In ppanko/irtPlot: General IRT Plotting

polyPlot is a flexible utility used to create publication-quality graphics for various aspects of polytomous IRT models. This function also provides several useful utilities to facilitate graphic creation. The general flow of tasks within the function is as follows:

• Use the ltm package to estimate a model based on the given data.
• Calculate outcome (e.g., probability, information) using the estimated parameters and supplied theta values.
• Plot the resultant outcome against the theta values using ggplot2 for each item.
• Press Enter to toggle between plots.
• Save the plots if desired.

Getting Started

Essential

polyPlot is part of the irtPlot package available on github. To download the package, install the devtools package and run the following lines:

library(devtools)
install_github("ppanko/irtPlot")

Before using the polyPlot function, the user must provide two primary arguments. First, the user must declare a data set for which he or she wishes to create graphics. Second, a sequence of theta values should be selected in order to plot the parameters. In the example below, the data are drawn from ltm package and the theta values are chosen accordingly using the seq function.

library(irtPlot)
data <- Science[,c("Future", "Benefit", "Work")]
theta <- seq(-4,4, 0.01)

Next, the user must choose which model and type to create. The models that this function currently supports are Samejima's Graded Response Model ("grm") as well as the Partial Credit Model ("pcm"). The supported plot types are Cumulitive Response Probability ("crp"), Item Characteristic Curves ("icc"), Item Charactersitc Function ("icf"). See relevant sections below for more information on types.

Note: to reduce clutter, polyPlot creates a separate graphic for each item. Subsequent to running the polyPlot function, a new device will be opened to display the plots. After the user presses Enter or clicks on the screen using the left mouse button, the first plot will be printed to the device along with its corresponding plot number printed in the terminal.

In the current example, polyPlot will first print the "Future" item along with a [[1]] in the terminal. After the user continues to the next plot either by clicking or pressing Enter, the "Benefit" item will be displayed in the device with a corresponding [[2]]. Lastly, after a final click, the "Work" item will be shown with a [[3]] in the terminal.

Options

Having dispensed with the required arguments, the user is free to select a number of optional commands to personalize the desired plot. Most of the the arguments in this section are used to tweak default setting for the save utility.

• ddir: A character string specifying the directory in which to save plots. Default is the current working directory.

• save: A logical value specifying whether or not to save the resultant graphics. Requires TRUE/FALSE; default is FALSE.

• title: A character string to specify the name of the plot. Default behavior is to specify the plot type and the column name of each item in the model.

• filename: A vector of character strings indicating the desired filename for each item to save. The default filenames are based on the model, type and the column name of each item.

• dpi: An integer value specifying the resolution of the saved graphic. Default value is 300; recommended value for standard poster-size figures is 600.

• width: A numeric value for the desired width of the figure in inches. Default is 8.5 inches.

• height: A numeric value for the desired height of the figure in inches. Default is 10 inches.

• itmNam: A vector of character strings specifying the name of the items displayed in the legend. Default values are the names of the columns in the specified data set.

• silent: A logical value specifying whether or not to print graphics to screen. This feature is primarily used when the user simply wants to save a plot without examining it. Default is FALSE.

The user is encouraged to view the help files using the ?polyPlot function.

I. Cumulitive Response Probabilities

The first plot type available in the polyPlot function is used to calculate the probability of a response being greater than a given threshold. For a graded response model, the probability of observing a response greater than x for item i can be stated as shown below:

$$P(Y_i > x|(\theta_s) = \frac{exp(1.7a_i(\theta_s - b_{ij}))}{1 + exp(1.7a_i(\theta_s - b_{ij}))}$$

For the partial credit model, the cumulative probabilities are calculated based on the category response probabilities. The probability of observing a response greater than 1 for item i can be expressed as:

$$P(Y_{i} > 1|\theta_s) = \frac{P_{i2}}{P_{i1} + P_{i2}}$$

The graphics below represent the "crp" type plots for the "Future", "Benefit" and "Work" items relative to their respective model: first "grm" then "pcm".

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Graded Response Model

polyPlot(data, theta, model = "grm", type = "crp")

Partial Credit Model

polyPlot(data, theta, model = "pcm", type = "crp")

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II. Item Characteristic Curves

This plot type represents the probability of responding in each of the categories for a given $\theta_s$. To calculate the probability of each response, the probability of responding in a certain threshold must be subtracted from the probability of the adjacent category as shown below:

$$P(Y_{i}= 1|\theta_s) = P_{i1}(\theta_s) - P_{i2}(\theta_s)$$

The formula differs for the first and last categories (in this case 0 and n, respectively) as can be seen in the following:

$$P(Y_{i}= 0|\theta_s) = 1 - P_{i1}(\theta_s)$$ $$P(Y_{i}= n|\theta_s) = P_{in}(\theta_s) - 0$$

The calculation for an Item Characteristic Curve differs for the Partial Credit Model. In this case, the probability of answering in a given category x is given by:

$$P(Y_{i} = x|\theta_s) = \frac{exp[\sum_{j=0}^x (\theta_s-\beta_{ij})]}{\sum_{r=0}^{m_i} [exp \sum_{j=0}^x (\theta_s-\beta_{ij})]}$$

Examples of each "icc" plot type are shown below for each of the three "Science" items.

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Graded Response Model

polyPlot(data, theta, model = "grm", type = "icc")

Partial Credit Model

polyPlot(data, theta, model = "pcm", type = "icc")

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III. Item Characteristic Function

A useful feature of Item Characteristic Curves for polytomous items is that they can be combined to form a measure of the expected score at each level of $\theta_s$. The expected score on item j is expressed below:

$$E(X_i|\theta_s) = \sum_{x=0}^{m_j} xP_{ix} (\theta_s)$$

The expected scores for the three "Science" items are shown below for each item separately.

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Graded Response Model

polyPlot(data, theta, model = "grm", type = "icf")

Partial Credit Model

polyPlot(data, theta, model = "pcm", type = "icf")

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References

Embertson, S.E., & Reise, S. P. (2000). Item Response Theory for Psychologists. Hillsdale, NJ: Erlbaum.

Lee, J. (2016). Lectures on Item Response Theory. Personal Collection of J. Lee, Texas Tech University, Lubbock, TX.

Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149-174.

Rizopoulos D. (2006). ltm: An R package for Latent Variable Modelling and Item Response Theory Analyses. Journal of Statistical Software, 17(5), 1-25. URL http://www.jstatsoft.org/v17/i05/

Samejima, F. (1969). Estimation of Latent Ability Using a Response Pattern of Graded Scores (Psychometric Monograph No. 17). Richmond, VA: Psychometric Society. Retrieved from http://www.psychometrika.org/journal/online/MN17.pdf

Wickham, H. (2009). ggplot2: Elegant Graphics for Data Analysis. New York: Springer-Verlag.

See Also

• difPlot
• irtPlot

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Please leave comments at the project repository: https://www.github.com/ppanko/irtPlot

ppanko/irtPlot documentation built on May 25, 2019, 11:24 a.m.