Posts on psychometrics: The Science of Assessment

what is Psychometrics

Psychometrics is the science of educational and psychological assessment, using data to ensure that tests are fair and accurate.  Ever felt like you took a test which was unfair, too hard, didn’t cover the right topics, or was full of questions that were simply confusing or poorly written?  Psychometricians are the people who help organizations fix these things using data science, as well as more advanced topics like how to design an AI algorithm that adapts to each examinee.

Psychometrics is a critical aspect of many fields.  Having accurate information on people is essential to education, human resources, workforce development, corporate training, professional certifications/licensure, medicine, and more.  It scientifically studies how tests are developed, delivered, and scored.  Wondering how it does this?  Let’s dive into the details.

What is Psychometrics?

Psychometrician Qualities

Psychometrics is the study of assessment itself, regardless of what type of test is under consideration. In fact, many psychometricians don’t even work on a particular test, they just work on psychometrics itself, such as new methods of data analysis.  Most professionals don’t care about what the test is measuring, and will often switch to new jobs at completely unrelated topics, such as moving from a K-12 testing company to psychological measurement to an Accountant certification exam.  We often refer to whatever we are measuring simply as “theta” – a term from item response theory.

Psychometrics tackles fundamental questions around assessment, such as how to determine if a test is reliable or if a question is of good quality, as well as much more complex questions like how to ensure that a score today on a university admissions exam means the same thing as it did 10 years ago.

Psychometrics is a branch of data science.  In fact, it’s been around a long time before that term was even a buzzword.  Don’t believe me?  Check out this Coursera course on Data Science, and the first example they give as one of the foundational historical projects in data science is… psychometrics!  (early research on factor analysis of intelligence)

Even though assessment is everywhere and Psychometrics is an essential aspect of assessment, to most people it remains a black box, and professionals are referred to as “psychomagicians” in jest. However, a basic understanding is important for anyone working in the testing industry, especially those developing or selling tests.

Psychometrics is NOT limited to very narrow types of assessment.  Some people use the term interchangeably with concepts like IQ testing, personality assessment, or pre-employment testing.  These are each but tiny parts of the field!  Also, it is not the administration of a test.


Why do we need Psychometrics?

This purpose of tests is providing useful information about people, such as whether to hire them, certify them in a profession, or determine what to teach them next in school.  Better tests mean better decisions.  Why?  The scientific evidence is overwhelming that tests provide better information for decision makers than many other types of information, such as interviews, resumes, or educational attainment.  Thus, tests serve an extremely useful role in our society.

The goal of psychometrics is to provide validity: evidence to support that interpretations of scores from the test are what we intended.  If a certification test is supposed to mean that someone passing it meets the minimum standard to work in a certain job, we need a lot of evidence about that, especially since the test is so high stakes in that case.  Meta-analysis, a key tool in psychometrics, aggregates research findings across studies to provide robust evidence on the reliability and validity of tests. By synthesizing data from multiple studies, meta-analysis strengthens the validity claims of tests, especially crucial in high-stakes certification exams where accuracy and fairness are paramount.


What does Psychometrics do?

Building and maintaining a high-quality test is not easy.  A lot of big issues can arise.  Much of the field revolves around solving major questions about tests: what should they cover, what is a good question, how do we set a good cutscore, how do we make sure that the test predicts job performance or student success, etc.  Many of these questions align with the test development cycle – more on that later.test development cycle job task analysis psychometrics

How do we define what should be covered by the test? (Test Design)

Before writing any items, you need to define very specifically what will be on the test.  If the test is in credentialing or pre-employment, psychometricians typically run a job analysis study to form a quantitative, scientific basis for the test blueprints.  A job analysis is necessary for a certification program to get accredited.  In Education, the test coverage is often defined by the curriculum.

How do we ensure the questions are good quality? (Item Writing)

There is a corpus of scientific literature on how to develop test items that accurately measure whatever you are trying to measure.  A great overview is the book by Haladyna.  This is not just limited to multiple-choice items, although that approach remains popular.  Psychometricians leverage their knowledge of best practices to guide the item authoring and review process in a way that the result is highly defensible test content.  Professional item banking software provides the most efficient way to develop high-quality content and publish multiple test forms, as well as store important historical information like item statistics.

How do we set a defensible cutscore? (Standard Setting)

Test scores are often used to classify candidates into groups, such as pass/fail (Certification/Licensure), hire/non-hire (Pre-Employment), and below-basic/basic/proficient/advanced (Education).  Psychometricians lead studies to determine the cutscores, using methodologies such as Angoff, Beuk, Contrasting-Groups, and Borderline.

How do we analyze results to improve the exam? (Psychometric Analysis)

Psychometricians are essential for this step, as the statistical analyses can be quite complex.  Smaller testing organizations typically utilize classical test theory, which is based on simple mathematics like proportions and correlations.  Large, high-profile organizations typically use item response theory (IRT), which is based on a type of nonlinear regression analysis.  Psychometricians evaluate overall reliability of the test, difficulty and discrimination of each item, distractor analysis, possible bias, multidimensionality, linking multiple test forms/years, and much more.  Software such as  Iteman  and  Xcalibre  is also available for organizations with enough expertise to run statistical analyses internally.  Scroll down below for examples.

How do we compare scores across groups or years? (Equating)

This is referred to as linking and equating.  There are some psychometricians that devote their entire career to this topic.  If you are working on a certification exam, for example, you want to make sure that the passing standard is the same this year as last year.  If you passed 76% last year and this year you passed 25%, not only will the candidates be angry, but there will be much less confidence in the meaning of the credential.

How do we know the test is measuring what it should? (Validity)

Validity is the evidence provided to support score interpretations.  For example, we might interpret scores on a test to reflect knowledge of English, and we need to provide documentation and research supporting this.  There are several ways to provide this evidence.  A straightforward approach is to establish content-related evidence, which includes the test definition, blueprints, and item authoring/review.  In some situations, criterion-related evidence is important, which directly correlates test scores to another variable of interest.  Delivering tests in a secure manner is also essential for validity.


Where is Psychometrics Used?


In certification testing, psychometricians develop the test via a documented chain of evidence following a sequence of research outlined by accreditation bodies, typically: job analysis, test blueprints, item writing and review, cutscore study, and statistical analysis.  Web-based item banking software like  FastTest  is typically useful because the exam committee often consists of experts located across the country or even throughout the world; they can then easily log in from anywhere and collaborate.


In pre-employment testing, validity evidence relies primarily on establishing appropriate content (a test on PHP programming for a PHP programming job) and the correlation of test scores with an important criterion like job performance ratings (shows that the test predicts good job performance).  Adaptive tests are becoming much more common in pre-employment testing because they provide several benefits, the most important of which is cutting test time by 50% – a big deal for large corporations that test a million applicants each year. Adaptive testing is based on item response theory, and requires a specialized psychometrician as well as specially designed software like  FastTest.

K-12 Education

Most assessments in education fall into one of two categories: lower-stakes formative assessment in classrooms, and higher-stakes summative assessments like year-end exams.  Psychometrics is essential for establishing the reliability and validity of higher-stakes exams, and on equating the scores across different years.  They are also important for formative assessments, which are moving towards adaptive formats because of the 50% reduction in test time, meaning that student spend less time testing and more time learning.


Universities typically do not give much thought to psychometrics even though a significant amount of testing occurs in higher education, especially with the move to online learning and MOOCs.  Given that many of the exams are high stakes (consider a certificate exam after completing a year-long graduate program!), psychometricians should be used in the establishment of legally defensible cutscores and in statistical analysis to ensure reliable tests, and professionally designed assessment systems used for developing and delivering tests, especially with enhanced security.


Have you ever taken a survey at your doctor’s office, or before/after a surgery?  Perhaps a depression or anxiety inventory at a psychotherapist?  Psychometricians have worked on these.


The Test Development Cycle

Psychometrics is the core of the test development cycle, which is the process of developing a strong exam.  It is sometimes called similar names like assessment lifecycle.

You will recognize some of the terms from the introduction earlier.  What we are trying to demonstrate here is that those questions are not standalone topics, or something you do once and simply file a report.  An exam is usually a living thing.  Organizations will often be republishing a new version every year or 6 months, which means that much of the cycle is repeated on that timeline.  Not all of it is; for example, many orgs only do a job analysis and standard setting every 5 years.

Consider a certification exam in healthcare.  The profession does not change quickly because things like anatomy never change and medical procedures rarely change (e.g., how to measure blood pressure).  So, every 5 years it does a job analysis of its certificants to see what they are doing and what is important.  This is then converted to test blueprints.  Items are re-mapped if needed, but most likely do not need it because there are probably only minor changes to the blueprints.  Then a new cutscore is set with the modified-Angoff method, and the test is delivered this year.  It is delivered again next year, but equated to this year rather than starting again.  However, the item statistics are still analyzed, which leads to a new cycle of revising items and publishing a new form for next year.


Example of Psychometrics in Action

Here is some output from our Iteman software.  This is deeply analyzing a single question on English vocabulary, to see if the student knows the word alleviate.  About 70% of the students answered correctly, with a very strong point-biserial.  The distractor P values were all in the minority and the distractor point-biserials were negative, which adds evidence to the validity.  The graph shows that the line for the correct answer is going up while the others are going down, which is good.  If you are familiar with item response theory, you’ll notice how the blue line is similar to an item response function.  That is not a coincidence.

FastTest Iteman Psychometrics Analysis

Now, let’s look at another one, which is more interesting.  Here’s a vocab question about the word confectioner.  Note that only 37% of the students get it right… even though there is a 25% chance just of guessing!!!  However, the point-biserial discrimination remains very strong at 0.49.  That means it is a really good item.  It’s just hard, which means it does a great job to differentiate amongst the top students.

Confectioner confetti

Psychometrics looks fun!  How can I join the band?

You will need a graduate degree.  I recommend you look at the NCME website ( with resources for students.  Good luck!

Already have a degree and looking for a job?  Here’s the two sites that I recommend:

NCME – Also has a job listings page that is really good (

Horizon Search – Headhunter for Psychometricians and I/O Psychologists

Test response function 10 items Angoff

Setting a cutscore on a test scored with item response theory (IRT) requires some psychometric knowledge.  This post will get you started.

How do I set a cutscore with item response theory?

There are two approaches: directly with IRT, or using CTT then converting to IRT.

  1. Some standard setting methods work directly with IRT, such as the Bookmark method.  Here, you calibrate your test with IRT, rank the items by difficulty, and have an expert panel place “bookmarks” in the ranked list.  The average IRT difficulty of their bookmarks is then a defensible IRT cutscore.  The Contrasting Groups method and the Hofstee method can also work directly with IRT.
  2. Cutscores set with classical test theory, such as the Angoff, Nedelsky, or Ebel methods, are easy to implement when the test is scored classically.  But if your test is scored with the IRT paradigm, you need to convert your cutscores onto the theta scale.  The easiest way to do that is to reverse-calculate the test response function (TRF) from IRT.

The Test Response Function

The TRF (sometimes called a test characteristic curve) is an important method of characterizing test performance in the IRT paradigm.  The TRF predicts a classical score from an IRT score, as you see below.  Like the item response function and test information function (item response and test information function), it uses the theta scale as the X-axis.  The Y-axis can be either the number-correct metric or proportion-correct metric.

Test response function 10 items Angoff

In this example, you can see that a theta of -0.3 translates to an estimated number-correct score of approximately 7, or 70%.

Classical cutscore to IRT

So how does this help us with the conversion of a classical cutscore?  Well, we hereby have a way of translating any number-correct score or proportion-correct score.  So any classical cutscore can be reverse-calculated to a theta value.  If your Angoff study (or Beuk) recommends a cutscore of 7 out of 10 points (70%), you can convert that to a theta cutscore of -0.3 as above.  If the recommended cutscore was 8 (80%), the theta cutscore would be approximately 0.7.

Because IRT works in a way that it scores examinees on the same scale with any set of items, as long as those items have been part of a linking/equating study.  Therefore, a single study on a set of items can be equated to any other linear test form, LOFT pool, or CAT pool.  This makes it possible to apply the classically-focused Angoff method to IRT-focused programs.  You can even set the cutscore with a subset of your item pool, in a linear sense, with the full intention to apply it on CAT tests later.

Note that the number-correct metric only makes sense for linear or LOFT exams, where every examinee receives the same number of items.  In the case of CAT exams, only the proportion correct metric makes sense.

How do I implement IRT?

Interested in applying IRT to improve your assessments?  Download a free trial copy of  Xcalibre  here.  If you want to deliver online tests that are scored directly with IRT, in real time (including computerized adaptive testing), check out  FastTest.

Equation editor item type

Technology-enhanced items are assessment items (questions) that utilize technology to improve the interaction of a test question in digital assessment, over and above what is possible with paper.  Tech-enhanced items can improve examinee engagement (important with K12 assessment), assess complex concepts with higher fidelity, improve precision/reliability, and enhance face validity/sellability. 

To some extent, the last word is the key one; tech-enhanced items simply look sexier and therefore make an assessment platform easier to sell, even if they don’t actually improve assessment.  I’d argue that there are also technology-enabled items, which are distinct, as discussed below.

What is the goal of technology enhanced items?

The goal is to improve assessment, by increasing things like reliability/precision, validity, and fidelity. However, there are a number of TEIs that is actually designed more for sales purposes than psychometric purposes. So, how to know if TEIs improve assessment?  That, of course, is an empirical question that is best answered with an experiment.  But let me suggest one metric address this question: how far does the item go beyond just reformulating a traditional item format to use current user-interface technology?  I would define the reformulating of traditional format to be a fake TEI while going beyond would define a true TEI.

An alternative nomenclature might be to call the reformulations technology-enhanced items and the true tech usage to be technology-enabled items (Almond et al, 2010; Bryant, 2017), as they would not be possible without technology.

A great example of this is the relationship between a traditional multiple response item and certain types of drag and drop items.  There are a number of different ways that drag and drop items can be created, but for now, let’s use the example of a format that asks the examinee to drag text statements into a box. 

An example of this is K12 assessment items from PARCC that ask the student to read a passage, then ask questions about it.

drag drop sequence

The item is scored with integers from 0 to K where K is the number of correct statements; the integers are often then used to implement the generalized partial credit model for final scoring.  This would be true regardless of whether the item was presented as multiple response vs. drag and drop. The multiple response item, of course, could just as easily be delivered via paper and pencil. Converting it to drag and drop enhances the item with technology, but the interaction of the student with the item, psychometrically, remains the same.

Some True TEIs, or Technology Enabled Items

Of course, the past decade or so has witnessed stronger innovation in item formats. Gamified assessments change how the interaction of person and item is approached, though this is arguably not as relevant for high stakes assessment due to concerns of validity. There are also simulation items. For example, a test for a construction crane operator might provide an interface with crane controls and ask the examinee to complete a tasks. Even at the K-12 level there can be such items, such as the simulation of a science experiment where the student is given various test tubes or other instruments on the screen.

Both of these approaches are extremely powerful but have a major disadvantage: cost. They are typically custom-designed. In the case of the crane operator exam or even the science experiment, you would need to hire software developers to create this simulation. There are now some simulation-development ecosystems that make this process more efficient, but the items still involve custom authoring and custom scoring algorithms.

To address this shortcoming, there is a new generation of self-authored item types that are true TEIs. By “self-authored” I mean that a science teacher would be able to create these items themselves, just like they would a multiple choice item. The amount of technology leveraged is somewhere between a multiple choice item and a custom-designed simulation, providing a compromise of reduced cost but still increasing the engagement for the examinee. A major advantage of this approach is that the items do not need custom scoring algorithms, and instead are typically scored via point integers, which enables the use of polytomous item response theory.

Are we at least moving forward?  Not always!

There is always pushback against technology, and in this topic the counterexample is the gridded item type.  It actually goes in reverse of innovation, because it doesn’t take a traditional format and reformulate it for current UI. It actually ignores the capabilities of current UI (actually, UI for the past 20+ years) and is therefore a step backward. With that item type, students are presented a bubble sheet from a 1960s style paper exam, on a computer screen, and asked to fill in the bubbles by clicking on them rather than using a pencil on paper.

Another example is the EBSR item type from the artist formerly known as PARCC. It was a new item type that intended to assess deeper understanding, but it did not use any tech-enhancement or -enablement, instead asking two traditional questions in a linked manner. As any psychometrician can tell you, this approach ignored basic assumptions of psychometrics, so you can guess the quality of measurement that it put out.

How can I implement TEIs?

It takes very little software development expertise to develop a platform that supports multiple choice items. An item like the graphing one above, though, takes substantial investment. So there are relatively few platforms that can support these, especially with best practices like workflow item review or item response theory. 

modified-Angoff Beuk compromise

A modified-Angoff method study is one of the most common ways to set a defensible cutscore on an exam.  It therefore means that the pass/fail decisions made by the test are more trustworthy than if you picked an arbitrary round number like 70%. If your doctor, lawyer, accountant, or other professional has passed an exam where the cutscore has been set with this method, you can place more trust in their skills.

What is the Angoff method?

The Angoff method is a scientific way of setting a cutscore (pass point) on a test.  If you have a criterion-referenced interpretation, it is not legally defensible to just conveniently pick a round number like 70%; you need a formal process.  There are a number of acceptable methodologies in the psychometric literature for standard-setting studies, also known as cutscores or passing points.  Some examples include Angoff, modified-Angoff, Bookmark, Contrasting Groups, and Borderline.  The modified-Angoff approach is by far the popular approach.  It is used especially frequently for certification, licensure, certificate, and other credentialing exams.

It was originally suggested as a mere footnote by renowned researcher William Angoff, at Educational Testing Service.

How does the Angoff approach work?

First, you gather a group of subject matter experts, and have them define what they consider to be a Minimally Competent Candidate (MCC).  Next, you have them estimate the percent of minimally competent candidates that will answer each item correctly.  You then analyze the results for outliers or inconsistencies, and have the experts discuss then re-rate the items to gain better consensus.  The average final rating is then the expected percent-correct score for a minimally competent candidate.

Advantages of the Angoff method

  1. It is defensible.  Because it is the most commonly used approach and is widely studied in the scientific literature, it is well-accepted.
  2. You can implement it before a test is ever delivered.  Some other methods require you to deliver the test to a large sample first.
  3. It is conceptually simple, easy enough to explain to non-psychometricians.
  4. It incorporates the judgment of a panel of experts, not just one person or a round number.
  5. It works for tests with both classical test theory and item response theory.
  6. It does not take long to implement – if a short test, it can be done in a matter of hours!
  7. It can be used with different item types, including polytomously scored items (multi-points).

Disadvantages of the Angoff method

  1. It does not use actual data, unless you implement the Beuk method alongside.
  2. It can lead to the experts overestimating the performance of entry-level candidates, as they forgot what it was like to start out 20-30 years ago.  This is one reason to use the Beuk method as a “reality check” by showing the experts that if they stay with the cutscore they just picked, the majority of candidates might fail!

FAQ about the Angoff approach

[sc_fs_multi_faq headline-0=”h3″ question-0=”How do I calculate the Angoff cutscore and inter-rater reliability?” answer-0=”Download our Angoff Analysis Tool.” image-0=”” headline-1=”h3″ question-1=”What is the difference between Angoff and modified-Angoff?” answer-1=”The original approach had the experts only say whether they thought an MCC would get it right, not the percentage.” image-1=”” headline-2=”h3″ question-2=”Why do I need to do an Angoff study?” answer-2=”If the test is used to make decisions, like hiring or certification, you are not allowed to pick a round number like 70% with no justification.” image-2=”” headline-3=”h3″ question-3=”What if the experts disagree?” answer-3=”You will need to evaluate inter-rater reliability and agreement, then re-rate the items. More info below.” image-3=”” headline-4=”h3″ question-4=”How many experts do I need?” answer-4=”The bare minimum is 6; 8-10 is better.” image-4=”” headline-5=”h3″ question-5=”Do I need to deliver the test first?” answer-5=”No, that is one advantage of this method – you can set a cutscore before you deliver to any examinees.” image-5=”” count=”6″ html=”true” css_class=””]


Example of the Modified-Angoff Approach

First of all, do not expect a straightforward, easy process that leads to an unassailably correct cutscore.  All standard-setting methods involve some degree of subjectivity.  The goal of the methods is to reduce that subjectivity as much as possible.  Some methods focus on content, others on examinee performance data, while some try to meld the two.

Step 1: Prepare Your Team

The modified-Angoff process depends on a representative sample of subject matter experts (SMEs), usually 6-20. By “representative” I mean they should represent the various stakeholders. For instance, a certification for medical assistants might include experienced medical assistants, nurses, and physicians, from different areas of the country. You must train them about their role and how the process works, so they can understand the end goal and drive toward it.

Step 2: Define The Minimally Competent Candidate (MCC)

This concept is the core of the modified-Angoff method, though it is known by a range of terms or acronyms, including minimally qualified candidates (MQC) or just barely qualified (JBQ).  The reasoning is that we want our exam to separate candidates that are qualified from those that are not.  So we ask the SMEs to define what makes someone qualified (or unqualified!) from a perspective of skills and knowledge. This leads to a conceptual definition of an MCC. We then want to estimate what score this borderline candidate would achieve, which is the goal of the remainder of the study. This step can be conducted in person, or via webinar.

Step 3: Round 1 Ratings

Next, ask your SMEs to read through all the items on your test form and estimate the percentage of MCCs that would answer each correctly.  A rating of 100 means the item is a slam dunk; it is so easy that every MCC would get it right.  A rating of 40 is very difficult.  Most ratings are in the 60-90 range if the items are well-developed. The ratings should be gathered independently; if everyone is in the same room, let them work on their own in silence. This can easily be conducted remotely, though.

Step 4: Discussion

This is where it gets fun.  Identify items where there is the most disagreement (as defined by grouped frequency distributions or standard deviation) and make the SMEs discuss it.  Maybe two SMEs thought it was super easy and gave it a 95 and two other SMEs thought it was super hard and gave it a 45.  They will try to convince the other side of their folly. Chances are that there will be no shortage of opinions and you, as the facilitator, will find your greatest challenge is keeping the meeting on track. This step can be conducted in person, or via webinar.

Step 5: Round 2 Ratings

Raters then re-rate the items based on the discussion.  The goal is that there will be a greater consensus.  In the previous example, it’s not likely that every rater will settle on a 70.  But if your raters all end up from 60-80, that’s OK. How do you know there is enough consensus?  We recommend the inter-rater reliability suggested by Shrout and Fleiss (1979), as well as looking at inter-rater agreement and dispersion of ratings for each item. This use of multiple rounds is known as the Delphi approach; it pertains to all consensus-driven discussions in any field, not just psychometrics.

Step 6: Evaluate Results and Final Recommendation

Evaluate the results from Round 2 as well as Round 1.  An example of this is below.  What is the recommended cutscore, which is the average or sum of the Angoff ratings depending on the scale you prefer?  Did the reliability improve?  Estimate the mean and SD of examinee scores (there are several methods for this). What sort of pass rate do you expect?  Even better, utilize the Beuk Compromise as a “reality check” between the modified-Angoff approach and actual test data.  You should take multiple points of view into account, and the SMEs need to vote on a final recommendation. They, of course, know the material and the candidates so they have the final say.  This means that standard setting is a political process; again, reduce that effect as much as you can.

Some organizations do not set the cutscore at the recommended point, but at one standard error of judgment (SEJ) below the recommended point.  The SEJ is based on the inter-rater reliability; note that it is NOT the standard error of the mean or the standard error of measurement.  Some organizations use the latter; the former is just plain wrong (though I have seen it used by amateurs).


modified angoff

Step 7: Write Up Your Report

Validity refers to evidence gathered to support test score interpretations.  Well, you have lots of relevant evidence here. Document it.  If your test gets challenged, you’ll have all this in place.  On the other hand, if you just picked 70% as your cutscore because it was a nice round number, you could be in trouble.

Additional Topics

In some situations, there are more issues to worry about.  Multiple forms?  You’ll need to equate in some way.  Using item response theory?  You’ll have to convert the cutscore from the modified-Angoff method onto the theta metric using the Test Response Function (TRF).  New credential and no data available? That’s a real chicken-and-egg problem there.

Where Do I Go From Here?

Ready to take the next step and actually apply the modified-Angoff process to improving your exams?  Sign up for a free account in our  FastTest item banker. You can also download our Angoff analysis tool for free.


Shrout, P. E., & Fleiss, J. L. (1979). Intraclass correlations: uses in assessing rater reliability. Psychological bulletin86(2), 420.

test response functions

Item response theory (IRT) is a family of machine learning models in the field of psychometrics, which are used to design, analyze, validate, and score assessments.  It is a very powerful psychometric paradigm that allows researchers to build stronger assessments, whether they work in Education, Psychology, Human Resources, or other fields.  It also solves critical measurement problems like equating across years, designing adaptive tests, or creating vertical scales.

Want to learn more about IRT, how it works, and why it is so important for assessment?  Read on.

What is Item Response Theory?

IRT is a family of models that try to describe how examinees respond to items on a test, hence the name.  These models can be used to evaluate item performance, because the descriptions are quite useful in and of themselves.  However, item response theory ended up doing so much more.Example Item response theory function

IRT is model-driven, in that there is a specific mathematical equation that is assumed, and we fit the models based on raw data, similar to linear regression.  There are different parameters (a, b, c) that shape this equation to different needs.  That’s what defines different IRT models.  This will be discussed at length below.

The models put people and items onto a latent scale, which is usually called θ (theta).  This represents whatever is being measured, whether IQ, anxiety, or knowledge of accounting laws in Croatia.  IRT helps us understand the nature of the scale, how a person answers each question, the distribution of item difficulty, and much more.  IRT used to be known as latent trait theory and item characteristic curve theory.

IRT requires specially-designed software.  Click the link below to download our software Xcalibre, which provides a user-friendly and visual platform for implementing IRT.


IRT analysis with Xcalibre


Why do we need Item Response Theory?

IRT represents an important innovation in the field of psychometrics. While now more than 50 years old – assuming the “birth” is the classic Lord and Novick (1969) text – it is still underutilized and remains a mystery to many practitioners.

Item response theory is more than just a way of analyzing exam data, it is a paradigm to drive the entire lifecycle of designing, building, delivering, scoring, and analyzing assessments.

  • IRT helps us determine if a test is providing accurate scores on people, much more so than classical test theory.
  • IRT helps us provide better feedback to examinees, which has far-reaching benefits for education and workforce development.
  • IRT reduces bias in the instrument, through advanced techniques like differential item functioning.
  • IRT maintains meaningful scores across time, known as equating.
  • IRT can connect multiple levels of content, such as Math curriculum from Grades 3 to 12 if that is what you want to measure, known as vertical scaling.
  • IRT is necessary to implement adaptive testing.
  • IRT is particularly adept at handling a variety of test formats, including both speeded tests and power tests.

IRT requires larger sample sizes and is much more complex than its predecessor, classical test theory, but is also far more powerful.  IRT requires quite a lot of expertise, typically a PhD.  So it is not used for small assessments like a final exam at universities, but is used for almost all major assessments in the world.


The Driver: Problems with Classical Test Theory

Classical test theory (CTT) is approximately 100 years old, and still remains commonly used because it is appropriate for certain situations, and it is simple enough that it can be used by many people without formal training in psychometrics.  Most statistics are limited to means, proportions, and correlations.  However, its simplicity means that it lacks the sophistication to deal with a number of very important measurement problems.  A list of these is presented later.

Learn more about the differences between CTT and IRT here.


Item Response Theory Parameters

The foundation of IRT is a mathematical model defined by item parametersA parameter is an aspect of a mathematical model that can change its shape or other aspects.  For dichotomous items (those scored correct/incorrect), each item has three parameters:


   a: the discrimination parameter, an index of how well the item differentiates low from top examinees; typically ranges from 0 to 2, where higher is better, though not many items are above 1.0.

   b: the difficulty parameter, an index of what level of examinees for which the item is appropriate; typically ranges from -3 to +3, with 0 being an average examinee level.

   c: the pseudo-guessing parameter, which is a lower asymptote; typically is focused on 1/k where k is the number of options.

These paramters are used in the formula below, but are also displayed graphically.

3PL irt equation

Item response function

These parameters are used to graphically display an item response function (IRF), which models the probability of a correct answer as a function of ability.  In the example IRF, the a parameter is approximately, 1.0, indicating a fairly discriminating test item.  The b parameter is approximately 0.0 (the point on the x-axis where the midpoint of the curve is), indicating an average-difficulty item; examinees of average ability would have a 60% chance of answering correctly.  The c parameter is approximately 0.20, like a 5-option multiple choice item.  Consider the x-axis to be z-scores on a standard normal scale.

In some cases, there is no guessing involved, and we only use and b.  This is called the two-parameter model.  If we only use b, this is the one-parameter or Rasch Model.  Here is how that is calculated.


Item parameters, which are crucial within the IRT framework, might change over time or multiple testing occasions, a phenomenon known as item parameter drift.


Example Item Response Theory calculations

Examinees with higher ability are much more likely to respond correctly.  Look at the graph above.  Someone at +2.0 (97th percentile) has about a 94% chance of getting the item correct.  Meanwhile, someone at -2.0 has only a 25% chance – barely above the 1 in 5 guessing rate of 20%.  An average person (0.0) has a 60% chance.  Why 60?  Because we are accounting for guessing.  If the curve went from 0% to 100% probability, then yes, the middle would be 50% change.  But here, we assume 20% as a baseline due to guessing, so halfway up is 60%.

five item response functions

Of course, the parameters can and should differ from item to item, reflecting differences in item performance.  The following graph shows five IRFs with the three-parameter model.  The dark blue line is the easiest item, with a b of -2.00.  The light blue item is the hardest, with a b of +1.80.  The purple one has a c=0.00 while the light blue has c=0.25, indicating that it is more susceptible to guessing.

These IRFs are not just a pretty graph or a way to describe how an item performs.  They are the basic building block to accomplishing those important goals mentioned earlier.  That comes next…


Applications of Item Response Theory to Improve Assessment

Item response theory uses the IRF for several purposes.  Here are a few.

test information function from item response theory

  1. Interpreting and improving item performance
  2. Scoring examinees with maximum likelihood or Bayesian methods
  3. Form assembly, including linear on the fly testing (LOFT) and pre-equating
  4. Calculating the accuracy of examinee scores
  5. Development of computerized adaptive tests (CAT)
  6. Post-equating
  7. Differential item functioning (finding bias)
  8. Data forensics to find cheaters or other issues

In addition to being used to evaluate each item individually, IRFs are combined in various ways to evaluate the overall test or form.  The two most important approaches are the conditional standard error of measurement (CSEM) and the test information function (TIF).  The test information function is higher where the test is providing more measurement information about examinees; if relatively low in a certain range of examinee ability, those examinees are not being measured accurately.  The CSEM is the inverse of the TIF, and has the interpretable advantage of being usable for confidence intervals; a person’s score plus or minus 1.96 times the SEM is a 95% confidence interval for their score.  The graph on the right shows part of the form assembly process in our  FastTest  platform.


Assumptions of Item Response Theory

Item response theory assumes a few things about your data.

  1. The latent trait you are measuring is unidimensional.  If it is multidimensional, there is multidimensional item response theory, or you can treat the dimensions as separate traits.
  2. Items have local independence, which means that the act of answering one is not impacted by others.  This affects the use of testlets and enemy items.
  3. The probability of responding correctly to an item (or in a certain response, in the case of polytomous like Likert), is a function of the examinee’s ability/trait level and the parameters of the model, following the calculation of the item response function, with some allowance for random error.  As a corollary, we are assuming that the ability/trait has some distribution, with some people having higher or lower levels (e.g., intelligence) and that we are trying to find those differences.

Many texts will only postulate the first two as assumptions, because the third is just implicitly assumed.


Advantages and Benefits of Item Response Theory

So why does this matter?  Let’s go back to the problems with classical test theory.  Why is IRT better?

  • Sample-independence of scale: Classical statistics are all sample dependent, and unusable on a different sample; results from IRT are sample-independent. within a linear transformation.  Two samples of different ability levels can be easily converted onto the same scale.
  • Test statistics: Classical statistics are tied to a specific test form.
  • Sparse matrices are OK: Classical test statistics do not work with sparse matrices introduced by multiple forms, linear on the fly testing, or adaptive testing.
  • Linking/equating: Item response theory has much stronger equating, so if your exam has multiple forms, or if you deliver twice per year with a new form, you can have much greater validity in the comparability of scores.
  • Measuring the range of students: Classical tests are built for the average student, and do not measure high or low students very well; conversely, statistics for very difficult or easy items are suspect.
  • Vertical scaling: IRT can do vertical scaling but CTT cannot.
  • Accounting for guessing: CTT does not account for guessing on multiple choice exams.
  • Scoring: Scoring in classical test theory does not take into account item difficulty.  With IRT, you can score a student on any set of items and be sure it is on the same latent scale.
  • Adaptive testing: CTT does not support adaptive testing in most cases.  Adaptive testing has its own list of benefits.
  • Characterization of error: CTT assumes that every examinee has the same amount of error in their score (SEM); IRT recognizes that if the test is all middle-difficulty items, then low or high students will have inaccurate scores.
  • Stronger form building: IRT has functionality to build forms to be more strongly equivalent and meet the purposes of the exam.
  • Nonlinear function: IRT does not assume linear function of the student-item relationship when it is impossible.  CTT assumes a linear function (point-biserial) when it is blatantly impossible.


Item Response Theory Models: One Big Happy Family

Remember: IRT is actually a family of models, making flexible use of the parameters.  In some cases, only two (a,b) or one parameters (b) are used, depending on the type of assessment and fit of the data.  If there are multipoint items, such as Likert rating scales or partial credit items, the models are extended to include additional parameters. Learn more about the partial credit situation here.

Here’s a quick breakdown of the family tree, with the most common models.


How do I analyze my test with Item Response Theory?

OK item fit

First: you need to get special software.  There are some commercial packages like  Xcalibre, or you can use packages inside platforms like R and Python.

The software will analyze the data in cycles or loops to try to find the best model.  This is because, as always, the data does not always perfectly align.  You might see graphs like the one below if you compared actual proportions (red) to the predicted ones from the item response function (black).  That’s OK!  IRT is quite robust.  And there are analyses built in to help you evaluate model fit.

Some more unpacking of the image above:

  • This was item #39 on the test
  • We are using the three parameter logistic model (3PL), as this was a multiple choice item with 4 options
  • 3422 examinees answered the item
  • 76.9 of them got it correct
  • The classical item discrimination (point biserial item-total correlation) was 0.253, which is OK but not very high
  • The a parameter was 0.432, which is OK but not very strong
  • The b parameter was -1.195, which means the item was quite easy
  • The c parameter was 0.248, which you would expect if there was a 25% chance of guessing
  • The Chi-square fit statistic rejected the null, indicating poor fit, but this statistic is susceptible to sample size
  • The z-Resid fit statistic is a bit more robust, and it did not flag the item for bad fit

The image here shows output from  Xcalibre  from the generalized partial credit model, which is a polytomous model often used for items scored with partial credit.  For example, if a question lists 6 animals and asks students to click on the ones that are reptiles, of which there are 3.  The possible scores are then 0, 1, 2, 3.

Here, the graph labels them as 1-2-3-4, but the meaning is the same.  Here is how you can interpret this.

  • Someone is likely to get 0 points if their theta is below -2.0 (bottom 3% or so of students).
  • A few low students might get 1 point (green)
  • Low-middle ability students are likely to get 2 correct (blue)
  • Anyone above average (0.0) is likely to get all 3 correct.

The boundary locations are where one level becomes more likely than another, i.e., where the curves cross.  For example, you can see that the blue and black lines cross at the boundary -0.339.

Where can I learn more?

For more information, we recommend the textbook Item Response Theory for Psychologists by Embretson & Riese (2000) for those interested in a less mathematical treatment, or de Ayala (2009) for a more mathematical treatment.  If you really want to dive in, you can try the 3-volume Handbook of Item Response Theory edited by van der Linden, which contains a chapter discussing ASC’s IRT analysis software,  Xcalibre.

Want to talk to one of our experts about how to apply IRT?  Get in touch!

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laptop data graph

Criterion-related validity is evidence that test scores are related to other data which we expect them to be.  This is an essential part of the larger issue of test score validity, which is providing evidence that test scores have the meaning we intend them to have.  If you’ve ever felt that a test doesn’t cover what it should be covering, or that it doesn’t reflect the skills needed to perform the job you are applying for – that’s validity.

What is criterion-related validity?

Criterion-related validity is an aspect of test score validity which refers to evidence that scores from a test correlate with an external variable that it should correlate with.  In many situations, this is the critical consideration of a test; for example, a university admissions exam would be quite suspect if scores did not correlate well with high school GPA or accurately predict university GPA.  That is literally its purpose for existence, so we want to have some proof that the test is performing that way.  A test serves its purpose, and people have faith in it, when we have such highly relevant evidence.

Incremental validity is a specific aspect of criterion-related validity that assesses the added predictive value of a new assessment or variable beyond the information provided by existing measures.  There are two approaches to establishing criterion-related validity: concurrent and predictive.  There are also two directions: discriminant and convergent.

Concurrent validity

The concurrent approach to criterion-related validity means that we are looking at variables at the same point in time, or at least very close.  In the example of university admissions testing, this would be correlating the test scores with high school GPA.  The students would most likely just be finishing high school at the time they took the test, excluding special cases like students that take a gap year before university.

Predictive validity

The predictive validity approach, as its name suggests, is regarding the prediction of future variables.  In the example of university admissions testing, we would be using test scores to predict university GPA or graduation rates.  A common application of this is pre-employment testing, where job candidates are testing with the goal of predicting positive variables like job performance, or variables that the employer might want to avoid, like counterproductive work behavior.  Which leads us to the next point…

Convergent validity

Convergent validity refers to criterion-related validity where we want a positive correlation, such as test scores with job performance or university GPA.  This is frequently the case with criterion-related validity studies.  One thing to be careful of in this case is differential prediction, also known as predictive bias.  This is where the validity is different for one group of examinees, often a certain demographic group, even though the average score might be the same for each group.

Here is an example of the data you might evaluate for predictive convergent validity of a university admissions test.

Predictive validity

Discriminant validity

Unlike convergent, discriminant validity is where we want to correlate negatively or zero with other variables.  As noted above, some pre-employment tests have this case.  An integrity or conscientiousness assessment should correlate negatively with instances of counterproductive work behavior, perhaps quantified as number if disciplinary marks on employee HR files.  In some cases, the goal might be to find a zero correlation.  That can be the case with noncognitive traits, where a measure of conscientiousness should not have a strong correlation in any direction with other members of the Big Five.

The big picture

Validity is a complex topic with many aspects.  Criterion-related validity is only one part of the picture.  However, as seen in some of the examples above, it is profoundly critical to some types of assessment, especially where the exam exists only to predict some future variables.

Want to delve further into validity?  The classic reference is Cronbach & Meehl (1955).  We also recommend work by Messick, such as this one.  Of course, check with relevant standards to your assessment, such as AERA/APA/NCME or NCCA.


Digital assessment (DA) aka e-Assessment or electronic assessment is the delivery of assessments, tests, surveys, and other measures via digital devices such as computers, tablets, and mobile phones.  The primary goal is to be able to develop items, publish tests, deliver tests, and provide meaningful results – as quickly, easily, and validly as possible.  The use of computers enables many modern benefits, from adaptive testing to tech-enhanced items.  To deliver digital assessment, an organization typically implements cloud-based digital assessment platforms.  Such platforms do much more than just the delivery though, and modules include:

test development cycle fasttest



Why Digital Assessment / e-Assessment?

Globalization and digital technology are rapidly changing the world of education, human resources, and professional development. Teaching and learning are becoming more learner-centric, and technology provides an opportunity for assessment to be integrated into the learning process with corresponding adjustments. Furthermore, digital technology grants opportunities for teaching and learning to move their focus from content to critical thinking. Teachers are already implementing new strategies in classrooms, and assessment needs to reflect these changes, as well.

Looking for such a platform?  Request a free account in ASC’s industry-leading e-Assessment ecosystem.


Free FastTest Account


Advantages of Digital Assessment


One of the main pros of DA is the ease-of-use for staff and learners—examiners can easily set up questionnaires, determine grading methods, and send invitations to examinees. In turn, examinees do not always have to be in a classroom setting to take assessments and can do it remotely in a more comfortable environment. In addition, DA gives learners the option of taking practice tests whenever they are available for that.


DA allows educators quickly evaluate performance of a group against an individual learner for analytical and pedagogical reasons. Report-generating capabilities of DA enable educators to identify learning problem areas on both individual and group levels soon after assessments occur in order to adapt to learners’ needs, strengths, and weaknesses. As for learners, DA provides them with instant feedback, unlike traditional paper exams.


Conducting exams online, especially those at scale, seems very practical since there is no need to print innumerable question papers, involve all school staff in organization of procedures, assign invigilators, invite hundreds of students to spacious classrooms to take tests, and provide them with answer-sheets and supplementary materials. Thus, flexibility of time and venue, lowered human, logistic and administrative costs lend considerable preeminence to electronic assessment over traditional exam settings.


In this digital era, our utmost priority should be minimizing detrimental effects on the environment that pen-and-paper exams bring. Mercilessly cutting down trees for paper can no longer be the norm as it has the adverse environmental impact. DA will ensure that organizations and institutions can go paper-free and avoid printing exam papers and other materials. Furthermore, DAs take up less storage space since all data can be stored on a single server, especially in respect to keeping records in paper.


Enhanced privacy for students is another advantage of digital assessment that validates its utility. There is a tiny probability of malicious activities, such as cheating and other unlawful practices that can potentially rig the system and lead to incorrect results. Secure assessment system supported by AI-based proctoring features makes students embrace test results without contesting them, which, in turn, fosters a more positive mindset toward institutions and organizations building a stronger mutual trust between educators and learners.


The benefits of DA include setting up an automated grading system, more convenient and time-efficient than standard marking and grading procedures, which minimizes human error. Automated scoring juxtaposes examinees’ responses against model answers and makes relevant judgements. The dissemination of technology in e-education and the increasing number of learners demand a sophisticated scoring mechanism that would ease teachers’ burden, save a lot of time, and ensure fairness of assessment results. For example, digital assessment platforms can include complex modules for essay scoring, or easily implement item response theory and computerized adaptive testing.


Those involved in designing, managing and evaluating assessments are aware of the tediousness of these tasks. Probably, the most routine process among assessment procedures is manual invigilation which can be easily avoided by employing proctoring services. Smart exam software, such as FastTest, features the options of automated item generation, item banking, test assembling and publishing, saving precious time that would otherwise be wasted on repetitive tasks. Examiners should only upload the examinees’ emails or ids to invite them for assessment. The best part about it all is instant exporting of results and delivering reports to stakeholders.

Public relations and visibility

There is a considerably lower use of pen and paper in the digital age. The infusion of technology has considerably altered human preferences, so these days an immense majority of educators rely more on computers for communication, presentations, digital designing, and other various tasks. Educators have an opportunity to mix question styles on exams, including graphics, to make them more interactive than paper ones. Many educational institutions utilize learning management systems (LMS) for publishing study materials on the cloud-based platforms and enabling educators to evaluate and grade with ease. In turn, students benefit from such systems as they can submit their assignments remotely.


Challenges of Implementing Digital Assessment

Difficulty in grading long-answer questions

DA copes brilliantly with multiple-choice questions; however, there are still some challenges with grading long-answer questions. This is where Digital e-Assessment intersects with the traditional one as subjective answers ask for manual grading. Luckily, technology in the education sector continues to evolve and even essays can already be marked digitally with a help of AI-features on the platforms like FastTest.

Need to adapt

Implementing something new always brings disruption and demands some time to familiarize all stakeholders with it. Obviously, transition from traditional assessment to DA will require certain investments to upgrade the system, such as professional development of staff and finances. Some staff and students might even resist this tendency and feel isolated without face-to-face interactions. However, this stage is inevitable and will definitely be a step forward for both educators and learners.

Infrastructural barriers & vulnerability

One of the major cons of DA is that technology is not always reliable and some locations cannot provide all examinees with stable access to electricity, internet connection, and other basic system requirements. This is a huge problem in developing nations, and still remains a problem in many areas of well-developed nations. In addition, integrating DA technology might be very costly in case of wrong strategies while planning assessment design, both conceptual and aesthetic. Such barriers hamper DA, which is why authorities should consider addressing them prior to implementing DA.



To sum up, implementing DA has its merits and demerits, as outlined above. Even though technology simplifies and enhances many processes for institutions and stakeholders, it still has some limitations. Nevertheless, all possible drawbacks can be averted by choosing the right methodology and examination software. We cannot reject the necessity to transit from traditional form of assessment to digital one, admitting that the benefits of DA outweigh its drawbacks and costs by far. Of course, it is up to you to choose whether to keep using hard copy assessments or go for online option. However, we believe that in the digital era all we need to do is to plan wisely and choose an easy-to-use and robust examination platform with AI-based anti-cheating measures, such as FastTest, to secure credible outcomes.



Wall, J. E. (2000). Technology-Delivered Assessment: Diamonds or Rocks? ERIC Clearinghouse on Counseling and Student Services.


Leadership assessments are more than just tools; they are crucial to identifying and developing effective organizational leadership. Plenty of options exist for “leadership assessments,” from off-the-shelf tools costing $15 to the incredible bespoke, intense, and, sometimes, invasive assessments that use multiple psychologists and can cost upwards of $50,000. In this blog, I’ll provide a framework for leadership assessments with the right amount of measurement rigor that won’t break the bank or your candidates. I’ll focus on leadership assessments in the selection (i.e., hiring) context, although the information extends to assessments for leadership development.

The Role of Job Analysis in Leadership Assessments

In any selection context, it’s important to begin with a job analysis so you understand what you are trying to measure. In other words, what does “good” look like? We find that using historical information (e.g., job description, ONET data, industry information, etc.) followed by detailed discussions with subject matter experts within an organization appropriately balances efficiency and comprehensiveness in pinpointing the essential skills and tasks necessary for a leader to succeed. 

Designing Robust Leadership Assessments

psychometric training and workshops

Senior female CEO and multicultural business people discussing company presentation at boardroom table. Diverse corporate team working together in modern meeting room office. Top view through glass

From the job analysis, we gather a list of the critical skills along with their importance ratings. With this information, we create a comprehensive leadership assessment incorporating several methodologies to evaluate potential leaders. These include:

  • Construct Validity: Utilizing psychometric tools that are rigorously tested and validated to measure specific leadership constructs effectively. For example, measuring a candidate’s ability to influence may involve assessments like the Hogan Insight Series and the Leadership Effectiveness Analysis, as they have proven their constructs with leadership samples over the past decades.
  • Content Validity: Structured interviews reflect the competencies identified during the job analysis stage. These interviews are crafted to probe deep into the candidate’s experiences and skills in areas critical to the role.
  • Criterion-Related Validity: The real test of an assessment’s value is its ability to predict actual job performance. Correlating assessment outcomes with real job performance metrics confirms the predictive validity of the assessment tools.

At People Strategies, we integrate these approaches to ensure our leadership assessments are not just theoretical but also practical and actionable.

Effective Reporting for Informed Decision-Making

Despite the complex underpinnings, the reporting of assessment results is simplified to aid quick and informed decision-making. Each leadership skill assessed is rated and integrated into a weighted fit score. This score reflects its importance as determined in the job analysis, providing a clear, quantifiable measure of a candidate’s suitability for the leadership role.


Effective leadership assessments are vital for nurturing and selecting the right leaders to meet an organization’s strategic goals. At People Strategies, we emphasize a scientific approach, integrating thorough validation processes to ensure that our assessments are accurate and deeply informative. This rigorous methodology allows us to deliver assessments that are both comprehensive in their analysis and practical in their application, ensuring that organizations can confidently make crucial leadership decisions.

About the Guest Author: David Dubin, PhD

David Dubin, PhD, is founder and principal of People Strategies.  People Strategies knows that a business strategy means nothing without the people. We help busy leaders and HR professionals find and implement the tools they need to bring the benefits of people science to their organizations so they can bring hiring and employee development to the next level.


parcc ebsr items

The Partnership for Assessment of Readiness for College and Careers (PARCC) is a consortium of US States working together to develop educational assessments aligned with the Common Core State Standards.  This is a daunting task, and PARCC is doing an admirable job, especially with their focus on utilizing technology.  However, one of the new item types has a serious psychometric fault that deserves a caveat with regards to scoring and validation.

What is an Evidence-Based Selected-­Response (EBSR) question?

The item type is an “Evidence-Based Selected-­Response” (PARCC EBSR) item format, commonly called a Part A/B item or Two-Part item.  The goal of this format is to delve deeper into student understanding, and award credit for deeper knowledge while minimizing the impact of guessing.  This is obviously an appropriate goal for assessment.  To do so, the item is presented as two parts to the student, where the first part asks a simple question and the second part asks for supporting evidence to their answer in Part A.  Students must answer Part A correctly to receive credit on Part B.  As described on the PARCC website:

In order to receive full credit for this item, students must choose two supporting facts that support the adjective chosen for Part A. Unlike tests in the past, students may not guess on Part A and receive credit; they will only receive credit for the details they’ve chosen to support Part A.

How EBSR items are scored

While this makes sense in theory, it leads to problem in data analysis, especially if using Item Response Theory (IRT). Obviously, this violates the fundamental assumption of IRT: local independence (items are not dependent on each other).  So when working with a client of mine, we decided to combine it into one multi-point question, which matches the theoretical approach PARCC EBSR items are taking.  The goal was to calibrate the item with Muraki’s Generalized Partial Credit Model (GPCM), which is the standard approach used to analyze polytomous items in K12 assessment (learn more here).  The GPCM tries to order students based on the points they earn: 0 point students tend to have the lowest ability, 1 point students of moderate ability, and 2 point students are of the highest ability.  Should be obvious, right?  Nope.

The first thing we noticed was that some point levels had very small sample sizes.  Suppose that Part A is 1 point and Part B is 1 point (select two evidence pieces but must get both).  Most students will get 0 points or 2 points.  Not many will receive 1 point.  We thought about it, and realized that the only way to earn 1 point is to guess Part A but select no correct evidence or only select one evidence point.  This leads to issues with the GPCM.

Using the Generalized Partial Credit Model

Even when there was sufficient N at each level, we found that the GPCM had terrible fit statistics, meaning that the item was not performing according to the model described above.  So I ran  Iteman, our classical analysis software, to obtain quantile plots that approximate the polytomous IRFs without imposing the GPCM modeling.  I found that in the 0-2 point items tend to have the issue where not many students get 1 point, and moreover the line for them is relatively flat.  The GPCM assumes that it is relatively bell-shaped.  So the GPCM is looking for where the drop-offs are in the bell shape, crossing with adjacent CRFs – the thresholds – and they aren’t there.  The GPCM would blow up, usually not even estimating thresholds in correct ordering.


So I tried to think of this from a test development perspective.  How do students get 1 point on these PARCC EBSR items?  The only way to do so is to get Part A right but not Part B.  Given that Part B is the reason for Part A, this means this group is students who answer Part A correctly but don’t know the reason, which means they are guessing.  It is then no surprise that the data for 1-point students is in a flat line – it’s just like the c parameter in the 3PL.  So the GPCM will have an extremely tough time estimating threshold parameters.

Why EBSR items don’t work

From a psychometric perspective, point levels are supposed to represent different levels of ability.  A 1-point student should be higher ability than a 0-point student on this item, and a 2-point student of higher ability than a 1-point student.  This seems obvious and intuitive.  But this item, by definition, violates the idea that a 1-point student should have higher ability than a 0-point student.  The only way to get 1 point is to guess the first part – and therefore not know the answer and are no different than the 0-point examinees whatsoever.  So of course the 1-point results look funky here.

The items were calibrated as two separate dichotomous items rather than one polytomous item, and the statistics turned out much better.  This still violates the IRT assumption but at least produces usable IRT parameters that can score students.  Nevertheless, I think the scoring of these items needs to be revisited so that the algorithm produces data which is able to be calibrated in IRT.

The entire goal of test items is to provide data points used to measure students; if the Evidence-Based Selected-­Response item type is not providing usable data, then it is not worth using, no matter how good it seems in theory!


Scaling is a psychometric term regarding the establishment of a score metric for a test, and it often has two meanings. First, it involves defining the method to operationally scoring the test, establishing an underlying scale on which people are being measured.  A common example is the T-score, which transforms raw scores into a standardized scale with a mean of 50 and a standard deviation of 10, making it easier to compare results across different populations or test forms.  It also refers to score conversions used for reporting scores, especially conversions that are designed to carry specific information.  The latter is typically called scaled scoring.

Examples of Scaling

You have all been exposed to this type of scaling, though you might not have realized it at the time. Most high-stakes tests like the ACT, SAT, GRE, and MCAT are reported on scales that are selected to convey certain information, with the actual numbers selected more or less arbitrarily. The SAT and GRE have historically had a nominal mean of 500 and a standard deviation of 100, while the ACT has a nominal mean of 18 and standard deviation of 6. These are actually the same scale, because they are nothing more than a converted z-score (standard or zed score), simply because no examinee wants to receive a score report that says you got a score of -1. The numbers above were arbitrarily selected, and then the score range bounds were selected based on the fact that 99% of the population is within plus or minus three standard deviations. Hence, the SAT and GRE range from 200 to 800 and the ACT ranges from 0 to 36. This leads to the urban legend of receiving 200 points for writing your name correctly on the SAT; again, it feels better for the examinee. A score of 300 might seem like a big number and 100 points above the minimum, but it just means that someone is in the 3rd percentile.

Now, notice that I said “nominal.” I said that because the tests do not actually have those means observed in samples, because the samples have substantial range restriction. Because these tests are only taken by students serious about proceeding to the next level of education, the actual sample is of higher ability than the population. The lower third or so of high school students usually do not bother with the SAT or ACT. So many states will have an observed average ACT of 21 and standard deviation of 4. This is an important issue to consider in developing any test. Consider just how restricted the population of medical school students is; it is a very select group.

How can I select a score scale?


For various reasons, actual observed scores from tests are often not reported, and only converted scores are reported.  If there are multiple forms which are being equated, scaling will hide the fact that the forms differ in difficulty, and in many cases, differ in cutscore.  Scaled scores can facilitate feedback.  They can also help the organization avoid explanations of IRT scoring, which can be a headache to some.

When deciding on the conversion calculations, there are several important questions to consider.

First, do we want to be able to make fine distinctions among examinees? If so, the range should be sufficiently wide. My personal view is that the scale should be at least as wide as the number of items; otherwise you are voluntarily giving up information. This in turn means you are giving up variance, which makes it more difficult to correlate your scaled scores with other variables, like the MCAT is correlated with success in medical school. This, of course, means that you are hampering future research – unless that research is able to revert back to actual observed scores to make sure all information possible is used. For example, supposed a test with 100 items is reported on a 5-point grade scale of A-B-C-D-F. That scale is quite restricted, and therefore difficult to correlate with other variables in research. But you have the option of reporting the grades to students and still using the original scores (0 to 100) for your research.

Along the same lines, we can swing completely in the other direction. For many tests, the purpose of the test is not to make fine distinctions, but only to broadly categorize examinees. The most common example of this is a mastery test, where the examinee is being assessed on their mastery of a certain subject, and the only possible scores are pass and fail. Licensure and certification examinations are an example. An extension of this is the “proficiency categories” used in K-12 testing, where students are classified into four groups: Below Basic, Basic, Proficient, and Advanced. This is used in the National Assessment of Educational Progress. Again, we see the care taken for reporting of low scores; instead of receiving a classification like “nonmastery” or “fail,” the failures are given the more palatable “Below Basic.”

Another issue to consider, which is very important in some settings but irrelevant in others, is vertical scaling. This refers to the chaining of scales across various tests that are at quite different levels. In education, this might involve linking the scales of exams in 8th grade, 10th grade, and 12th grade (graduation), so that student progress can be accurately tracked over time. Obviously, this is of great use in educational research, such as the medical school process. But for a test to award a certification in a medical specialty, it is not relevant because it is really a one-time deal.

Lastly, there are three calculation options: pure linear (ScaledScore = RawScore * Slope + Intercept), standardized conversion (Old Mean/SD to New Mean/SD), and nonlinear approaches like Equipercentile.

Perhaps the most important issue is whether the scores from the test will be criterion-referenced or norm-referenced. Often, this choice will be made for you because it distinctly represents the purpose of your tests. However, it is quite important and usually misunderstood, so I will discuss this in detail.

Criterion-Referenced vs. Norm-Referenced


This is a distinction between the ways test scores are used or interpreted. A criterion-referenced score interpretation means that the score is interpreted with regards to defined content, blueprint, or curriculum (the criterion), and ignores how other examinees perform (Bond, 1996). A classroom assessment is the most common example; students are scored on the percent of items correct, which is taken to imply the percent of the content they have mastered. Conversely, a norm-referenced score interpretation is one where the score provides information about the examinee’s standing in the population, but no absolute (or ostensibly absolute) information regarding their mastery of content. This is often the case with non-educational measurements like personality or psychopathology. There is no defined content which we can use as a basis for some sort of absolute interpretation. Instead, scores are often either z-scores or some linear function of z-scores.  IQ is historically scaled with a mean of 100 and standard deviation of 15.

It is important to note that this dichotomy is not a characteristic of the test, but of the test score interpretations. This fact is more apparent when you consider that a single test or test score can have several interpretations, some of which are criterion-referenced and some of which are norm-referenced. We will discuss this deeper when we reach the topic of validity, but consider the following example. A high school graduation exam is designed to be a comprehensive summative assessment of a secondary education. It is therefore specifically designed to cover the curriculum used in schools, and scores are interpreted within that criterion-referenced context. Yet scores from this test could also be used for making acceptance decisions at universities, where scores are only interpreted with respect to their percentile (e.g., accept the top 40%). The scores might even do a fairly decent job at this norm-referenced application. However, this is not what they are designed for, and such score interpretations should be made with caution.

Another important note is the definition of “criterion.” Because most tests with criterion-referenced scores are educational and involve a cutscore, a common misunderstanding is that the cutscore is the criterion. It is still the underlying content or curriculum that is the criterion, because we can have this type of score interpretation without a cutscore. Regardless of whether there is a cutscore for pass/fail, a score on a classroom assessment is still interpreted with regards to mastery of the content.  To further add to the confusion, Industrial/Organizational psychology refers to outcome variables as the criterion; for a pre-employment test, the criterion is typically Job Performance at a later time.

This dichotomy also leads to some interesting thoughts about the nature of your construct. If you have a criterion-referenced score, you are assuming that the construct is concrete enough that anybody can make interpretations regarding it, such as mastering a certain percentage of content. This is why non-concrete constructs like personality tend to be only norm-referenced. There is no agreed-upon blueprint of personality.

Multidimensional Scaling

camera lenses for multidimensional item response theory

An advanced topic worth mentioning is multidimensional scaling (see Davison, 1998). The purpose of multidimensional scaling is similar to factor analysis (a later discussion!) in that it is designed to evaluate the underlying structure of constructs and how they are represented in items. This is therefore useful if you are working with constructs that are brand new, so that little is known about them, and you think they might be multidimensional. This is a pretty small percentage of the tests out there in the world; I encountered the topic in my first year of graduate school – only because I was in a Psychological Scaling course – and have not encountered it since.

Summary of test scaling

Scaling is the process of defining the scale that on which your measurements will take place. It raises fundamental questions about the nature of the construct. Fortunately, in many cases we are dealing with a simple construct that has a well-defined content, like an anatomy course for first-year medical students. Because it is so well-defined, we often take criterion-referenced score interpretations at face value. But as constructs become more complex, like job performance of a first-year resident, it becomes harder to define the scale, and we start to deal more in relatives than absolutes. At the other end of the spectrum are completely ephemeral constructs where researchers still can’t agree on the nature of the construct and we are pretty much limited to z-scores. Intelligence is a good example of this.

Some sources attempt to delineate the scaling of people and items or stimuli as separate things, but this is really impossible as they are so confounded. Especially since people define item statistics (the percent of people that get an item correct) and items define people scores (the percent of items a person gets correct). It is for this reason that item response theory, the most advanced paradigm in measurement theory, was designed to place items and people on the same scale. It is also for this reason that item writing should consider how they are going to be scored and therefore lead to person scores. But because we start writing items long before the test is administered, and the nature of the construct is caught up in the scale, the issues presented here need to be addressed at the very beginning of the test development cycle.