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From |
"Mike Hollis" <mehla@earthlink.net> |

To |
<statalist@hsphsun2.harvard.edu> |

Subject |
RE: st: RE: Re: errors in outcome variables regression |

Date |
Sat, 5 Jul 2003 11:28:59 -0700 |

Assuming "well behaved" measurement error in the dependent variable, the OLS regression coefficient(s) will be unbiased. But the standard errors for the coefficients (and other statistics involving the variance of the dependent variable) will be wrong. Consider an extension of Mark's model where we add a single explanatory variable and his error term distinguishing between the "true" and "measurement" components of the error term: y = bo + b1X1 + u + u_m. Under standard assuptions, the residual variance of y is then V(u + u_m) = V(u) + V(u_m). If y is measured without error, the standard error of b1 would be sqrt( V(u) / SSx ), where u_m=0 and SSx is the usual mean-corrected sum of squares for X. With measurement error, the standard error for b1 would be sqrt ( V(u) + V (u_m) / SSx). Accordingly, in this case, measurement error causes the true standard error of b1 to be overstated by 1 + v(u-m)/v(m). Other statistics involving either the variance of y or the residual variance of y|x (e.g., simple correlation coefficient, R**2 for the equation, standardized regression coefficients) will likewise be incorrect. -----Original Message----- From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu]On Behalf Of Mark Schaffer Sent: Saturday, July 05, 2003 9:26 AM To: statalist@hsphsun2.harvard.edu; Mike Hollis Subject: Re: st: RE: Re: errors in outcome variables regression Mike et al., Quoting Mike Hollis <mehla@earthlink.net>: > Measurement error in the endogeneous variable will, however, cause > the > residual variance for the equation to be overstated, meaning, in > general, > that the standard errors for the regression coefficients will be too > large > and the estimated t- and F-statistics will be too small. Scott re-replied to Margaret's original post, so I'll re-reply to Mike's. I'm pretty sure Mike's point above isn't correct. So long as the measurement error satisfies the usual distributional assumptions that make OLS kosher (homoskedasticity, orthogonality etc.), and so long as the regressions error (the "non-measurement-error" error) also satisfies these assumptions, then OLS is fine. Intuitively, the reason is the following. Say the measurement error is u_m and the regression error is u. Define a new combined error term u_c = u + u_m. Now rewrite the regression equation with this single combined error term. It's not hard to see that so long as u_c satisfies the usual distributional assumptions (and it should if both u and u_m do so) then OLS is fine. For more details, see Scott's cite of Greene. That said, there will often be times that measurement error in the endogenous variable will not satisfy the usual assumptions and OLS will not be kosher. In particular, if the measurement error is heteroskedastic, then the SEs and the F-stat will not be consistent. But this is a heteroskedasticity problem, not a measurement error problem per se. Hope this helps. --Mark > > If you have a estimate of the reliability of the outcome variable, > you could > conceivable use this to adjust the standard errors and associated > statistics, although the quality of this adjustment obviously > depends on the > quality of your reliability estimate. (Note, however, that the > intra-class > correlation coefficient is a measure of non-independence. > Correcting for > measurement error in your case requires something like Chronbach's > alpha or, > if you're lucky enough to have them, multiple indicators for the > outcome > variable. See Ken Bollen's _Structural Equations with Latent > Variables_ for > a discussion of different strategies.) > > If the regression coefficients in your current model are > statitically > significant (i.e., you're not in a situation where you're trying to > correct > for measurement error to reduce standard errors in an attempt to > cause > statistically non-significant to become significant), you might > simply note > the fact that you suspect your outcome variable is affected by > measurement > error and that this will cause the significance level of the > regression > coefficients in your model to be underestimated. > > -----Original Message----- > From: owner-statalist@hsphsun2.harvard.edu > [mailto:owner-statalist@hsphsun2.harvard.edu]On Behalf Of Scott > Merryman > Sent: Friday, July 04, 2003 5:58 AM > To: statalist@hsphsun2.harvard.edu > Subject: st: Re: errors in outcome variables regression > > > ----- Original Message ----- > From: "Margaret May" <M.T.May@bristol.ac.uk> > To: <statalist@hsphsun2.harvard.edu> > Sent: Friday, July 04, 2003 5:32 AM > Subject: st: errors in outcome variables regression > > > > I have been looking at the command eivreg (errors in variables > regression) > > which corrects the effect estimate when independent variables are > measured > > with error. The problem I have is looking at differences in a > continuous > > outcome between exposure groups where the outcome variable is > measured > with > > error. I can estimate the reliability of the outcome measure as I > have > data > > from a validity study so can estimate the intra-class > correlation > > coefficient. Is there a method for correcting for measurement > error in > > outcome variables? > > > > Margaret May > > > > > A question concerning errors in the dependent variable came up on > March 6th > by Charlie Trevor with replies by myself and Mark Schaffer on March > 6th and > 7th. > > My reply was: > > Is this necessary? > >From Greene (4th ed. page 376): > "...assuming for the moment that only y* is measured with error... > this > result conforms completely to the assumption of the classical > regression > model. As long as the regressor is measured properly, measurement > error on > the dependent variable can be absorbed in the disturbance of the > regression > and ignored." > > Hope this helps, > Scott > > > > * > * For searches and help try: > * http://www.stata.com/support/faqs/res/findit.html > * http://www.stata.com/support/statalist/faq > * http://www.ats.ucla.edu/stat/stata/ > > > * > * For searches and help try: > * http://www.stata.com/support/faqs/res/findit.html > * http://www.stata.com/support/statalist/faq > * http://www.ats.ucla.edu/stat/stata/ > Prof. Mark Schaffer Director, CERT Department of Economics School of Management & Languages Heriot-Watt University, Edinburgh EH14 4AS tel +44-131-451-3494 / fax +44-131-451-3008 email: m.e.schaffer@hw.ac.uk web: http://www.sml.hw.ac.uk/ecomes ________________________________________________________________ DISCLAIMER: This e-mail and any files transmitted with it are confidential and intended solely for the use of the individual or entity to whom it is addressed. If you are not the intended recipient you are prohibited from using any of the information contained in this e-mail. In such a case, please destroy all copies in your possession and notify the sender by reply e-mail. Heriot Watt University does not accept liability or responsibility for changes made to this e-mail after it was sent, or for viruses transmitted through this e-mail. Opinions, comments, conclusions and other information in this e-mail that do not relate to the official business of Heriot Watt University are not endorsed by it. ________________________________________________________________ * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/ * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

**Follow-Ups**:**RE: st: RE: Re: errors in outcome variables regression***From:*Mark Schaffer <M.E.Schaffer@hw.ac.uk>

**References**:**Re: st: RE: Re: errors in outcome variables regression***From:*Mark Schaffer <M.E.Schaffer@hw.ac.uk>

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