True Score

CTT

CTT asserts that observed scores for a person Xp can be split into two parts: a true score (Tp) that is specific to the person and error scores (Ep):

$$X_p=T_p+E_P.$$

Although this equation is algebraically simple, it is complicated by the fact that neither the person’s true score nor the errors are directly observable. Indeed, Tp is undefined. Typically, this problem is circumvented by assuming that the expected value (E) of the errors is 0; that is,

$$E\left(E_P\right)=0.$$

Under this assumption,

$$\begin{array}{c}E\left(X_P\right)=E\left(T_P+E_P\right)=E\left(T_P\right)+E\left(E_P\right)\\ =E\left(T_P\right)=T_P,\end{array}$$

because Tp is a constant specific to the person. Although it is common practice to refer to a definition of true score, the aforementioned development shows that in CTT, true score is typically better viewed as a quantity derived from the model in Equation 1 and the assumption in Equation 2.

Another central assumption in CTT is that true scores and error scores are uncorrelated,

$${\sigma }_{TE}=0.$$

Under this assumption, the variance (over persons) of observed scores is simply:

$${\sigma }_X^2={\sigma }_T^2+{\sigma }_E^2.$$

True scores play a crucial role in reliability, which is defined canonically as the squared correlation between observed and true scores, ${\rho }_{XT}^2$. Given the assumption that σTE = 0, it is easy to show that ${\rho }_{XT}^2={\sigma }_T^2/{\sigma }_X^2=1-{\sigma }_E^2/{\sigma }_X^2$, where the last formula is the most frequently used basis for estimating reliability.

Returning to Equation 3, there is one very important unanswered question, that is, what constitutes the replications over which the expectations in Equation 3 are taken? In CTT, the traditional answer to this question is expectations are taken over forms of a test that have equal observed score means, variances, and covariances—called classically parallel forms. This is only one of many possible answers, however, and each different definition of replications can (and usually does) lead to different results. This is the principal reason why there are so many different formulas for estimating quantities such as reliability.

[Page 1725]It is particularly important to note that there is no “right” or “best” definition of replications. It follows that there is no universally right or best characterization of true score. Rather, an investigator must choose how the true score shall be viewed. So, for example, an investigator who chooses to use coefficient α to estimate reliability is assuming (knowingly or unknowingly) that replications consist of forms that are essentially τ equivalent, which is not quite the same as classically parallel.

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