Traditional hypothesis testing can be applied to check if additivity can be assumed. If color differences are additive, the perception of the difference of differences,
\(\Delta d =d_1 - d_2\), should not depend on the size of the individual differences. It follows that participant accuracy should not be affected by the average size of differences in a triad,
\(\bar{d} = 0.5*(d_1 + d_2)\). In this case, a linear regression,
\begin{equation}
\widehat{acc} = h(\Delta d, \bar{d}) = \beta _0 + \beta _1 |\Delta d| + \beta _2 \bar{d},
\end{equation}
built to include this term should return a
\(\beta _2\) coefficient close to zero. A coefficient for the average difference term significantly different from zero supports the claim that small differences are not additive for equivalent large differences. This can be formalized as a pair of null and alternative hypotheses,
\begin{eqnarray}
\begin{array}{l}H_0: \beta _2 = 0, \\[.5em]
H_1: \beta _2 \ne 0, \end{array} \quad
\end{eqnarray}
where the alternative hypothesis is the general case of nonadditivity. In the case of diminishing returns, accuracy is expected to decrease with increasing average difference,
\(\bar{d}\), resulting in a significantly negative coefficient. In the case of increasing returns, the opposite is expected. For the purposes of this study, we determine significance using a
t-test, which is conveniently built into the regression in R. The hypothesis testing is meant as a preliminary analysis as it makes the strong assumption that the perceptual scale is well known. This assumption is likely to fail, even if a perceptual scale has previously been estimated assuming additivity. However, in application, other statistical tests can be used to determine whether the regression suggests nonadditivity such as a likelihood ratio test.