Standard errors for slope and residual variance with measurement error in Y

Question

I am using R to estimate the error in the slope of a regression line. (I will later use the slope to calculate something.) I have some data, call them $x$ and $y$, and will fit a linear regression such as $y = mx+c$. If y has some error associated with it, say $\delta y = 0.1$ for example, what is the error in the slope?

The residual standard error given by the lm() function in R does not take this error in y into account. So how do I take the residual standard error and add the error in y that I have?

Answer 1

Let's consider the simplest case: There is measurement error in $Y$ only. It is normally distributed, centered on the true value of each $y_i$, and independent (of the value of $y_i$, $x_i$, etc.). We'll say the standard deviation of the measurement error is $0.1$. That seems to be the situation you have in mind.

In this case, the variance of the residuals will be inflated relative to the true variance of the errors. (Note that "errors" is the traditional—unfortunate—name of the random part of the data generating process, they are not erroneous in the everyday sense, and they are not the same as the measurement errors.) Specifically, variances add, so the expected variance of the residuals will be the true variance of the errors plus the variance of the measurement error. What R calls the "residual standard error" is (somewhat bizarrely) the standard deviation of the residuals (cf., here), so you need to square that to get the variance. If you knew the variance of the measurement error a-priori, or had some independent estimate of it, you could simply subtract that value from the result. If you'd like, you could take the square root of the difference to get an estimate of the SD of the errors.

On the other hand, the standard error of the slope is based on the variability in your observed data. Since this is what was used to fit the slope, and slopes fit on data with measurement error should bounce around more widely that slopes fit on data measured without measurement error. As a result, I would not advise you to try to remove the variance of the measurement error from the standard errors of your regression parameters. They are correct already.