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f_ICC.m
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f_ICC.m
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function ICC = f_ICC(data,alpha)
% Computes the Intra Class Correllation Coefficients ICC1, ICC2, ICC3,
% ICC1k, ICC2k, and ICC3k.
% Based on the development by Shrout1979, presentation by McGraw including
% errata corrections
% Data is returned as presented in the form returned by the ICC function in the
% R package 'DescTools'
% Shrout, Patrick E. and Fleiss, Joseph L. Intraclass correlations: uses in
% assessing rater reliability. Psychological Bulletin, 1979, 86, 420-3428.
% McGraw, Kenneth O. and Wong, S. P. (1996), Forming inferences about some
% intraclass correlation coefficients. Psychological Methods, 1, 30-46.
% and errata in Psychological Methods, 4, page 390.
% Syntax:
% Input:
% data is an k by m matrix of k targets by m raters
% alpha is the alpha level for significance using the confidence intevals
% rho0 is the hypothesised value of ICC (set to zero if unsure)
rho0 = 0;
% Output:
% ICC
% F-value
% degrees of freedom df1, df2
% p-value corresponding to the F-Value
% confidence limits at the significance given in alpha
% Example: (data from McGraw Table 6,
% data = [103,119;% 82,65;116,106;102,102;99,105;98,100;104,107;
% 62,85;97,101;107,110];
% alpha = 0.05;
% ICC = f_ICC(data,alpha);
% Verification from R:
% Intraclass correlation coefficients
% type est F-val df1 df2 p-val lwr.ci upr.ci
% Single_raters_absolute ICC1 0.722 6.18 9 10 0.00437 0.241 0.922
% Single_random_raters ICC2 0.720 6.00 9 9 0.00677 0.228 0.922
% Single_fixed_raters ICC3 0.714 6.00 9 9 0.00677 0.197 0.920
% Average_raters_absolute ICC1k 0.838 6.18 9 10 0.00437 0.389 0.959
% Average_random_raters ICC2k 0.837 6.00 9 9 0.00677 0.371 0.959
% Average_fixed_raters ICC3k 0.833 6.00 9 9 0.00677 0.329 0.959
% RPMatthew 20180410
[numTargets,numJudges]=size(data);
meanTarget = mean(data,2);
meanJudge = mean(data);
meanTotal = mean(data(:));
% Within Targets Mean Square (MSW in McGraw1996)
tmp = (data - repmat(meanTarget,1,numJudges)).^2;
WSS = sum(tmp(:));
WMS = WSS / (numTargets*(numJudges - 1));
% Within Judges (Raters) Mean Square (MSC in McGraw1996)
RSS = sum((meanJudge - meanTotal).^2) * numTargets;
CMS = RSS / (numJudges - 1);
% Between Targets Mean Square (MSR in McGraw1996)
BSS = sum((meanTarget - meanTotal).^2) * numJudges;
RMS = BSS / (numTargets - 1);
% Residual Mean Square (MSE in McGraw1996)
ESS = WSS - RSS;
EMS = ESS / ((numJudges - 1) * (numTargets - 1));
ICC_1 = (RMS-WMS)/(RMS+(numJudges-1)*WMS);
ICC_2 = (RMS-EMS)/(RMS+(CMS-EMS)*numJudges/numTargets+(numJudges-1)*EMS);
ICC_3 = (RMS-EMS)/(RMS+(numJudges-1)*EMS);
ICC_1k = (RMS-WMS)/(RMS);
ICC_2k = (RMS-EMS)/(RMS+(CMS-EMS)/numTargets);
ICC_3k = (RMS-EMS)/(RMS);
FObs_1 = RMS/WMS;%anova1(data,[],'off');
FTabled_1L = finv((1-0.5*alpha),numTargets-1,numTargets*(numJudges-1));
FTabled_1U = finv((1-0.5*alpha),numTargets*(numJudges-1),numTargets-1);
FL_1 = FObs_1/FTabled_1L;
FU_1 = FObs_1*FTabled_1U;
ICC_1CI = [(FL_1-1)/(FL_1+(numJudges-1)),(FU_1-1)/(FU_1+(numJudges-1))];
ICC_1kCI = [1-1/FL_1,1-1/FU_1];
ICC_1F = RMS/WMS*((1-rho0)/(1+(numJudges-1)*rho0));
ICC_1pVal = 1-fcdf(ICC_1F,numTargets-1,numTargets*(numJudges-1));
sprintf('%2.5f',ICC_1pVal);
ICC_1df1 = numTargets-1;
ICC_1df2 = numTargets*(numJudges-1);
ICC_1kF = RMS/WMS*(1-rho0);
ICC_1kpVal = 1-fcdf(ICC_1kF,numTargets-1,numTargets*(numJudges-1));
sprintf('%2.5f',ICC_1kpVal);
ICC_1kdf1 = numTargets-1;
ICC_1kdf2 = numTargets*(numJudges-1);
FObs_2 = RMS/EMS;
a_2 = (numJudges*ICC_2)/(numTargets*(1-ICC_2));
b_2 = 1+(numJudges*ICC_2*(numTargets-1))/(numTargets*(1-ICC_2));
c_2 = ICC_2/(numTargets*(1-ICC_2));
d_2 = 1+(ICC_2*(numTargets-1))/(numTargets*(1-ICC_2));
v_2 = ((a_2*CMS+b_2*EMS)^2)/((a_2*CMS)^2/(numJudges-1)+((b_2*EMS)^2/((numTargets-1)*(numJudges-1))));
FLstar_2 = finv((1-0.5*alpha),numTargets-1,v_2);
FUstar_2 = finv((1-0.5*alpha),v_2,numTargets-1);
ICC_2CI = [(numTargets*(RMS-FLstar_2*EMS))/(FLstar_2*(numJudges*CMS+(numJudges*numTargets-numJudges-numTargets)*EMS)+numTargets*RMS),...
(numTargets*(FUstar_2*RMS-EMS))/(numJudges*CMS+(numJudges*numTargets-numJudges-numTargets)*EMS+numTargets*FUstar_2*RMS)];
FObs_2k = RMS/EMS;
c_2k = ICC_2k/(numTargets*(1-ICC_2k));
d_2k = 1+(ICC_2k*(numTargets-1))/(numTargets*(1-ICC_2k));
v_2k = (c_2k*CMS+d_2k*EMS)^2/((c_2k*CMS)^2/(numJudges-1)+((d_2k*EMS)^2/((numTargets-1)*(numJudges-1))));
FLstar_2k = finv((1-0.5*alpha),numTargets-1,v_2k);
FUstar_2k = finv((1-0.5*alpha),v_2k,numTargets-1);
ICC_2kCI = [(numTargets*(RMS-FLstar_2k*EMS))/(FLstar_2k*(CMS-EMS)+numTargets*RMS),...
(numTargets*(FUstar_2k*RMS-EMS))/(CMS-EMS+numTargets*FUstar_2k*RMS)];
a = (numJudges*rho0)/(numTargets*(1-rho0));
b = 1+(numJudges*rho0*(numTargets-1))/(numTargets*(1-rho0));
c = (rho0)/(numTargets*(1-rho0));
d = 1+(rho0*(numTargets-1))/(numTargets*(1-rho0));
ICC_2F = (RMS)/(a*CMS+b*EMS);
ICC_2pVal = 1-fcdf(ICC_2F,numTargets-1,(numTargets-1)*(numJudges-1));
sprintf('%2.5f',ICC_2pVal);
ICC_2df1 = numTargets-1;
ICC_2df2 = (numTargets-1)*(numJudges-1);
ICC_2kF = (RMS)/(c*CMS+d*EMS);
ICC_2kpVal = 1-fcdf(ICC_2kF,numTargets-1,(numTargets-1)*(numJudges-1));
sprintf('%2.5f',ICC_2kpVal);
ICC_2kdf1 = numTargets-1;
ICC_2kdf2 = (numTargets-1)*(numJudges-1);
FObs_3 = RMS/EMS;%anova1(data,[],'off');
FTabled_3L = finv((1-0.5*alpha),numTargets-1,(numTargets-1)*(numJudges-1));
FTabled_3U = finv((1-0.5*alpha),(numTargets-1)*(numJudges-1),numTargets-1);
FL_3 = FObs_3/FTabled_3L;
FU_3 = FObs_3*FTabled_3U;
ICC_3CI = [(FL_3-1)/(FL_3+(numJudges-1)),(FU_3-1)/(FU_3+(numJudges-1))];
ICC_3kCI = [1-1/FL_3,1-1/FU_3];
ICC_3F = RMS/EMS*((1-rho0)/(1+(numJudges-1)*rho0));
ICC_3pVal = 1-fcdf(ICC_3F,numTargets-1,(numTargets-1)*(numJudges-1));
sprintf('%2.5f',ICC_3pVal);
ICC_3df1 = numTargets-1;
ICC_3df2 = (numTargets-1)*(numJudges-1);
ICC_3kF = RMS/EMS*(1-rho0);
ICC_3kpVal = 1-fcdf(ICC_3kF,numTargets-1,(numTargets-1)*(numJudges-1));
sprintf('%2.5f',ICC_3kpVal);
ICC_3kdf1 = numTargets-1;
ICC_3kdf2 = (numTargets-1)*(numJudges-1);
clear ICC
ICC{1}.nameMcGraw = 'Single Raters Absolute';
ICC{1}.nameShrout = 'ICC(1,1)';
ICC{1}.est = ICC_1;
ICC{1}.FValue = ICC_1F;
ICC{1}.df1 = ICC_1df1;
ICC{1}.df2 = ICC_1df2;
ICC{1}.pVal = ICC_1pVal;
ICC{1}.confInterval = ICC_1CI;
ICC{2}.nameMcGraw = 'Single Random Raters';
ICC{2}.nameShrout = 'ICC(2,1)';
ICC{2}.est = ICC_2;
ICC{2}.FValue = ICC_2F;
ICC{2}.df1 = ICC_2df1;
ICC{2}.df2 = ICC_2df2;
ICC{2}.pVal = ICC_2pVal;
ICC{2}.confInterval = ICC_2CI;
ICC{3}.nameMcGraw = 'Single Fixed Raters';
ICC{3}.nameShrout = 'ICC(3,1)';
ICC{3}.est = ICC_3;
ICC{3}.FValue = ICC_3F;
ICC{3}.df1 = ICC_3df1;
ICC{3}.df2 = ICC_3df2;
ICC{3}.pVal = ICC_3pVal;
ICC{3}.confInterval = ICC_3CI;
ICC{4}.nameMcGraw = 'Average Raters Absolute';
ICC{4}.nameShrout = 'ICC(1,k)';
ICC{4}.est = ICC_1k;
ICC{4}.FValue = ICC_1kF;
ICC{4}.df1 = ICC_1kdf1;
ICC{4}.df2 = ICC_1kdf2;
ICC{4}.pVal = ICC_1kpVal;
ICC{4}.confInterval = ICC_1kCI;
ICC{5}.nameMcGraw = 'Average Random Raters';
ICC{5}.nameShrout = 'ICC(2,k)';
ICC{5}.est = ICC_2k;
ICC{5}.FValue = ICC_2kF;
ICC{5}.df1 = ICC_2kdf1;
ICC{5}.df2 = ICC_2kdf2;
ICC{5}.pVal = ICC_2kpVal;
ICC{5}.confInterval = ICC_2kCI;
ICC{6}.nameMcGraw = 'Average Fixed Raters';
ICC{6}.nameShrout = 'ICC(3,k)';
ICC{6}.est = ICC_3k;
ICC{6}.FValue = ICC_3kF;
ICC{6}.df1 = ICC_3kdf1;
ICC{6}.df2 = ICC_3kdf2;
ICC{6}.pVal = ICC_3kpVal;
ICC{6}.confInterval = ICC_3kCI;
end