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Color Research and Application, Vol.45, No.2, 196-207, 2020

Testing the differences between two color measurement probability distributions using Hotelling's T-2 test and the permutation test

It is common practice in statistics to test the equality of two population means using, for example, the Student's t test, in the univariate case, or the Hotelling's T-2 Test in the multivariate case. However, tests on the equality of population means are not well developed for testing the difference between two populations of color measurements. Methods for analyzing populations of spectral reflectance and L*a*b* measurements have been described for applications such as analyzing inter-instrument agreement and repeatability. Methods have also been proposed for the analysis of color differences, but there are little written about techniques for testing whether two samples have the same probability distribution. This article focuses on testing the difference between color measurement probability distributions based on color difference. In addition, a metric is proposed called the threshold for color difference discrimination (TCDD, in units of Delta E), the color difference at which two populations can be considered to have different population distributions. A lower TCDD means smaller color differences between two samples can be resolved. Two parametric tests based on Hotelling's T-2 test and a nonparametric permutation test were used to determine the TCDD for populations of color measurements with different variances and sample sizes. The TCDD was found to be smaller by tests using the Hotelling's T-2 statistic, compared with a permutation test performed directly on color difference. It was also found, as expected, that larger sample sizes led to smaller TCDDs, as did smaller population variances.