Item response theory (IRT) for continuous responses I hope this message finds everyone safe and well. 
I want to estimate the Rasch item difficulty parameters for my test items (dataset below). However, I have two challenges:
(1) Item scores are continuous between 0 and 1 (item_score)
(2) Test is adaptive and thus, items (item_id) vary across persons (person_id)
Is there an R package to take (1) and (2) into account (I highly appreciate a demonstration)?
--Many thanks
#===== Dataset in R
dat <- read.csv('https://raw.githubusercontent.com/izeh/n/master/g.csv', stringsAsFactors = F)

 A: Edit: This answer is not correct. It's not clear to me what OP is looking for, but take a look at the discussion at https://chat.stackexchange.com/rooms/107527/discussion-between-jeremy-miles-and-user7148318 before answering. 
There is no package that can do this (as far as I know).
The lavaan package for sem does not give standard errors for latent variable estimates (and I don't know you can predict a model with new data).
You might be able to do this with merTools::predictInterval(), however I'm having trouble understanding your data. What identifies the question? 
You see to have very large number of questions (~ 16000?) - each of which is not answered by very many people, so your standard errors will be high. Have I understood that correctly?
My guess is that the will look like:
library(dplyr)
library(lme4)
library(merTools)
dat <- read.csv('https://raw.githubusercontent.com/izeh/n/master/g.csv', stringsAsFactors = F)

dat$item_id <- factor(item_id)

dat_no_1 <- dat[dat$person_id != 1, ]
dat_no_1 <- dat[dat$person_id > 1, ]

fit1 <- lme4::lmer(item_score ~ as.factor(item_id) +
                     (1 | person_id),
                   data = dat_no_1)
system.time(
  pred1 <-
    merTools::predictInterval(fit1,
                              newdata = dat_no_1,
                              n.sims = 999)
)

I can't test this though, as I run out of memory.
Note that this ignores the non-normality issue.
A: As noticed in the other answer by Jeremy Miles, IRT models can be thought as a care of random-effect models. This is discussed in greater detail by De Boeck et al:

De Boeck, P., Bakker, M., Zwitser, R., Nivard, M., Hofman, A.,
Tuerlinckx, F., & Partchev, I. (2011). The Estimation of Item Response
Models with the lmer Function from the lme4 Package in R. Journal of
Statistical Software, 39(12), 1–28.
https://doi.org/10.18637/jss.v039.i12

In such a case 1PL model can be estimated in lme4 using the command
lmer(ir ~ -1 + item + (1 | id), data = DataSet, family = "binomial")

Since the items are in the unit interval, you still can use the binomial link as with binary responses. Crossed effects are also supported by mixed-effects models.
