Some options for models for this type of data include the following:
- Use beta-regression (response distribution is a beta distribution, which takes values in 0 to 1).
- Transform your data to the real-line e.g. using $x' = \log(x) - \log(1-x)$ as your response variable, then use a standard regression model (predictions & confidence/prediction intervals can be back-transformed using $\exp(x')/(1+\exp(x'))$). In this case you are assuming that the regression residuals after taking account other model covriates follow a normal distribution.
- There's likely other reasonable options. E.g. if you just want to test some hypothesis in a randomized experiment, you could just use rank-based non-parametric methods.
Which of these makes sense will depend on which models fit the data reasonably well (you can of course make such models more complex e.g. by introducing random effects, if e.g. the observed data are overdispersed vs. what your model says), as well as what you want to achieve. If you need to get sensible predictions, taking into account that your data are bounded into (0,1) is important, but for example when modeling binary data (which can only be 0 or 1) some people use the so-called "linear-probability model" that models the response probability using linear relationships and allows predicted probabilities <0 or >1 (that can work okay for some purposes, but not so much as a prediction model).
It will depend a lot on the specifics of what you want to use your model for, how much a mis-specification of a model would matter (and depending on your specific setting you may have more or less tolerance for certain issues with a model).
For example, look at this tweet that models a percentage difference in votes (can only be between -100 to 100%) using linear regression. Clearly, this model is not perfect, in theory (given different data) capable of making absurd predictions of one candidate being ahead once all votes are counted by 110% or something like that, ignores the correlation in the observations and ignores that the underlying data are a finite number of votes (including potentially for a third candidate). However, in this particular case it provided a sort of sensible local linear extrapolation of a trend that was useful. Or to quote the much used "All models are wrong, some models are useful", in this case the model was a bit wrong, but probably quite useful to get an idea for where things were going.
