Item Response Theory

What’s wrong with the old approach? Classical test theory –Sample dependent –Parallel test form issue Comparing examinee scores Reliability –No predictability –“Error” is the same for everybody

So, what is IRT? A family of mathematical models that describe the interaction between examinees and test items Examinee performance can be predicted in terms of the underlying trait Provides a means for estimating scores for people and characteristics of items Common framework for describing people and items

Some Terminology “Ability” –We use this as a generic term used to describe the “thing” that we are trying to measure –The “thing” can be any old “thing” and we need not concern ourselves with labeling the “thing”, but examples of the “thing” include: Reading ability Math performance Depression

The ogive Natural occurring form that describes something about people Used throughout science, engineering, and the social sciences Also, used in architecture, carpentry, photograph, art, and so forth

The ogive

The Item Characteristic Curve (ICC) This function really does everything: –Scales items & people onto a common metric –Helps in standard setting –Foundation of equating –Some meaning in terms of student ability

The ICC Any line in a Cartesian system can be defined by a formula The simplest formula for the ogive is the logistic function:

The ICC Where b  is the item parameter, and  is the person parameter The equation represents the probability of responding correctly to item i given the ability of person j.

b is the inflection point Item i b i =0.125

We can now use the item parameter to calculate p Let’s assume we have a student with  =1.0, and we have our b  = Then we can simply plug in the numbers into our formula

Using the item parameters to calculate p p =  i =1.00

Wait a minute What do you mean a student with an ability of 1.0?? Does an ability of 0.0 mean that a student has NO ability? What if my student has a reading ability estimate of -1.2?

The ability scale Ability is on an arbitrary scale that just so happens to be centered around 0.0 We use arbitrary scales all the time: –Fahrenheit –Celsius –Decibels –DJIA

Scaled Scores Although ability estimates are centered around zero – reported scores are not However, scaled scores are typically a linear transformation of ability estimates Example of a linear transformation: –(Ability x Slope) + Intercept

The need for scaled scores ½ the kids will have negative ability estimates

The Two Scales of Measurement Reporting Scale (Scaled Scores) –Student/parent level report –School/district report –Cross year comparisons –Performance level categorization The Psychometric Scale (  ) –IRT item and person parameters –Equating –Standard setting

Unfortunately, life can get a lot worse Items vary from one another in a variety of ways: –Difficulty –Discrimination –Guessing –Item type (MC vs. CR)

Items can vary in terms of difficulty Ability of a student Easier item Harder item

Items can vary in terms of discrimination Discrimination is reflected by the “pitch” in the ICC Thus, we allow the ICCs to vary in terms of their slope

Good item discrimination 2 close ability levels Noticeable difference in p

Poor item discrimination smaller difference Same 2 ability levels

Guessing This item is asymptotically approaching 0.25

Constructed Response Items

Items and people Interact in a variety of ways We can use IRT to show that there exists a nice little s-shaped curve that shows this interaction As ability increases – the probability of a correct response increases

Advantages of IRT Because of the stochastic nature of IRT there are many statistical principles we can take advantage of A test is a sum of its parts

The test characteristic curve A test is made up of many items The TCC can be used to summarize across all of our items The TCC is simply the summation of ICCs along our ability continuum For any ability level we can use the TCC to estimate the overall test score for an examinee

Several ICCs are on a test

The test characteristic curve

From an observed test score (i.e., a student’s total test score) we can estimate ability The TCC is used in standard setting to establish performance levels The TCC can also be used to equate tests from one year to the next The test characteristic curve

Estimating Ability Total score = 3 Ability≈0.175

Psychometric “Information” The amount that an item contributes to estimating ability Items that are close to a person’s ability provide more information than items that are far away An item is most informative around the point of inflection

Item Information Item is most informative here because this is where we can discriminate among nearby  values

Item Information Item is much less informative at points along  where there is little slope in the ICC

Test Information Test information is the sum of item information Tests are also most “informative” where the slope of the TCC is the greatest Information (like everything else in IRT) is a function of ability Test information really is test “precision”

Let’s start with a TCC

Information Functions We can evaluate information at a given cutpoint BP/P

Information and CTT CTT has reliability and of course the famous  coefficient IRT has the test information function –Test quality can be evaluated conditionally along the performance continuum In IRT information is, conveniently, reciprocally related to standard error

Standard Error as a function of ability  SE = 0.25

Standard Error of Ability Total score = 3 Ability≈0.175

Standard Error of Ability Total score = 3 Ability≈0.175  Confident region of ability estimate

Item Response Theory A vast kingdom of equations, and dizzying array of complex concepts Ultimately, we use IRT to explain the interaction between students and test items The cornerstone to IRT is the ICC which depicts that as ability increases the chances of getting an item correct increases

Item Response Theory Everything in IRT can be studied conditionally along the performance continuum The CTT concept of reliability is what we call test information, and we can think of this as being a function of test precision SE is related to information and can also be studied along 

The Utility of Item Response Theory Can be used to estimate characteristics of items and people Can be used in the test development process to maximize information (minimize SE) at critical points along  Can even be used for test administration purposes