Standard Formulation
    Score = True Score plus Error or X = T + E
    The true score is assumed to be uncorrelated with the errors of 
       measurement.

Definition of Components
    Classical
       True score: the meaningful portion of variance or the average of all
          possible measures of the true score
       Measurement error: irrelevant sources of variance or the score minus
          the true score
    Modern: The variance of the score consists of many different 
       components.  In a particular research context, some of these
       components are meaningful (the true score) and others are not
       (error).

Reliability
    The proportion of variance due to true score -- V(T)/V(X)
    Do not confuse reliability with how to measure it (e.g., internal
       consistency or test-retest).	
    Reliability refers not just to the measure, but to sample and context
       of measurement.
    Learn about a computer program for reliability computation.

Standardized Models
    The path from true score to measured variable when both variables are 
       standardized equals the square root of the reliability.  Thus, the
       correlation of the measure with the true score equals the square
       root of the reliability. 

Correction for Attenuation
    A correlation needs to be divided by the square root of the product
       of the two variables' reliability to determine what the correlation
       between the two variables would be if the variables' reliabilities 
       were perfect.  

Terminology
    Indicator: a measure in structural model than contains measurement 
       error; usually has a square around it
    Construct or latent variable: a theoretical variable in a model which is
       tapped by indicators; usually circled 

Correlated Measurement Errors
    Two indicators of the same construct may share variance because they are
       measured by a common method.

The Effects of Unreliability in Causal Models
    If a causal variable has measurement error, the estimate of its 
       effect is biased, as well as the effects of other variables in the 
       structural equation.  Measurement error in the effect variable does 
       not bias its coefficient unless the variables are standardized.
       In this case the bias is that the true beta equal the measured beta
       divided by the square root of the endogenous variable's reliability.
  
    For a causal variable X, measurement error biases the estimate of 
       another causal variable Z that is in the equation when:

          Causal variable X has measurement error.
          Variables X and Z are correlated.
          X affects the endogenous variable.

    These three factors multiply to produce bias and so if any one is 
       missing, there is no bias.

    There are three ways to remove the biasing effects of unreliability in 
       the causal variable:

             Instrumental Variable Estimation
             Multiple Indicator Latent Variable Models
             Correction for Attenuation

       The last strategy is problematic because it presumes that 
       reliabilities are exactly known which is never true and sometimes 
       estimation breaks down because the correlation matrix is 
       ill-conditioned.  To implement the strategy, one would input the 
       correlation or covariance matrix and fix the measure's error variance 
       to 1 - reliability for the correlation matrix and (1 - reliability)
       times the measure's variance for the covariance matrix.

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