Appendix 1: Is it possible to have one high reliable scale with low ‘responsiveness’, and a low
reliable scale with high ‘responsiveness’ measuring the same construct?
Consider two scales A and B measuring the same construct  before and after an effective
intervention. Before intervention  has expectation 1 and variance  12 . After the intervention  has
expectation 2 and variance  22 . The covariance between values before and after intervention is
denoted as 12.
Theta () is a latent construct and is quantified through scales A and B, assuming that both scales
conform to the classical test theory. This means that observed values with scale A and B at baseline
(Y1,A and Y1,B) may be seen as Y1,A= 1+e1,A and Y1,B= 1+e1,B, where e1,A and e1,B denote
measurement-error each with expectation zero, and variances  12, A and  12,B . Reliability of scales A
and B at baseline equals  12 /( 12   12, A ) , and  12 /( 12   12,B ) , and obviously, if B is more reliable
than A, this means that measurement-error of A varies more than that of B:  12,B <  12, A . After
intervention, classical test theory applies too: Y2,A= 2+e2,A and Y2,B= 2+e2,B, with error-variances
 22, A and  22,B .
Consider now the standardized response mean (SRM) as a measure of responsiveness. SRM for
scale A is defined as
y1, A  y 2, A
d
SRM A 

,
sd
s12, A  s22, A  2 s12, A
where d is the average change, and sd the standard deviation of the changes, and where
y1, A and y 2 , A are the observed means of scale A before and after intervention, and s12, A and s22, A the
observed variances of scale A before and after intervention, and s12,A the covariance between the
observed scale-scores. Obviously, expectation of the numerator of the SRMA equals 1-2, and the
same is the case for the numerator of SRMB. Hence, any difference between SRMA and SRMB must
be due to their denominators. The expected variances s12, A and s22, A are, off course, equal to
s12, A =  12 +  12, A , and s22, A =  22 +  22, A . The expected covariance s12,A equals S12,A= 12 + cov(1,e1,A) +
cov(2,e2,A) + cov(e1,A,e2,A). Since e1,A and e2,A are measurement-error terms (before and after
intervention), they will have zero covariance, both with  and with each other. Hence, the
denominator of SRMA equals (  12 +  22 -212)+  12, A +  22, A , and using similar arguments the
denominator of SRMB equals (  12 +  22 -212)+  12,B +  22,B .
Inspection of the denominators of SRMA and SRMB shows that SRMA can only be larger than
SRMB, when the measurement-error variance of scale A after intervention is smaller than that of
scale B:  22, A <<  22, B . Although this is not impossible, the occurrence is unlikely, especially when
the reverse is the case before intervention. It is clear moreover that the difference in responsiveness
is only a reflection of a difference in reliability between the two scales, since  22, A <<  22, B implies
that the reliability of scale B after intervention is lower than of scale A.