13

that they can use to help develop more reliable tests. It is a desirable quality of any
measurement instrument to be as free as possible from measurement error because
measurement error adds variability to the data, which makes it more difficult to
measure change over time or to identify differences between groups. Most fre-
quently, the observed score (O) is defined as the sum of 2 unobservable, or latent
variables: the true score (T) and the error score (E); i.e.,

XT=+E

The associated variances are similarly related:

ssxes

(^22) =+ 2
t
Since T and E are unknown, it has to be further assumed that (1) T and E are uncor-
related, (2) measurement errors occur at random, and (3) error scores on parallel
tests are uncorrelated. As demonstrated by Lord, however, these assumptions more
or less follow from the definition of T and E [ 48 ]. Because of the assumption that
the expected value of E equals 0, the presence of measurement error will not sys-
tematically distort the expected value of O away from the expected value of T.
Consequently, O is an unbiased estimate of T. More problematic is the relationship
between the variances of X, T, and E of which only sx^2 is known. Many CTT mod-
els are concerned with obtaining information regarding st^2. Various ways to obtain
reliability coefficients have been proposed over time that expresses the ratio of true
score variance to the total variance of test scores:
R t
x
t
te

• s
  s
  s
  ss
  2
  2
  2
  22
  The most principled way to obtain the reliability coefficient would be to calculate
  the correlation coefficient between two parallel forms of an instrument, which
  would involve the availability of parallel versions of a PRO instrument that is a PRO
  on which patients have the same true score and with equal errors of measurement
  across instruments. Of course, the construction of parallel forms is a requirement
  that can never be exactly met. A lot of work has therefore been directed at develop-
  ing methods that can be used not only to evaluate reliability without parallel forms,
  such as generalizability theory but also to the practice of test–retest reliability [ 49 ].
  The CTT framework is very useful from a theoretical and practical point of view
  because it provides an explanation for various statistical phenomena related to
  measurement errors, such as regression toward the mean. CTT-based reliability
  coefficients can also be applied, for example, to identify, from a set of PRO instru-
  ments for a given outcome domain, the instrument that would be least affected by
  distortion due to measurement error. Finally, in the development of PRO instru-
  ments the theory provides means for scale developers to ensure the reliability of
  their instrument. An advantage of CTT over more complicated approaches is that
  CTT is based on relatively weak assumptions that are likely to be met under practi-
  cal situations and that these approaches are generally well-known in the field of
  1 PROMs and Quality of Care