Why does my beta regression look linear when plotted? Im using a beta regression to model since my dependent variable is confined between [0,1]. However, when I plot the relationship between x and y via my beta regression, the effect looks linear. When doing a linear scatterplot, the lines are almost identical. This is unlike any other beta regressions that I've seen so far, and I wonder what I might be doing wrong.
My code looks like this:
library(betareg)

vb=betareg(dep ~ pctDFNB, data=v, link="loglog")
vb2=betareg(dep ~ pctDFNB, data=v, link="logit")

# PLOT
ggplot(v, aes(x = pctDFNB, y = dep)) +
  geom_point(size = 4, aes(fill = party), shape = 21) +
  scale_fill_grey() +
  geom_line(aes(y = predict(vb, v),
                colour = "log-log", linetype = "log-log")) +
  geom_line(aes(y = predict(vb2, v), 
                colour = "logit", linetype = "logit")) +
  scale_colour_manual("", values = c("red", "blue")) +
  scale_linetype_manual("", values = c("solid", "dashed")) +
  theme_bw()

And the plot I receive looks like this:

For information, my beta distribution looks like this: 
EDIT: Thanks for your answers, they are truly insightful and appreciated!

When the outcomes are very close to 0.5, the model on the response scale is nearly linear. The plot below demonstrates this.  In black, I have plotted the inverse of the logit function (which is what your model would look like were you to extrapolate).  in red is a linear approximation centred around 0.
See how the red and black line are on top of one another approximately between y=0.4 and y=0.6?  The logit function is very nearly linear in that neighbourhood, which is where your data lay.  Try extrapolating the predictions out farther.  You will see the model bottom out at 0 and top out at 1 when your predictor is large enough.

