Sánchez-Cubillo et al. (Reference Sánchez-Cubillo, Periáñez, Adrover-Roig, Rodríguez-Sánchez, Ríos-Lago, Tirapu and Barceló2009) conducted a study in which 41 healthy older subjects performed a battery of neuropsychological tests, including the Trail Making Test (TMT), the Digit Symbol subtest (WAIS-III), the Digits Forward and Backward subtests (WAIS-III), a Finger Tapping Test, a Stroop Test, and a task-switching paradigm akin to the Wisconsin Card Sorting Test (cf. Strauss, Sherman, and Spreen, Reference Strauss, Sherman and Spreen2006). The results of correlation and regression analyses suggested that TMTA requires mainly visuo-perceptual abilities, TMTB primarily reflects working memory and task-switching abilities, while the TMTB−A difference score provides a relatively pure indicator of task-switching abilities. The use of the TMTB−A difference score should help clinicians to interpret abnormal performance in terms of a failure of this specific cognitive mechanism.
Unfortunately, there is the danger that the reliability of difference scores will be unacceptably low because the reliability of a difference score is simply a function of the average reliability of its two components and of the correlation between them (Crawford, Sutherland, and Garthwaite, Reference Crawford, Sutherland and Garthwaite2008). Under the circumstance of a common standard deviation of the two components used to form the difference, the formula for the reliability of a difference score, r(B−A), is:
![\[--><$$>r(B\, - \,A) = \frac{{\frac{{r(AA)\, + \,r(BB)}}{2}\, - \,r(BA)}}{{1\, - \,r(BA)}}\eqno<$$><!--\]](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20160901070446819-0922:S1355617711000361_eqnU1.gif?pub-status=live){kind=small}
,where r(AA) and r(BB) are the reliabilities of the two components, and r(BA) is the correlation between them (Crawford et al., Reference Crawford, Sutherland and Garthwaite2008). Thus, if difference scores compare measures of two related constructs, the correlation between the components will be substantial, and it may eventually approach the reliabilities of the components in its magnitude. Given this situation, the variance of the difference score will predominantly be measurement error variance, simply because the numerator of Equation 1 will approach zero.
The available data point in this direction. Reynolds’ (Reference Reynolds2002) estimates of internal consistencies and correlations that were obtained in the normative sample of the Comprehensive Trail Making Test (CTMT) are presented in Table 1. Applying Reynolds’ (Reference Reynolds2002) data to Eq. 1, a reliability of the difference score TMTB−A of r(B−A) = 0.32 is predicted, which is unacceptably low for any clinical purpose. Crawford et al. (Reference Crawford, Sutherland and Garthwaite2008) obtained even lower estimates for the reliabilities of TMT difference scores from the Delis-Kaplan Executive Functioning System (D-KEFS; Delis, Kaplan, and Kramer, Reference Delis, Kaplan and Kramer2001). Specifically, the reliability estimates of the difference score D-KEFS-TMTNumber–Letter–Switching (TMTB analogue) minus D-KEFS-TMTNumber–Sequencing (TMTA analogue) equaled in three different age groups .10, −.06, and −.08, respectively (Crawford et al., Reference Crawford, Sutherland and Garthwaite2008).
Table 1 Estimates of CTMT internal consistencies and correlations in the normative sample (N = 1.664; Reynolds, Reference Reynolds2002)
Note. TMTA is defined here as a Number-Sequencing task. TMTB is defined as a Number-Letter-Switching task. Trail 1 essentially mimics the TMTA, whereas Trails 2 and 3 are similar to the TMTA but introduce distractor items. The inclusion of distractor items is deemed to be of no relevance at this point. The internal consistency estimates of the three TMTA analogues thus average to .74.
aTrail 4 of the CTMT does not fit into the TMTA/TMTB dichotomy.
bIn the sample of Sánchez-Cubillo et al. (Reference Sánchez-Cubillo, Periáñez, Adrover-Roig, Rodríguez-Sánchez, Ríos-Lago, Tirapu and Barceló2009), the correlation between the TMTA and the TMTB scores amounted to .73.
Adequate reliability is fundamental whenever the cognitive status of an individual is assessed. The advocated TMTB−A difference score should be considered un-interpretable in those contexts, due to its expectably unacceptable reliability, and, despite its superior construct validity. We deliberately designed the Brunswick Trail Making Test (BTMT) to maximize internal consistency (above the level of .90), by adjusting test length (Kopp, Rösser, and Wessel, Reference Kopp, Rösser and Wessel2008). To conclude, neuropsychologists should be very reluctant to use difference scores which compare measures of two related constructs, due to the potential trade-off between gains in validity and losses in reliability. Needless to emphasize that this argument applies to all domains of neuropsychological assessment.
