# CTT

Classical Test Theory (CTT), also known as true fraction theory. The earliest mathematical measurement theory.

## Catalogues

### Origin

Classical Test Theory (CTT), which began to rise from the end of the 19th century, formed a relatively complete system in the 1930s and gradually matured. In the 1950s, Glickson’s work gave it a complete mathematical theoretical form, and in 1968, Lord and Nowick’s “Statistical Theory of Psychological Test Scores” developed the classic test theory to its peak and realized In order to convert to modern measurement theory.

### Proper fraction

For the convenience of research, psychologists introduced the concept of true scores. True score is the true value or objective value when there is no measurement error in the measurement. The operation definition is the average value of countless measurement results. In actual measurement, the error is inevitable. When the error is close to true When scoring, we say that the error is small. The true score is usually expressed by T.

### Mathematical model

The observation score is represented by X, and E is the measurement error, then the basic equation of the true score is: X = T + E. Observe that there is a linear relationship between the score and the true score. The error here includes only the random error, and the systematic error is included in the true score.

### Suppose

According to the formula, we can derive three interrelated hypotheses: 　 First, observe N times repeatedly, the average error is zero, that is, the true score is equal to the average of the real score T=E(X) or E(E)= 0. 　 Second, the true score and measurement error are independent of each other. ρ(T,E)=0 　 Third, the error of each parallel test is zero. ρ(E1,E2)=0 　In practical applications, it is not feasible to repeatedly measure the same psychological traits of the same person with parallel tests, because parallel tests not only require the same traits to be measured, but also the subject, number, difficulty, and differentiation. Be consistent. This increases the difficulty of preparation. In general, we use the same test to measure a group. The error of everyone in the group can be assumed to be random and subject to a normal distribution. The variance of the measured score, true score and error score of the tested group has the following relationship, SX=ST+SE. Only random errors are involved in the formula, and the variance of the systematic error is included in the true fractional variance. This means that the true fractional variance contains the variance (SV) related to the measurement purpose and the variance (SI) independent of the measurement purpose. Thus, the formula can become SX=SV+SI+SE.

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