The Quality prediction based on SQP information
The coefficients predicted by SQP are presented in the measurement model of a simple concept ($F$) below where all variables are standardized except the random error term ($e^{'}$). The method effect ($M$) and the random error term ($e^{'}$) are assumed to be independents of each other and of the concept ($F$).
This is a measurement model where $Y$ is the standardized observed variable, $F$ is the standardized simple concept, $T$ is the standardized true score, $M$ is the standardized method factor which affects $T$, and $e^{´}$ is the random error affecting $Y$. The coefficient $v$ represents the validity coefficient, $r$ the reliability coefficient and and $\mu$ the method effect. All coefficients are standardized coefficients.
This model can be formulated in two equations:
$T = v F + {\mu} M $
$Y = r T + e^{´}$
or by substitution
$Y = q F + m M + e^{´}$
where $q = r v$ and $ m = r {\mu}$
This gives the interpretation that $q$ represents the effect of the concept $F$ on the observed variable and that $q^{2}$ is the explained variance in $Y$ by $F$ or the quality of the measure $Y$ for $F$.
In the same way m represents the systematic effect of the method factor on the observed variable $Y$ and $m^{2}$ is equal to the explained variance in $Y$ caused by the method used.
Using this notation the following equivalent measurement model can be specified:
We have seen that SQP provides estimates of $r$ and $v$ therefore $q$ is also known.
Since $ m = r {\mu}$ and ${\mu}^{2}=1-v^{2}$ it follows that $m$ is also known if $r$ and $v$ are known for a question.
Therefore SQP can provide the basic information for the evaluation of the quality of a standardized observed variable for its concept of interest.
Given that $q^{2}$ is the explained variance in $Y$ it also follows that variance of the unstandardized variable $f$ denoted by ${\sigma}_{ff}=q^{2}{\sigma}^{2}_{yy}$ where ${\sigma}_{yy}$ is the variance of the unstandardized variable $y$.
For a complete overview of the approach we refer to:
Saris W.E. and I.N. Gallhofer (Second Edition, 2014). Design, evaluation and analysis of questionnaires for survey research. Hoboken, Wiley
Since $ m = r {\mu}$ and ${\mu}^{2}=1-v^{2}$ it follows that $m$ is also known if $r$ and $v$ are known for a question.
Therefore SQP can provide the basic information for the evaluation of the quality of a standardized observed variable for its concept of interest.
Given that $q^{2}$ is the explained variance in $Y$ it also follows that variance of the unstandardized variable $f$ denoted by ${\sigma}_{ff}=q^{2}{\sigma}^{2}_{yy}$ where ${\sigma}_{yy}$ is the variance of the unstandardized variable $y$.
For a complete overview of the approach we refer to:
Saris W.E. and I.N. Gallhofer (Second Edition, 2014). Design, evaluation and analysis of questionnaires for survey research. Hoboken, Wiley
