As I stated in the comments above, missing data can be handled by either the ltm or mirt package when the data is MCAR. Here is an example of how to use both on a dataset with missing values:
> library(ltm)
> library(mirt
> set.seed(1234)
> dat <- expand.table(LSAT7)
> dat[sample(1:(nrow(dat)*ncol(dat)), 150)] <- NA
> head(dat)
Item.1 Item.2 Item.3 Item.4 Item.5
[1,] 0 0 0 0 0
[2,] 0 0 0 0 0
[3,] 0 0 0 0 0
[4,] 0 0 0 0 0
[5,] 0 0 0 0 0
[6,] 0 0 0 0 NA
> (ltmmod <- ltm(dat ~ z1))
Call:
ltm(formula = dat ~ z1)
Coefficients:
Dffclt Dscrmn
Item.1 -1.891 0.967
Item.2 -0.720 1.147
Item.3 -1.008 1.885
Item.4 -0.671 0.760
Item.5 -2.554 0.729
Log.Lik: -2572.402
> (mirtmod <- mirt(dat, 1))
Iteration: 22, Log-Lik: -2572.402, Max-Change: 0.00010
Call:
mirt(data = dat, model = 1)
Full-information item factor analysis with 1 factors
Converged in 22 iterations with 41 quadrature.
Log-likelihood = -2572.402
AIC = 5164.805; AICc = 5165.027
BIC = 5213.882; SABIC = 5182.122
> coef(mirtmod)
$Item.1
a1 d g u
par 0.967 1.829 0 1
$Item.2
a1 d g u
par 1.148 0.826 0 1
$Item.3
a1 d g u
par 1.886 1.902 0 1
$Item.4
a1 d g u
par 0.76 0.51 0 1
$Item.5
a1 d g u
par 0.729 1.863 0 1
$GroupPars
MEAN_1 COV_11
par 0 1
It's also possible to impute missing values given a good estimate of $\theta$ for obtaining things like model and item fit statistics (should do this several times if the amount of missingness is non-trivial, and it's even better to jitter the $\hat{\theta}$ values as a function of the respective $SE_{\hat{\theta}}$ values for more reasonable imputation results).
> Theta <- fscores(mirtmod, full.scores = TRUE, scores.only = TRUE)
> fulldat <- imputeMissing(mirtmod, Theta)
> head(fulldat)
Item.1 Item.2 Item.3 Item.4 Item.5
1 0 0 0 0 0
2 0 0 0 0 0
3 0 0 0 0 0
4 0 0 0 0 0
5 0 0 0 0 0
6 0 0 0 0 0
I think the difference primarily is a philosophical one when choosing Rasch/1PL models (the emphases on what measurement means is slightly different in that literature, and hence researchers try their best to obtain these special items), and an empirical/design one when deciding between using 2PL and 3PL models.
Since the slopes are all equal in 1PL models determining a persons location amounts to finding the optimal location where respondents have a P = 0.5 chance of answering correctly by simply choosing items with the best intercepts to get an estimate of $\theta$, whereas in 2- and 3PL models it's slightly more complicated due to the unequal slopes and lower bound parameters for guessing. As a consequence, 2-3PL models often require more advanced adaptive item selection procedures such as the Kullback–Leibler/Fisher information to select the next best item for honing in on $\theta$.
Speaking purely from a design perspective if the adaptive testing items contain a finite number of responses then the 3PL seems like the better option, but if it's more of a fill in the blank style answer (e.g., 2 + 3 = __.) then the 1PL and 2PL models would, at least theoretically, be more reasonable.
Best Answer
Yes, there are several: eRm, mirt, ltm etc. In fact, most IRT models can handle dichotomous items as a special case of polytomous items. E.g. in ltm, you can use grm on polytomous and dichotomous items:
We see the Science data are polytomous (4-levels). Let's first fit the graded response model to the original items
Now I will dichotomize the first item, Comfort, like so
We see now that the first item has levels "strongly disagree" and "disagree" combined into "negative" and "strongly agree" and "agree" combined into "positive".
So, you see, this is possible for all packages that support polytomous items. See sections 2.1. to 2.4. in this review on IRT packages in R.