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Suppose that my application's users are asked to give their opinion about the probability that a statement is true. They are presented with a slider widget that goes from 0% to 100%, but let's say the slider goes from 0 to 1. Due to the discrete nature of computing, there are a finite number of possibilities for their answer. Perhaps it is even preferred to limit the number of ticks on the slider to, say, 101 (including a tick for 0), because it's unlikely that the precision of our estimates for the probability that a statement is true allows for, say, an estimate like 0.9888888834.

Suppose that I want to construct a latent variable representation of this ordinal data, and the latent variable is the probability that a statement is true. Well, standard probit regression would have me assume that the latent variable is normally distributed. But the latent variable can only be approximately normally distributed because the latent variable is a probability of truth, which is constrained to being between 0 and 1. So my question is whether it is worth it to investigate other probability distributions for the latent variable, or if the normal case is good enough for most purposes. And for a more specific question, I wonder if I could represent it with a logit link.

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    $\begingroup$ I think I'd look at what the distribution of the variable. $\endgroup$ – Jeremy Miles Aug 15 '13 at 0:02
  • $\begingroup$ Also, in the model, what's outcome and what's predictor? $\endgroup$ – Jeremy Miles Aug 15 '13 at 0:02
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    $\begingroup$ You might consider beta regression. This is designed for a variable that is distributed between 0 and 1. orfe.princeton.edu/~alaink/NJ_aTaxiOrf467F12/Papers/… $\endgroup$ – Jeremy Miles Aug 15 '13 at 0:15
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    $\begingroup$ Just for clarification. Standard probit (or logit) regression is not suited for your data, because it requires a binary DV. Ordered regression may also not be the best choice, because your DV may have at max 101 levels. So probably beta regression (@JeremyMiles) is a good idea. Note that if you transform your DV, e.g., using a probit transform, it is no longer bounded between 0 and 1, but goes from -Inf to Inf, possibly normally distributed. $\endgroup$ – hplieninger Aug 15 '13 at 12:28
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    $\begingroup$ Actually, fractional logit and probit models exist, e.g. the frm package. The beta regression also handles fractional dependent variables. The DV should clearly be a fraction. That said, I strongly suspect that the response isn't truly fractional - people may chunk their responses at, for example, units of 0.1 or 0.05. I wouldn't throw out a suggestion to treat the response as ordinal, although I agree I'd be inclined to use a fractional model first. $\endgroup$ – Weiwen Ng Jun 11 '19 at 20:51

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