I want to use a risk score (RS) as an instrument for an exposure on a clinical outcome. However, I wont have access to data on the outcome for some time, and wish to examine whether this risk score would serve as a decent instrument. It is advised to report both the R-squared (as amount of variance explained by RS) and the F-statistic, but I'm having trouble understanding the following (basic) issues:

  • Am I correct in assuming that R-squared is the squared semi-partial correlation, and is this simply the difference in R-squared with and without adding the instrument to the model in linear regression? (model would be: exposure ~ RS + age + sex)
  • Running linear regression with the previously mentioned model, an F-statistic is produced. Is this the preferred F-estimate of instrument strength?

The R-squared you refer to is usually a partial R-squared, i.e. the "squared partial correlation" between the excluded instruments and the endogenous regressor. The relevant reference for this test would be

Shea, J., 1997, "Instrument Relevance in Multivariate Linear Models: A Simple Measure", Review of Economics and Statistics, Vol. 79, No. 2, May, 348-352

From there you will also see how to get the critical values for the test. Some statistical software like Stata produce these values automatically like the command ivreg2, for instance.

The F-test you refer to also is related to the first stage, i.e. the regression of your endogenous variable on the instrument(s) and the exogenous variables. In case of one instrument, the F-statistic is the t-statistic squared. As a rule of thumb you should be worried about weak instruments if your F-statistic is below 10. The reference is

Stock J, Yogo M. Testing for Weak Instruments in Linear IV Regression. In Andrews DWK Identification and Inference for Econometric Models New York: Cambridge University Press; 2005. pp. 80-108

  • $\begingroup$ I was under the impression that data on the outcome would be needed to run (for example) ivreg2, which is why I was trying to obtain the R-squared and the F-statistic with linear regression. I'll be sure to check out these references. Thank you for your detailed and clear explanation! $\endgroup$ – AbelSP Oct 19 '14 at 20:15
  • $\begingroup$ The first stage from which the mentioned statistics come is just an OLS regression of the endogenous variable on the instrument(s) and the exogenous variables. So if you are only interested in the strength of your instrument(s), then you don't need the outcome for this. For the final estimation you will need the outcome variable, too. $\endgroup$ – Andy Oct 19 '14 at 21:04
  • $\begingroup$ Thanks again! One final question, if you don't mind. Taking the square root of the t-statistic gave a much smaller F-statistic than I would have expected, could it possible be that the F-statistic (when using a single instrument) is actually the square of the t-statistic? $\endgroup$ – AbelSP Oct 19 '14 at 21:12
  • $\begingroup$ Yes, I mixed that up. The t-statistic squared gives you the F-statistic. Sorry for that, I updated the answer. Thanks for spotting this! $\endgroup$ – Andy Oct 19 '14 at 21:21

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