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## Interdisciplinaria

*versión On-line* ISSN 1668-7027

#### Resumen

ATTORRESI, Horacio Félix; AGUERRI, María Ester; LOZZIA, Gabriela Susana
y
GALIBERT, María Silvia. Intervalos de confianza para las puntuaciones verdaderas: Explicitación de sus supuestos.* Interdisciplinaria* [online].
2004,
vol.21, n.1, pp. 29-51.
ISSN 1668-7027.

The classical test theory was developed by starting with the linear model to which certain assumptions are added to allow the deduction of its fundamental properties. Since one of the measurement objectives is to estimate the individuals' true scores, the different inferential techniques for the true scores based on the observed ones, result of interest. Depending on the technique chosen, it will be necessary to add assumptions about the distribution of the classical linear model components. In the psychometric literature, two types of intervals are usually seen: one type is based on the measurement error e; and the other, on the error of estimation obtained from the prediction of true scores are predicted from a regression equation on the observed ones. Both types of intervals are set down on two models, each one with its own assumptions, and the models should be differentiated. In general, the authors do not state the assumptions clearly or, at times, they do not show how the assumptions are involved in the confidence interval construction. For example, Lord and Novick (1968) mentioned the need of variance homogeneity, without explaining how that assumption is used in the process. The assumption of variance homogeneity when the confidence intervals are built based on the error of estimation is not needed as it is shown later on. The greatest difficulties arise when the regression equation of true scores on observed scores is used. In that case, the assumptions for the usual regression model with fixed independent variables are used, while in the linear classical model, both scores vary jointly. The authors refer to the linear regression assumptions only in a slightly way, for example, about the homocedasticity, as if they took the true score confidence interval computation as a simple particular application of the Regression Theory on General Statistics. Nevertheless, in this context the regression model is something different. In the general regression model, all the variables are observable and it is applied to predict the value of one variable through the value of other or others of different nature. Two regression models can be distinguished depending whether the independent variable is considered as fixed or as random; the homocedasticity is a necessary assumption only for the first case. When the independent variable is random, the model is linked to the probability distribution of a bidimensional random vector; and therefore, there would not exist a dependent and another independent variable. The regression of the true scores on the observed ones is included in this case, since both variables vary jointly. Consequently, two different features regarding the regression model in the general statistic theory arise, and they confer certain peculiarity to the use of regression in the context of the classical test theory. The two differences are: - One of the two variables, the true score, is not observable. - For each individual, the true score and the observed score are variables of the same nature; actually, both are the same characteristic expressed in terms of their true values, and these are estimated by the observed ones. All these considerations must be taken into account when calculating the confidence intervals by application of the regression model, in order to base their deduction and to understand its meaning and limitations. Such is the objective of this article. To accomplish this, it is necessary to make explicit the assumptions on which the methods to infer the true scores are based. It deals with two models, which allow the deduction of two types of intervals and they will be treated as two different cases.

**Palabras llave
:
**Classical test theory; Confidence intervals; True scores; Assumptions.