The True score MTMM model
This decomposition can be made by a different parameterization of the Classical MTMM model. Now the basic structure of the model is split in two parts by introducing the so called true score (t) which is equal to the observed variable except for the random errors. So the first equation is as in the classical test theory (Lord and Novick 1956):
Y = t + e (1)
The second equation specifies the relationship between the true score (t), the latent traits (F) and the systematic errors related to the method used:
t = a + b.F + M (2)
Standardizing these two equations we get:
Y = r.t + e (3)
t = v.F + mM (4)
In equation (3) r is the strength of the relationship between the true score and the observed variable. In the literature r2 is the reliability of the observed variable which is equal to 1 - var(e).
In equation 4 the coefficient v indicates the strength of relationship between the latent trait and the true score(t) and v2 is called the validity of the observed variable.
The coefficient m is called the method effect and since var(t)=var(F)=var(M)=1 it follows that v2=1-m2. This means that they are each other complement.
Substituting equation (4) in (3) we get:
Y = r.v.F + r.mM + e (5)
The last equation shows that this equation is the same as the basic equation of the classical MTMM model with q= r.v. and s= r.m while the var(e) is the same in both MTMM models.
We see that q2 what we have called before the quality of the questions is now equal to the product of the reliability and the validity (r2.v2). We will continue to do so.
On the other hand we see that s2 what was called the variance of the systematic error has been changed because it is equal to (r2.m2). We prefer to use the coefficient m for the method effect because it is pure estimate of the method effect and not the product of the method effect and the reliability coefficient.
On the data presented in the section about the classical MTMM approach we also can estimate the parameters of the True score MTMM model.
It is form this more detailed information that Saris and Andrews have introduced the True score model for the analysis of data of MTMM experiments.
Saris W. E., and F. M. Andrews 1991. Evaluation of measurement instruments using a structural modeling approach. In P. P. Biemer, R. M. Groves, L. E. Lyberg, N. Mathiowetz and S. Sudman (eds.), Measurement Errors in Surveys, New York: Wiley, 575─599.
Saris W. E., and C. Aalberts 2003. Different explanations for correlated errors in MTMM studies. Structural Equation Modeling, 10, 193─214.
Saris W.E. and I.N.Gallhofer (2014) Design, evaluation and analysis of questionnaires for survey research. Hoboken, Wiley, (chapter 9 and 10)