# quantitative relation between standard error of measurement and validity

Is there a function to estimate the gain in predictive validity with the decrease of the standard error of measurement?

E.g. If the short version of a test has predictive validity of 0.35 and standard error of measurement of 0.4 what can we expect the predictive validity of the long version (with some new items added) to be, if we reduced the Sem to 0.3 ? (Lets presume that we measure exactly the same construct with the two versions, only the reliability is different.)

My understanding of validity is that it is not directly related to the variance of your measures around a factor. Validity refers to whether or not you are actually measuring what it is you think you are measuring.

For example, if I have three math questions and I use people's scores on these questions to measure the factor of "mathematical capability" I could find other questions that load onto this supposed factor extremely well. However, if I add a "language proficiency question" to my survey, and this loads well onto my supposed factor, maybe I was mistaken when I thought I was measuring "mathematical capability", maybe I'm actually measuring "general academic capability". this would suggest that my questions are not valid measures of "Mathematical capability" and that I have to go back to the drawing board.

Validity and variance are not necessarily related, because you could have a highly variable measure that points directly do the factor you want to measure (i.e., "unbiased") or an extremely precise measure that is off from what you want to be measuring.

I am unsure as to what test you are using to acquire a "validity" metric, but given that my understanding of your question is correct, I would say that validity is not related to the variance around a factor.

• There are many different types of validity - eg. construct validity, face validity - here I am only considering predictive validity. I believe that when using a unidimensional instrument (eg. a typical ability test) that has some predictive validity the less error you have measuring that dimension the higher the predictive validity will be. Zero reliability certainly means zero validity. Commented May 31, 2017 at 9:52
• I took a look online and read that predictive validity is typically measured as the correlation between a measure and the outcome that you desire to predict. If this is the same measure of predictive validity that you are interested in, then we are essentially discussing a multiple regression model where we are adding more predictor variables (though this could be complicated if you are interested in running a factor analysis first, and then using a derived factor to predict your outcome). If this is the same metric that you are interested in then I would be happy to discuss further. Commented May 31, 2017 at 21:10
• Thanks, but at this point I am only interested in what happens when we use a single predictor but increase its reliability.All other things being the same when we increase the reliability of the predictor we should get a better prediction. I would like to know how much better our prediction becomes when we increase the reliability of our predictor by X. Commented Jun 1, 2017 at 12:55
• I believe I found what I was looking for: en.wikipedia.org/wiki/Correction_for_attenuation Commented Jun 1, 2017 at 20:56
• Nice. that does look precisely like what you were looking for. If you presume that the reliability of Y is constant, then your correlation for attenuation will vary exclusively with the reliability of X. Sorry for the confusion. Commented Jun 1, 2017 at 21:20