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Thread: Measurement error and correlation coefficients: a social-sciency approach...

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    Measurement error and correlation coefficients: a social-sciency approach...

    hello everyone! yesterday's chat on measurement error and regression made me think that it would be nice to demonstrate (from a social-science-like point of view) the infulence that measurement error has on correlations (and by extension on regression coefficients) and why it was an important motivator for the development of areas like structural equation modeling.

    i say this is from a 'social science' perspective because i will make use of the classical test theory model (and its assumptions) but i'm sure its much more generalizable to other areas of statistics. it really doesn't need more than understanding the basic properties of covariance algebra and i took the bulk of it from the Bollen (1989) book, although this is the univariate case (he deals mostly with the multivariate approach and the matrix algebra can get in the way of understanding things, so i worked on these slides for when i was a TA last year).

    in any case, here we go!

    let us define to observed scores X_{i} and X_{j} as:

    X_{i} = T_{i} + E_{i}
    X_{j} = T_{j} + E_{j}

    so T stands for 'true score' and E for 'error' with the properties that the error has a mean of 0, a variance \sigma_{e}^{2} and some people like to say it's normally distributed (<--- not a necessary assumption but makes some things nicer later on). it is also important to keep in mind that E is UNCORRELATED with T, so the errors and the true score have a correlation/covariance of 0. the errors are also UNCORRELATED among themselves. in the kind of research that we do, we want to say stuff about T, the true score, because that is the variable that measures the construct of interest. everything else gets in the way so we want to minimize its impact.

    this leads to some basic (yet illustrative) ideas:

    Var(X_{i}) = Var(T_{i}+E_{i})=
    Var(T_{i}) + Var(E_{i}) + 2Cov(T_{i},E_{i}) =
    Var(T_{i}) + Var(E_{i})+0=Var(T_{i}) + Var(E_{i})

    a similar argument can be made for X_{j} and we get our first interesting result: measurement error INFLATES variances.

    now, what about covariances?

    Cov(X_{i},X_{j})=Cov(T_{i}+E_{i}, T_{j}+E_{j})=



    uhm... interesting. it appears that, under the previously described model, measurement error has no influence in the covariances of the observed scores X_{i} and X_{j}

    now, with these elements, we can re-express the correlation coefficient as:

    \frac{Cov(T_{i},T_{j})}{\sqrt{Var(T_{i}+E_{i})Var(X_{j}+E_{j})}}\leq \frac{Cov(T_{i},T_{j})}{\sqrt{Var(T_{i})Var(T_{j})}}

    so the correlation between the observed scores is always less than or equal to the correlation between the true scores.

    and a similar development can be made for regression coefficients in the case of Y = bX+e trying to estimate the true regression T_{Y}=b*T_{X}+e* to show that the regression coefficient b is attenuated and does not estimate b* even if the sample size grows to infinity.

    now, i know the model for classical test theory sounds constricted and artificial. but then again we're social scientists so we get a free pass!
    for all your psychometric needs! https://psychometroscar.wordpress.com/about/

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