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For example, can the results of the t test on y1 and y2 be interpreted in the usual way (i.e., like the results of the t test on x1 and x2)? If not, how should I go about testing whether or not y1 and y2 are drawn from the same population?

#Error-free measurements follow a N(10,1)
#I don't know these values
x1 <- rnorm(100,10,1)
x2 <- rnorm(100,10,1)

#Do a t test on the values I don't have
t.test(x1,x2)

#Add 10% relative measurement error
#These are the values I know, and I also know
#that they have a relative uncertainty of 10% 
y1 <- x1 + rnorm(100,0,0.1*x1)
y2 <- x2 + rnorm(100,0,0.1*x2)

#Do the t test on the measurements with error
t.test(y1,y2)
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  • $\begingroup$ Don't all data have errors? $\endgroup$ – Stumpy Joe Pete Mar 6 '13 at 19:12
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    $\begingroup$ There is "error" caused by natural variability in the population of the thing we are trying to measure and "error" caused by our inability to precisely measure the thing that we are trying to measure. $\endgroup$ – Tom Mar 6 '13 at 20:06
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    $\begingroup$ If you have a model $y = \mu + \xi +\eta$ where $\xi$ is a zero-mean error term representing underlying variation about the population mean and $\eta$ is a zero-mean error term representing measurement error, what's the difference from a model where $y = \mu + \epsilon$ where $\epsilon$ is the variation in the observations around the population mean? (i.e. where $\epsilon = \xi +\eta$) $\endgroup$ – Glen_b Mar 6 '13 at 21:45
  • $\begingroup$ I see no difference. So, it sounds like the answer to my question is yes, the two t tests are interpreted in the same way. Further, it appears that my knowledge of eta is of no value in the test, which might be why the textbooks don't differentiate between the two types of error. $\endgroup$ – Tom Mar 7 '13 at 10:53
  • $\begingroup$ This would be problematic if the two errors were not independent. This often happens in actual data. $\endgroup$ – Peter Flom Mar 7 '13 at 12:42

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