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!
 A: 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.
A: 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.

A: As the other answers have already noted, your relationship is almost linear, and this is picked up by the beta regression. (More precisely, by the logit link. This is where any "curved" shape would come from, not from the shape of the beta distribution as such. You may be misunderstanding the roles of the logit link function and the distributional assumptions.)
Compare a beta regression model for the built-in ReadingSkills dataset:

We see how the relationship between iq and accuracy is almost linear, as in your data, for participants with dyslexia - but it is noticeably curved for participants without dyslexia.
R code:
library(betareg)
data(ReadingSkills)
iq_predict <- with(ReadingSkills,seq(min(iq),max(iq),by=.01))   # for plotting predictions

with(ReadingSkills,plot(iq,accuracy,pch=19,las=1))
with(subset(ReadingSkills,dyslexia=="yes"),points(iq,accuracy,pch=19,col="red"))
with(subset(ReadingSkills,dyslexia=="no"),points(iq,accuracy,pch=19,col="blue"))
model <- betareg(accuracy~iq*dyslexia,data=ReadingSkills,link="logit")
lines(iq_predict,predict(model,data.frame(iq=iq_predict,dyslexia="no")),col="blue")
lines(iq_predict,predict(model,data.frame(iq=iq_predict,dyslexia="yes")),col="red")
legend("bottomright",lwd=1,pch=19,col=c("blue","red"),legend=c("No dyslexia","dyslexia"))

