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I am looking at this paper by the College Board on the validity of SAT score and High School GPA (HSGPA) for First Year GPA (FYGPA). They state they are using a straight forward linear regression to analyse the data. However, I have a question about how a correlation is affected by a couple of factors for these examples:

  1. Does the fact that the SAT score is a enforced normal distribution and GPA is not likely a normal distrubution affect the validity of the correlation?
  2. Should they report whether they are using normalized scores and does the lack of reporting mean that the different scales were not normalized?
  3. Does the difference in the distribution mean that the variance will be skewed? In other words, doesn't the fact that one metrics is forced normal distribution and the other is not imply a heteroskedasticity that makes a simple linear regression not the best tool for comparing these sets. So looking at the below picture which is representative of the distributions I can see how a linear regression can go from HS GPA to College GPA and still have an even distrubution. However, going from a perfect normal curve to totally skewed distribution with a linear colleration would have to have a very variance at the edges (upper part error will be below line while lower part error will be above line).

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  • $\begingroup$ There is no meaning to stating that the mean and variance are skewed. The Educational Testing Service (ETS) has a long history of competence in statistics as they administrate the scholastic aptitude test. $\endgroup$ Mar 26, 2017 at 0:21
  • $\begingroup$ what is the link between normal distribution and heteroskedasticity? Moreover you have several things in mind that are obscured. More background and raising a specific issue may be helpful to a person like me with no background in Educational Testing. $\endgroup$
    – user10619
    Mar 26, 2017 at 3:58

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  1. I guess by "the validity of the correlation" you just mean "the correlation". You could examine this question by looking at the correlation of raw scores with GPA rather than the scaled scores with GPA. A priori, it's not obvious how the correlation would change; it might not change much at all.
  2. Yes, transformations of variables should be reported, so if no transformation is mentioned, you're justified in assuming that none was used.
  3. No, I don't see how the change from raw scores to scaled scores would make a homoscedastic relationship heteroscedastic, nor vice versa. Think about it like this: homoscedasticity means that the conditional variance of GPA for each possible SAT score is the same, right? So any injective transformation of the SAT scores shouldn't make a difference. If the conditional variances were all equal before the transformation, they should still be all equal after.
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