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!

The first thing I notice is that the relationship between the input and output is approximately linear, so from an empirical point of view I'd expect the fitted model to replicate this relationship.
If you look at the range of your dependent variable, the range is around $0.4$ to $0.6$. The non-linearity in beta regression  typically occurs  is most evident when the outcome is closer to $0$ or $1$. In fact, if we plot the link functions you used, we can see that they are very close to linear for $0.4<\text{dependent}<0.6$, thus the non-linearity induced by the link function is very minor for your model. Although the relationship is approximately linear, I think that a beta regression is still an appropriate model since your dependent appears to be a proportion.
