Calculate uncertainty of the slope when dependent variable in a linear regression has substantial error? I have a dataset in which the dependent variable (y) has known and substantial error, and yet the observations happen to line up quite well along a line when plotted against the independent variable (x). Fitting a linear regression seems to substantially overestimate the precision of the slope estimate for y vs. x.
How can one appropriately propagate the known error in y through to the estimate of the slope?
I think there is part of an answer here, but it assumes the point fit a linear regression exactly: Calculate uncertainty of linear regression slope based on data uncertainty
As a reproducible example in R:
# the data
set.seed(5)
dat <- data.frame(x = 0:8, y = seq(0,16, length.out=9)+rnorm(9, 0, 0.5), y.se = 3)

# fit a naive model, not considering error in y
mod <- lm(y ~ x, dat)
summary(mod)
preds <- predict(mod, se.fit = TRUE)

plot(dat$x, dat$y, ylim=c(-7,22))
arrows(dat$x, dat$y-1.96*dat$y.se, dat$x, dat$y+1.96*dat$y.se, length=0)

# plot the confidence interval on the linear regression
polygon(c(dat$x, rev(dat$x)), c(preds$fit+preds$se.fit, rev(preds$fit-preds$se.fit)), col = 'grey')


The slope is estimated very precisely near 2.0:
Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.04670    0.32783   0.142    0.891    
x            1.97546    0.06886  28.689 1.61e-08 ***

Visually, however, a slope as low as 0.7 or as high as 3.3 would still fit through the error bounds of y quite well.

I'm not quite sure what your problem actually is. If there's random error in y but not x, the regression of y on x will be unbiased. The uncertainty due to the error will be properly reflected in the standard error of the parameter estimate.
In your example, the y.se variable is only used to draw the "arrows" in the graph. It's not used to calculate y, so it's not reflected in the lm results. Your y and x are almost perfectly correlated (r>.99), so one can predict the other almost perfectly.
