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First Draft of the paper - University of Toronto

First Draft of the paper - University of Toronto

sign, in which two

sign, in which two independent measurements are taken of each independentvariable.The test-retest approach solves the identifiability problem for linear modelswith additive measurement error, yet at present, very few existing datasets follow this standard. An exception is provided by the strand of researchin Psychology employing Campbell and Fiske’s (1959) “ multi-traitmulti-method matrix” in which several constructs (for example, personalitytraits) are each measured by several methods on the same subjects. This isa potential source of sample data sets.6.1 Implications for Teaching and ConsultingWe have some suggestions, which we deliberately advance in a forceful wayin the hope of encouraging debate. We do not imagine that we have the rightto tell people what to do, but we do believe we have stumbled across a veryimportant practical problem, and we believe it should be taken seriously.Everyone knows there are problems when ordinary regression is applied todata with measurement error in the independent variables. The simulationresults presented here bring the problems into sharp focus, showing thatthe usual methods fail to protect adequately against Type I error undercircumstances that could easily be encountered in practice. Our conclusionis that traditional regression methods are inappropriate for observationaldata in general, and should no longer be taught or presented to clients in thecustomary way.Let us consider some counter-arguments. The first is that while we knowthat statistical models are never exactly correct, they still can be useful indrawing conclusions from data. Our response is to say that this is true, butnot in the present case. We are not advocating measurement error modelsjust because they are a bit more realistic. The traditional methods have beenshown to be misleading in the worst way. Clear solutions are available, andwe should be telling our students and clients about them.Another counter-argument is that while correcting for potential confoundingvariables with traditional methods is imperfect, it results in less bias thanleaving the variables out of the model altogether (McCallum, 1972; Wickens,1972). In our view, this is akin to saying that an improperly secured car seatis better than no car seat at all. It is true as far as it goes, but it does notjustify complacency.A third consideration, which is very powerful in consulting situations,44

is that the client has data, and likely has expended much time and effortcollecting and coding them. But the independent variables are measuredwith error, and measured only once, so that no reasonable model that includesmeasurent error will be identified. One might hope that the situation couldbe rescued if certain covariance matrices are “known” from the literature, butthis is a fantasy that could be entertained only by a professional statistician.The client will say that everything is different from what has come before;that’s why the study was done in the first place. So, what should one do?The temptation to apply standard methods is hard to resist, becausethe client will be satisfied, no one will ever know, and surely it must bebetter than nothing. Our advice is to resist the temptation, explain that theusual way of “controlling” for a variable that is measured with error doesnot completely work, and that unfortunately, the data have been collectedin a way that makes it impossible to do an appropriate analysis. A referenceto the present article might help, since the simulation results are somethingthat empirical scientists can judge for themselves, without having to rely ontradition or authority.If measurement error is to be incorporated into statistical models, it mustfirst be acknowledged. When asked to take measurement error into accountin their statistical analyses, the reaction of some scientists may be defensive.While a statistician might view measurement error as natural andunavoidable, and a psychologist might readily acknowledge that almost nomeasurement has perfect reliability, we have spoken with some biologists whoseemed to view the issue as a threat to the reputations of their laboratories.There was particular reluctance to quantify imprecision in measurement andto report it in published papers. This perspective may be a significant barrierto adoption of statistical models and data collection protocols that allow formeasurement error. It will help if statisticians cease supporting the statusquo.In teaching, we suggest that the usual linear model should be presentedonly as an aproach to analyzing data from designed experiments, a steppingstone to the treatment of regression with measurement error, or as a tool forbrutal and thoughtless prediction, with an explicit warning that the coefficientsare not to be interpreted. We know from our own experience that thisis not easy, especially because there seem to be almost no textbook treatmentsof measurement error regression at the undergraduate level, and goodsample data sets are difficult to find. We expect this situation to improveover time.45

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