knitr::opts_chunk$set(echo = TRUE, eval = TRUE, message = FALSE, warning = FALSE)

The Rasch model (i.e., a fully descriptive model) can be estimated using `eirm`

. The following example shows how to estimate Rasch item parameters for the verbal aggression data set (see `?VerbAgg`

for further details). A preview of the `VerbAgg`

data set is shown below:

library("eirm")

data("VerbAgg") head(VerbAgg)

To estimate the Rasch model, a regression-like formula must be defined: `formula = "r2 ~ -1 + item + (1|id)"`

. In the formula,

`r2`

is the variable for dichotomous item responses`-1`

removes the intercept from the model and yields parameter estimates for all items in the data set. With`1`

(instead of`-1`

), an intercept representing the parameter of the first item and relative parameters for the remaining items (i.e., distance from the parameter of the first item) would be estimated.`item`

is the variable representing item IDs in the data set`(1|id)`

refers to the random effects for persons represented by the`id`

column in the data set.

The output for the Rasch model is shown below:

mod1 <- eirm(formula = "r2 ~ -1 + item + (1|id)", data = VerbAgg) print(mod1)

By default, the `eirm`

function returns the **easiness** parameters because the function uses a regression model parameterization where positive parameters indicate positive association with the dependent variable. In order to print the difficulty parameters (instead of easiness), `print(mod1, difficulty = TRUE)`

must be used:

print(mod1, difficulty = TRUE)

The `mod1`

object is essentially a `glmerMod`

-class object from the `lme4`

package (Bates, Maechler, Bolker, & Walker (2015)). All `glmerMod`

results for the estimated model can seen with `mod1$model`

. For example, estimated random effects for persons (i.e., theta estimates) can be obtained using:

theta <- ranef(mod1$model)$id head(theta)

The following example shows how to use item-related and person-related explanatory variables to explain dichotomous responses in the verbal aggression data set.

mod2a <- eirm(formula = "r2 ~ -1 + situ + btype + mode + (1|id)", data = VerbAgg) print(mod2a)

It is possible to visualize the parameters using an item-person map using `plot(mod2a)`

, which returns the following plot. Note that this plot is a modified version of the `plotPImap`

function from the `eRm`

package).

```
plot(mod2a)
```

Aesthetic elements such as axis labels and plot title can be added to the plot. For example, the following code updates the x-axis label and the main plot title (see `?plot.eirm`

for further details).

plot(mod2a, difficulty = TRUE, main = "Verbal Aggression Example", latdim = "Verbal Aggression")

This plot wpuld show the difficulty parameters (instead of easiness), change the main title above the plot, and change the x-axis -- the name for the latent trait being measured.

Also, it is possible to compare nested explanatory models with each other. The following example shows the estimation of a more compact version of `mod2a`

with one less variable and compares the two models (i.e., `mod2a`

vs. `mod2b`

).

mod2b <- eirm(formula = "r2 ~ -1 + situ + btype + (1|id)", data = VerbAgg) anova(mod2a$model, mod2b$model)

It is also possible to use the `eirm`

function with polytomous item responses as well. Because the function only accepts dichotomous responses (i.e., binomial distribution), polytomous data must be reformatted first. To reformat the data, the `polyreformat`

function can be used. The following example demonstrates how polytomous responses (no, perhaps, and yes) in the verbal aggression data set can be reformatted into a dichotomous form:

VerbAgg2 <- polyreformat(data=VerbAgg, id.var = "id", long.format = FALSE, var.name = "item", val.name = "resp") head(VerbAgg2)

In the reformatted data, `polyresponse`

is the new dependent variable (i.e., pseudo-dichotomous version of the original response variable `resp`

) and `polycategory`

represents the response categories. Based on the reformatted data, each item has two rows (number of response categories - 1) based on the following rules (Stanke & Bulut (2019)) for further details on this parameterization):

- If
`polycategory`

= "cat_perhaps" and`resp`

= "no", then`polyresponse`

= 0 - If
`polycategory`

= "cat_perhaps" and`resp`

= "perhaps", then`polyresponse`

= 1 - If
`polycategory`

= "cat_perhaps" and`resp`

= "yes", then`polyresponse`

= NA

and

- If
`polycategory`

= "cat_yes" and`resp`

= "no", then`polyresponse`

= NA - If
`polycategory`

= "cat_yes" and`resp`

= "perhaps", then`polyresponse`

= 0 - If
`polycategory`

= "cat_yes" and`resp`

= "yes", then`polyresponse`

= 1

**NOTE:** Although `polyreformat`

is capable of reshaping wide-format data into long-format and reformat the long data for the analysis with `eirm`

, a safer option is to transform the data from wide to long format before using `polyreformat`

. The `melt`

function from the `reshape2`

package can easily transform wide data to long data.

Several polytomous models can be estimated using the reformatted data:

**Model 1:** It explains only the first threshold (i.e., threshold from no to perhaps) based on explanatory variables:

mod3 <- eirm(formula = "polyresponse ~ -1 + situ + btype + mode + (1|id)", data = VerbAgg2)

**Model 2:** It explains the first threshold (i.e., threshold from no to perhaps) and second threshold (perhaps to yes) based on explanatory variables:

mod4 <- eirm(formula = "polyresponse ~ -1 + btype + situ + mode + polycategory + polycategory:btype + (1|id)", data = VerbAgg2)

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