraclass correlation coefficient can have values between 0 and 1. A value of 0 indicates no variance between the means of the different groups while a value of 1 indicates that the sample variance is due to the variance between groups rather than the variance within groups. So the intraclass correlation indicates the relative importance of the "between group variance" (values closer to 1) and "within group variance" (values closer to 0).

The correlation ratio is the square root of the intraclass correlation coefficient.

LET <par> = INTRACLASS CORRELATION <y> <x>
                        <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
            <x> is the group-id variable;
            <par> is a parameter where the intraclass correlation value is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET <par> = CORRELATION RATIO <y> <x>
                        <SUBSET/EXCEPT/FOR qualification>
where <y> is the response variable;
            <x> is the group-id variable;
            <par> is a parameter where the correlation ratio value is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET ICC = INTRACLASS CORRELATION Y X
LET CR = CORRELATION RATIO Y X
LET CR = CORRELATION RATIO Y X SUBSET X > 2

Dataplot's built-in statistics can be used with a number of commands. For details, enter HELP STATISTIC.

None

None

CORRELATION	=	Compute the correlation coefficient.
REPEATABILITY SD	=	Compute the repeatability standard deviation.
REPRODUCABILITY SD	=	Compute the reproducability standard deviation.
ANOVA	=	Perform a fixed effects analysis of variance.
LINEAR CORRELATION	=	Compute the correlation from a linear least squares fit.

One Factor Analysis

2019/08

 
. Step 1:   Create some data (from Wikipedia page on Correlation Ratio)
.
read x y
 1   45
 1   70
 1   29
 1   15
 1   21
 2   40
 2   20
 2   30
 2   42
 3   65
 3   95
 3   80
 3   70
 3   85
 3   73
end of data
.
. Step 2:   Compute the statistics
.
let cr  = correlation ratio y x
let icc = intraclass correlation y x
set write decimals 4
print cr icc
    

 
 PARAMETERS AND CONSTANTS--

    CR      --         0.8386
    ICC     --         0.7033