Equivalence (test theory)

The equivalence (equivalency) of measurements is the basis for a precise estimate of reliability . Equivalence is a term from test theory .

The classical test theory assumes that the observed value is a measurement of the actual true value and an error. The errors of several measurements are independent and unsystematic. Reliability estimates assume that the quality of the measurements is always the same (or were carried out under comparable conditions). However, this assumption is not always valid.

For example, the quality of repeated measurements can be influenced by fatigue or training effects. This means that the same test and error values ​​cannot always be assumed. The equivalence (equivalence) of the measurement therefore determines which formula can be used to estimate the reliability.

Types of equivalence of measurements

Strictly parallel

Two measurements are strictly parallel if they have the same true values ​​and the same error variance:

${\displaystyle {\begin{aligned}T\prime &=T\\\sigma _{E}^{2}\prime &=\sigma _{E}^{2}\end{aligned}}}$

The following applies:

${\displaystyle {\begin{aligned}T\prime &={\text{Wahrer Wert in Test A}}\\T&={\text{Wahrer Wert in Test B}}\\\sigma _{E}^{2}\prime &={\text{Fehlervarianz in Test A}}\\\sigma _{E}^{2}&={\text{Fehlervarianz in Test B}}\end{aligned}}}$

This means that a person in tests A and B will get the same true value and the measurement errors of both tests will be the same. It is therefore true that both tests measure the same property with the same scale and equally well for all persons.

Essentially parallel

Two measurements are essentially parallel if they shift the same true values ​​by a constant and have the same error variance:

${\displaystyle {\begin{aligned}T\prime &=T+\gamma \\\sigma _{E}^{2}\prime &=\sigma _{E}^{2}\end{aligned}}}$

The following applies:

${\displaystyle {\begin{aligned}\gamma &={\text{Additive Konstante}}\end{aligned}}}$

This means that a person's true worth is just shifted by one constant.

Only in the case of a strictly parallel or essentially parallel measurement does the correlation between the two tests correspond to an estimate of the reliability. The Spearman-Brown formula can be used for this, or if the sample size is small, the Kristof formula .

Tau equivalent

Although both measurements show the same true values, the error variances differ:

${\displaystyle {\begin{aligned}T\prime &=T\\\sigma _{E}^{2}\prime &\neq \sigma _{E}^{2}\end{aligned}}}$

The reliability must therefore differ between the individual measurements, since the reliability indicates the relationship between the true measured value and the observed measured value (according to the classical test theory, the sum of the true value and the measurement error).

Essential / Essentially tau-equivalent

In contrast to the tau-equivalent measurement, the true value is shifted by an additive constant:

${\displaystyle {\begin{aligned}T\prime &=T+\gamma \\\sigma _{E}^{2}\prime &\neq \sigma _{E}^{2}\end{aligned}}}$

Guttman's formula can be used to calculate the reliability with tau equivalence . If the two test parts are of different sizes, the Feldt formula is used.

Congeneric

Congeneric measurements are shifted by an additive constant and have a unit of measurement that differs by a multiplicative constant:

${\displaystyle {\begin{aligned}T\prime &=\beta \cdot T+\gamma \\\sigma _{E}^{2}\prime &\neq \sigma _{E}^{2}\end{aligned}}}$

The following applies:

${\displaystyle {\begin{aligned}\beta &={\text{Multiplikative Konstante}}\end{aligned}}}$

Both measurements still represent the same measured variable . Mean value, error variance and unit of measure are different, but the true values ​​correlate perfectly with each other.

literature

• M. Bühner: Introduction to test and questionnaire construction. 3rd, updated edition. Pearson Studies, Munich 2010, ISBN 978-3-86894-033-6 .

Web links

• Introduction to psychometric theory (reliability) (English; PDF; 2.3 MB)