Errors of Measurement: Attenuation

Defining Attenuation and the Correction

Attenuation is defined by the distance (as measured by the reliability of the variable, typically using Cronbach’s α) between the observed variable (operationalized) and the conceptualized or described variable. The relationship is between the various elements represented in Figure 1.

Figure 1 Diagram for Attenuation

Figure 1 displays the conceptual or desired variables X and Y. The correlation of (XY) is the desired outcome sought by the investigator. If the reliabilities of X and Y are both 1.00, indicating perfect measurement for both variables, then the value of the correlation (XY) has no attenuation when measured by the operationalized variables x and y. The distance between the conceptual or desired indication of X and the operationalized measurement of x, under conditions of perfect reliability, are considered as perfect and no attenuation occurs. Typically, the measurement of the desired variable X and Y use measurement instruments with less than perfect reliability. The result is that there is a distance between the desired X and the operationalized x (the same is true with the relationship between Y and y). The association between X and x can be represented by the reliability, often represented as rxx; or with Y to y, ryy. This association mathematically is represented by the reliability. In the diagram, the desired X or Y is causing the indication of x and y.

If the relationship between the variables becomes represented, the reason that x measures X becomes a sense of causality such that the level of X that exists causes the indication of x measured by the instrument, as is true that Y causes the indicator y. The result of the operationalization process is such that the relationship actually observed and measured is the correlation of (xy) as the best estimate of the desired correlation (XY). The question is the relationship between the observed correlation (xy) and the actual correlation (XY) between the two variables of interest. When the reliability of both X and Y is perfect, 1.00, then the two correlations (rXY and rxy) are identical. As the reliability reduces and lowers, the observed correlation will predictably reduce. Using the diagram in Figure 1, the mathematical relationships described create the following relationship permitting a correction to estimate the desired correlation (XY) using the reliability for each variable and the observed correlation (xy).

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