Converting Rasch units (RITs) to thetas

How do you place Measures of Academic Progress (MAP) Rash unit scores (RITs) on the normal theta metric (and vice versa)? I searched the Web but couldn’t find a straightforward answer. What I’ve gathered from various sources is that RIT scores are just logits multiplied by 10 and centered on 200. From there, logits can be transformed to thetas based on the probabilities of the respective distributions, as shown in the code below.

RIT scores are new to me because I’ve worked mostly with the 3-parameter model until recently. I’ve noticed that a lot psychometricians who use item response theory (IRT) fall into one of two camps: Rasch or 3PL. My personal preference is somewhere in the middle. I think 1) it’s unrealistic to assume items discriminate equally and 2) estimates of a are useful for test construction and adaptive selection. The c parameter, on the other hand, is costly to estimate and not entirely essential. So I guess you could say I’m a dispassionate 2PL guy.

```> #Simulate some thetas (z-scores).
> thetas <- sort(rnorm(20))
> thetas
[1] -1.34485345 -1.08206871 -1.04673807 -0.98990444
[5] -0.96802821 -0.81766629 -0.56671883 -0.18795885
[9] -0.17779446 -0.08314090 -0.04615256  0.13611619
[13]  0.13796440  0.43172163  0.47472683  0.57965345
[17]  0.64315758  0.71571489  1.10205842  1.16067809
>
> #Convert thetas to probabilities then logits.
> p <- pnorm(thetas)
> logits <- log(p / (1 - p))
>
> #Convert logits to RITs.
> RITs <- logits * 10 + 200
> RITs
[1] 176.7823 181.8148 182.4653 183.5001 183.8947 186.5552
[7] 190.8243 196.9958 197.1587 198.6728 199.2634 202.1739
[13] 202.2035 206.9477 207.6533 209.3914 210.4565 211.6874
[19] 218.5559 219.6538
>
> #Place RITs on normal theta metric.
> funk.RITs.thetas <- function(x) {
+   qnorm(plogis((x - 200) / 10))
+ }
> funk.RITs.thetas(RITs)
[1] -1.34485345 -1.08206871 -1.04673807 -0.98990444
[5] -0.96802821 -0.81766629 -0.56671883 -0.18795885
[9] -0.17779446 -0.08314090 -0.04615256  0.13611619
[13]  0.13796440  0.43172163  0.47472683  0.57965345
[17]  0.64315758  0.71571489  1.10205842  1.16067809
```

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