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The True Score Model is a measurement model developed to analyse the quality of survey questions, which is defined as “the strength of the relationship between the latent variable of interest and the observed answers to the survey question asked to measure this latent concept”. According to this model, the observed response variable is determined by the true score of the latent concept being measured and the random and systematic errors that occur during measurement. Using this model, researchers can correct the answers obtained from surveys (the observed response variable) for measurement errors and, thus, obtain the true score. An advantage of the more complex True Score Model used by is that it can distinguish between two components of question quality, these being validity and reliability. This distinction is important because these components can be affected by different factors.

Mathematical formula behind the true score model:
Formula obtained from :

(1) $$ p(y_{1j}, y_{2j}) = r_{1j} v_{1j} p(f_1,f_2) v_{2j} r_{2j} + r_{1j} m_{1j} m_{2j} r_{2j} $$

Where: p(y1j,y2j) = correlation between the observed variables. p(f1,f2) = correlation between the latent concepts being measured. rij = reliability coefficients. vij = validity coefficients. mij = method effects, which can be calculated as follows :

(2) $$ m_{ij}^2=1-v_{ij}^2 $$

How to correct for measurement errors:
In order to correct for measurement errors, one option is to correct the correlations between the variables in the matrix using the following formula presented by :

(3) $$ Corrected r_{ij}= \frac{observed r_{ij} - cmv}{q_i q_j} $$

Where: Observed rij = correlation between the answers to two survey questions. qij = quality coefficients of the variables. cmv = common method variance between the two variables, which can be calculated as follows.

(4) $$ cmv = r_i m_i m_j r_j $$

Where: rij = reliability coefficients of the variables, and mij = method effects on variables.

In general, the quality estimates and the cmv for simple concepts measured with a single item, or concepts-by-intuition, can be obtained using the software SQP 2.1 (http://sqp.upf.edu/).

Nevertheless, when the model includes complex concepts measured with several items, or concepts-by-postulation, which are analysed using composite scores, researchers have to calculate the quality estimates of these composite scores using the following formulas presented in :

Model with concepts-by-postulation with reflective indicators:
(5) $$ p(CP_1,S_1) = \sum_{i=1}^k \frac{q_i w_i}{sd(S_1)} = \lbrace\frac{1}{sd(S_1)}\rbrace \sum_{i=1}^k q_i w_i $$

Where: CP1 = concept-by-postulation. S1 = composite score. qi = quality coefficient of each indicator used to create the composite score. wi = weights of each indicator. sd(S1) = standard deviation of the composite score.

Model with concepts-by-postulation with formative indicators:
(6) $$ Quality of S = 1 - \frac{var(e_s)}{var(S)} $$

Where: S = composite score. Var (S) = variance of the composite score. Var (es) = variance of the errors in the composite score, which can be calculated as follows :

(7) $$ Var(e_s) = \sum w_i^2 var(e_i) + 2\sum w_i w_j cov(e_i e_j) $$

Where: wij = weights of the indicators used to create the composite score, var(ei) = error variance of each indicator and cov (eiej) = error covariance.

References:
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