Let's say that I am fitting a linear regression model to predict random variable $Y$ based on random variable $X$, and I know as a fact that the only random variable that detemines the value of $Y$ is $X$. The value of $R^2$ describes how much of the variance in the samples from $Y$ is reduced by my model. The reason $R^2$ is not $1.0$ is two fold:
- The linear model does not fully capture how $X$ affects $Y$
- Measurement error
I want to remove the effect of measurement error from $R^2$, i.e. I like to know how much of the unexplained variance is caused because of the insufficiency of the model's complexity.
Let's say that for each $x$, I have multiple measurements of $y$ in the dataset. Can I use this to estimate the variance caused by measurement error and correct the value of $R^2$ accordingly? How does your answer generalize to deviance, for instance if I use GLM with Poisson distribution instead of the linear regression?