$$Pr(Y_{ij} = 1 | a_i, b_i, c_i, d_i, \theta_j) = c_i + (d_i - c_i)\frac{exp^{(\alpha(\theta_j-\beta_i) ) }}{1 + exp^{(\alpha(\theta_j-\beta_i) ) }}$$

• $\alpha$ is the "discrimination" parameter
• $\beta$ is the "difficulty" parameter
• $c$ is the "pseudoguessing" parameter
• $d$ is the upper asymptote or highest probability of a correct response
• $\theta$ is the "ability" parameter

• Do to time contraints, we'll only talk about one of the ways to estimate 1PL models.
• More specifically, we'll be talking about fitting a Rasch model using the Joint Maximum Likelihood Estimator (JMLE) to the data.
• However, for those interested, if you view the slides and click your down arrow, there are some brief explanations of other IRT models that are appropriate for other contexts.

$$Pr(Y_{ij} = k | \theta_j) = \frac{exp^{ (\Sigma_{t=1}^k\alpha(\theta_j-\beta_{it}) ) }}{1 + \Sigma_{s = 1}^K exp^{ (\Sigma_{s=1}^s\alpha(\theta_j-\beta_{it}) ) } } $$

• The $\alpha$ & $\theta$ parameters have the same meeting as the had from the other models.
• The $\beta$ parameter is the difficulty associated with the $t^{th}$ response option on the $i^{th}$ item
• The difference is that here we are predicting the probability of the respondent selecting the $k^{th}$ option from the response set if they have an ability of $\theta_j$
• $\theta$ is also assumed to be $N(0, 1)$

$$Pr(Y_{ij} = k | \alpha, \beta_i, \theta_j) = \frac{exp^{ (\Sigma_{t=1}^k\alpha(\theta_j-\beta_{it}) ) }}{1 + \Sigma_{s = 1}^K exp^{ (\Sigma_{s=1}^s\alpha(\theta_j-\beta_{it}) ) } } $$

• There are some subtle but important differences between the Rating Scale and Partial Credit Models
• The distance between the category difficulties are also constrained to be equal (e.g., the difference between scoring a 3 or 4 is the same as the difference in scoring a 2 or 3)

$$Pr(Y_{ij} \geq k | \theta_j) = \frac{exp^{ (\Sigma_{t=1}^k\alpha_i(\theta_j-\beta_{ik}) ) }}{1 + \Sigma_{s = 1}^K exp^{ (\Sigma_{s=1}^s\alpha_i(\theta_j-\beta_{ik}) ) } } $$

• Here the interpretation of the $\beta$ parameter changes to indicate the difficulty of endorsing category $k$ or higher for the $i^{th}$ item
• Additionally, unlike the PCM, the item discriminations (i.e., the $\alpha$) parameters are freely estimated

$$Pr(Y_{ij} = k | \theta_j) = \frac{exp^{ (\alpha_{ik}(\theta_j-\beta_{ik}) ) }}{\Sigma_{h = 1}^K exp^{ (\alpha_{ih}(\theta_j-\beta_{ih}) ) } } $$

• You can think of this as the categorical analog to the GRM.
• These models would be used in cases where the response choice has no inherent value that could be ordered (e.g., what is your favorite ice cream flavor?)

• Java-based application for Psychometric analysis of data
• Freely available (e.g., does not cost you anything to use it and you can even modify it if you so desire)
• Some functionality has been integrated into other software platforms like Stata (the raschjmle program)
• The source code for all of the math and user interface is publicly available:

All the data used for these examples, and the source code used to simulate it is publicly available.

To get the files, go to https://github.com/wbuchanan/kaacSlideDeck/tree/gh-pages

The file itemResponses.csv contains the .CSV file used for the examples.

The file simulateItemResponses.R contains the source code used to generate the simulated data in the .CSV above.

• Griggs v Duke Power Co (1971) restricts the use of assessments for hiring to only those which are "demonstrably a reasonable measure of job performance".
• Watson v Fort Worth Bank & Trust (1988) sets a clear need to have statistical evidence for use of assessment in hiring/firing decisions

• Bad Measurement = Bad Decisions = Bad Outcomes
• We should enable the adults working with children to make the best decisions based on the best possible data.
• Bad Measurement + Correct Decision = Bad Decisions = Bad Outcomes
• The quality of your measurement is empirical, not opinion or feeling.
• Just because you think you are measuring something doesn't mean you are measuring it.