This is the first of a series of powerpoints presented at a CAT/IRT workshop at the University of Brasilia in 2012. It provides an introduction to item response theory (IRT), tying it to classical test theory and describing some of the major IRT models. Learn more at www.assess.com.
This presentation covers the intricacies of the Item Response Theory. I made this presentation to explain the concepts of IRT to my lab research group at the University of Minnesota. I have taken the contents from various sources so apologies for the poor design of the presentation.
Using Item Response Theory to Improve AssessmentNathan Thompson
This is the second of a series of powerpoints presented at a CAT/IRT workshop at the University of Brasilia in 2012. It provides a discussion on how IRT is applied to developing better assessments, including item and test information functions, standard error of measurement, and use of Xcalibre. Learn more at www.assess.com.
This is the third of a series of powerpoints presented at a CAT/IRT workshop at the University of Brasilia in 2012. It provides an introduction to item response theory (IRT), discussing advanced topics like linking & equating, scaling, differential item functioning, polytomous models, and dimensionality. Learn more at www.assess.com.
Introduction to Computerized Adaptive Testing (CAT)Nathan Thompson
These slides are from a short workshop I taught at the 2015 Conference for the International Association for Computerized Adaptive Testing (IACAT, www.iacat.org). Interested in CAT? I'd love to hear from you on LinkedIn, or visit www.assess.com to learn more.
This presentation covers the intricacies of the Item Response Theory. I made this presentation to explain the concepts of IRT to my lab research group at the University of Minnesota. I have taken the contents from various sources so apologies for the poor design of the presentation.
Using Item Response Theory to Improve AssessmentNathan Thompson
This is the second of a series of powerpoints presented at a CAT/IRT workshop at the University of Brasilia in 2012. It provides a discussion on how IRT is applied to developing better assessments, including item and test information functions, standard error of measurement, and use of Xcalibre. Learn more at www.assess.com.
This is the third of a series of powerpoints presented at a CAT/IRT workshop at the University of Brasilia in 2012. It provides an introduction to item response theory (IRT), discussing advanced topics like linking & equating, scaling, differential item functioning, polytomous models, and dimensionality. Learn more at www.assess.com.
Introduction to Computerized Adaptive Testing (CAT)Nathan Thompson
These slides are from a short workshop I taught at the 2015 Conference for the International Association for Computerized Adaptive Testing (IACAT, www.iacat.org). Interested in CAT? I'd love to hear from you on LinkedIn, or visit www.assess.com to learn more.
Topic: What is Reliability and its Types?
Student Name: Kanwal Naz
Class: B.Ed 1.5
Project Name: “Young Teachers' Professional Development (TPD)"
"Project Founder: Prof. Dr. Amjad Ali Arain
Faculty of Education, University of Sindh, Pakistan
It talks about the different types of validity in assessment.
* Face Validity
* Content Validity
* Predictive Validity
* Concurrent Validity
* Construct Validity
Good items are the basic building blocks of any good test or assessment. This presentation covers best practices in developing high-quality items for better psychometrics.
Topic: What is Reliability and its Types?
Student Name: Kanwal Naz
Class: B.Ed 1.5
Project Name: “Young Teachers' Professional Development (TPD)"
"Project Founder: Prof. Dr. Amjad Ali Arain
Faculty of Education, University of Sindh, Pakistan
It talks about the different types of validity in assessment.
* Face Validity
* Content Validity
* Predictive Validity
* Concurrent Validity
* Construct Validity
Good items are the basic building blocks of any good test or assessment. This presentation covers best practices in developing high-quality items for better psychometrics.
A Simple Guide to the Item Response Theory (IRT) and Rasch ModelingOpenThink Labs
This document, which is a practical introduction to Item Response Theory (IRT) and Rasch modeling, is
composed of five parts:
I. Item calibration and ability estimation
II. Item Characteristic Curve in one to three parameter models
III. Item Information Function and Test Information Function
IV. Item-Person Map
V. Misfit
Dedicated Training & Coaching Institute for GMAT, SAT, GRE, TOEFL, IELTS test preparation classes and admission consultant in Dubai, Sharjah, Abu Dhabi and Alain. Get personalized guidance, counseling, information, practice tests from Princeton Review ME for studying abroad.
(MST) Test Construction and Material
(class report(s)/discussion(s))
DISCLAIMER: I do not claim ownership of the photos, videos, templates, and etc used in this slideshow
CREDIT/s: education-portal
Data Science Interview Questions | Data Science Interview Questions And Answe...Simplilearn
This video on Data science interview questions will take you through some of the most popular questions that you face in your Data science interviews. It’s simply impossible to ignore the importance of data and our capacity to analyze, consolidate, and contextualize it. Data scientists are relied upon to fill this need, but there is a serious dearth of qualified candidates worldwide. If you’re moving down the path to be a data scientist, you need to be prepared to impress prospective employers with your knowledge. In addition to explaining why data science is so important, you’ll need to show that you're technically proficient with Big Data concepts, frameworks, and applications. So, here we discuss the list of most popular questions you can expect in an interview and how to frame your answers.
Why learn Data Science?
Data Scientists are being deployed in all kinds of industries, creating a huge demand for skilled professionals. The data scientist is the pinnacle rank in an analytics organization. Glassdoor has ranked data scientist first in the 25 Best Jobs for 2016, and good data scientists are scarce and in great demand. As a data, you will be required to understand the business problem, design the analysis, collect and format the required data, apply algorithms or techniques using the correct tools, and finally make recommendations backed by data.
You can gain in-depth knowledge of Data Science by taking our Data Science with python certification training course. With Simplilearn’s Data Science certification training course, you will prepare for a career as a Data Scientist as you master all the concepts and techniques. Those who complete the course will be able to:
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Install the required Python environment and other auxiliary tools and libraries
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Perform scientific and technical computing using the SciPy package and its sub-packages such as Integrate, Optimize, Statistics, IO and Weave
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Learn more at www.simplilearn.com/big-data-and-analytics/python-for-data-science-training
Top 100+ Google Data Science Interview Questions.pdfDatacademy.ai
Data science interviews can be particularly difficult due to the many proficiencies that you'll have to demonstrate (technical skills, problem solving, communication) and the generally high bar to entry for the industry.we Provide Top 100+ Google Data Science Interview Questions : All You Need to know to Crack it
visit by :-https://www.datacademy.ai/google-data-science-interview-questions/
An Adaptive Evaluation System to Test Student Caliber using Item Response TheoryEditor IJMTER
Computational creativity research has produced many computational systems that are
described as creative [1]. A comprehensive literature survey reveals that although such systems are
labelled as creative, there is a distinct lack of evaluation of the Creativity of creative systems [1].
Nowadays, a number of online testing websites exist but the drawback of these tests is that every
student who gives a particular test will always be given the same set of questions irrespective of their
caliber. Thus, a student with a very high Intelligence Quotient (IQ) may be forced to answer basic
level questions and in the same way weaker students may be asked very challenging questions which
they cannot response. This method of testing results into a wastage of time for the high IQ students
and can be quite frustrating for the weaker students. This would never benefit a teacher to understand
a particular student’s caliber for the subject under Consideration. Each learner has different learning
status and therefore different test items should be used in their evaluation. This paper proposes an
Adaptive Evaluation System developed based on an Item Response Theory and would be created for
mobile end user keeping in mind the flexibility of students to attempt the test from anywhere. This
application would not only dynamically customize questions for students based on the previous
question he/she has answered but also by adjusting the degree of difficulty for test questions
depending on student ability, a teacher can acquire a valid & reliable measurement of student’s
competency.
In a world of data explosion, the rate of data generation and consumption is on the increasing side, there comes the buzzword - Big Data.
Big Data is the concept of fast-moving, large-volume data in varying dimensions (sources) and
highly unpredicted sources.
The 4Vs of Big Data
● Volume - Scale of Data
● Velocity - Analysis of Streaming Data
● Variety - Different forms of Data
● Veracity - Uncertainty of Data
With increasing data availability, the new trend in the industry demands not just data collection,
but making ample sense of acquired data - thereby, the concept of Data Analytics.
Taking it a step further to further make a futuristic prediction and realistic inferences - the concept
of Machine Learning.
A blend of both gives a robust analysis of data for the past, now and the future.
There is a thin line between data analytics and Machine learning which becomes very obvious
when you dig deep.
Introduction to unidimensional item response modelSumit Das
Item response theory has become an important technique in the field of psychology and education. This slides gives a brief introduction to unidimensional item response models.
Similar to Introduction to Item Response Theory (20)
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Biological screening of herbal drugs: Introduction and Need for
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Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
Introduction to Item Response Theory
1. Day 1 AM: An Introduction to
Item Response Theory
Nathan A. Thompson
Vice President, Assessment Systems Corporation
Adjunct faculty, University of Cincinnati
nthompson@assess.com
2. Welcome!
Thank you for attending!
Introductions and important info now
Software… download or USB
Please ask questions
◦ Also, slow me down or ask for translation!
Goal: provide an intro on IRT/CAT to
those who are new
◦ For those with some experience, to
provide new viewpoints and more
resources/recommendations
3. Where I’m from, professionally
PhD, University of Minnesota
◦ CAT for classifications
Test development manager for
ophthalmology certifications
Psychometrician at Prometric (many
certifications)
VP at ASC
7. Introduce yourselves
Name
Employer/organization
Types of tests you do and/or why you
are interested in IRT/CAT
(There might be someone with similar
interests here)
8. Another announcement
Newly formed: International
Association for Computerized Adaptive
Testing (IACAT)
◦ www.iacat.org
◦ Free membership
◦ Growing resources
◦ Next conference: August 2012, Sydney
9. Welcome!
This workshop is on two highly related
topics: IRT and CAT
IRT is the modern paradigm for
developing, analyzing, scoring, and
linking tests
CAT is a next-generation method of
delivering tests
CAT is not feasible without IRT, so we
do IRT first
10. IRT – where are we going?
IRT, as many of you know, provides a
way of analyzing items
However, it has drawbacks (no
distractor analysis), so the main
reasons to use IRT are at the test level
It solves certain issues with classical
test theory (CTT)
But the two should always be used
together
11. IRT – where are we going?
Advantages
◦ Better error characterization
◦ More precise scores
◦ Better linking
◦ Model-based
◦ Items and people on same scale (CAT)
◦ Sample-independence
◦ Powerful test assembly
12. IRT – where are we going?
Keyword: paradigm or approach
◦ Not just another statistical analysis
◦ It is a different way of thinking about how
tests should work, and how we can
approach specific problems (scaling,
equating, test assembly) from that
viewpoint
13. Day 1
There will be four parts this morning,
covering the theory behind IRT:
◦ Rationale: A graphical introduction to IRT
◦ Models (dichotomous and polytomous) and
their response functions
◦ IRT scoring (θ estimation)
◦ Item parameter estimation and model fit
15. What is IRT?
Basic Assumptions
1. Unidimensionality
A unidimensional latent trait (1 at a time)
Item responses are independent of each
other (local independence), except for the
trait/ability that they measure
2. A specific form of the relationship
between trait level and probability of a
response
The response function, or IRT model
There are a growing number of models
16. What is IRT?
A theory of mathematical functions that
model the responses of examinees to
test items/questions
These functions are item response
functions (IRFs)
Historically, it has also been known as
latent trait theory and item
characteristic curve theory
The IRFs are best described by showing
how the concept is derived from classical
analysis…
17. Classical item statistics
CTT statistics are typically calculated
for each option
Option N Prop Rpbis Mean
1 307 0.860 0.221 91.876
2 25 0.070 -0.142 85.600
3 14 0.039 -0.137 83.929
4 11 0.031 -0.081 86.273
18. Classical item statistics
The proportions are often translated
to a figure like this, where examinees
are split into groups
19. Classical item statistics
The general idea of IRT is to split the
previous graph up into more groups,
and then find a mathematical model
for the blue line
This is what makes the item response
function (IRF)
21. The item response function
Reflects the probability of a given
response as a function of the latent
trait (z-score)
Example:
22. The IRF
For dichotomously scored items, it is
the probability of a correct or keyed
response
Also called Item Characteristic Curve (ICC) or
Trace Line
Only one curve (correct response), and all
other responses are grouped as (1-IRF)
For polytomous items (partial credit,
etc.), it is the probability of each
response
23. The IRF
How do we know exactly what the
IRF for an item is?
We estimate parameters for an
equation that draws the curve
For dichotomous IRT, there are three
relevant parameters: a, b, and c
24. The IRF
a: The discrimination parameter;
represents how well the item
differentiates examinees; slope of the
curve at its center
b: The difficulty parameter; represents
how easy or hard the item is with
respect to examinees; location of the
curve (left to right)
c: The pseudoguessing parameter;
represents the ‘base probability’ of
answering the question; lower asymptote
26. The IRF…
is the “basic building block” of IRT
will differ from item to item
can be one of several different
models (now)
can be used to evaluate items (now)
is used for IRT scoring (next)
leads to “information” used for test
design (after that)
is the basis of CAT (tomorrow)
28. IRT models
Several families of models
◦ Dichotomous
◦ Polytomous
◦ Multidimensional
◦ Facets (scenarios vs raters)
◦ Mixed (additional parameters)
◦ Cognitive diagnostic
◦ We will focus on first two
29. Dichotomous IRT models
There are 3 main models in use, as
mentioned earlier: 1PL, 2PL, 3PL
The “L” refers to “logistic”: which is
the type of equation
IRT was originally developed decades
ago with a cumulative normal curve
This means that calculus needed to
be used
30. The logistic function
An approximation was developed: the
logistic curve
No calculus needed
There are two formats based on D
If D = 1.702, then diff < 0.01
If D = 1.0, a little more difference;
called the true logistic form
Does not really matter, as long as you
are consistent
32. Item parameters
We add parameters to slightly modify
the shape to get it to match our data
For example, a 4-option multiple
choice item has a 25% chance of
being guessed correctly
So we add a c parameter as a lower
asymptote, which means that the
curve is “squished” so it never goes
below 0.25 (next)
34. Item parameters
We can also add a parameter (a) that
modifies the slope
And a b parameter that slides the
entire curve left or right
◦ Tells us which person z-score for which the
item is appropriate
Items can be evaluated based on these
just like with CTT statistics
A little more next…
35. Item parameters: a
The a parameter ranges from 0.0 to
about 2.0 in practice (theoretically to
infinity)
Higher means better discriminating
For achievement testing, 0.7 or 0.8 is
good, aptitude testing is higher
Helps you: Remove items with a<0.4?
Identify a>1.0 as great items?
36. Item parameters: b
For what person z-score is the item
appropriate? (non-Rasch)
Should be between -3 and 3
◦ 99.9% of students are in that range
0.0 is average person
1.0 is difficult (85th percentile)
-1.0 is easy (15th percentile)
2.0 is super difficult (98%)
-2.0 is super easy (2%)
37. Item parameters: b
If item difficulties are normally
distributed, where does this fall?
(Rasch)
0.0 is average item (NOT PERSON)
38. Item parameters: c
The c parameter should be about
1/k, where k is the number of options
If higher, this indicates that options
are not attractive
For example, suppose c = 0.5
This means there is a 50/50 chance
That implies that even the lowest
students are able to ignore two
options and guess between the other
two options
40. The (3PL) logistic function
Here is the equation for the 3PL, so you
can see where the parameters are
inserted
Item i, person j
Equivalent formulations can be seen in
the literature, like moving the (1-c)
above the line
( )
1
( 1| ) (1 )
1 i j ii i j i i Da b
P X c c
e
41. The (3PL) logistic function
ai is the item discrimination
parameter for item i,
bi is the item difficulty or location
parameter for item i,
ci is the lower asymptote, or
pseudoguessing parameter for item i,
D is the scaling constant equal to
1.702 or 1.0.
42. The (3PL) logistic function
The P is due primarily to (-b)
The effect due to a and c is not as
strong
That is, your probability of getting
the item correct is mostly due to
whether it is easy/difficult for you
◦ This leads to the idea of adaptive testing
43. 3PL
IRT has 3 dichotomous models
I’ll now go through the models with
more detail, from 3PL down to 1PL
The 3PL is appropriate for knowledge
or ability testing, where guessing is
relevant
Each item will have an a, b, and c
parameter
44. IRT models
Three 3PL IRFs, c = 0, 0.1, 0.2,
(b = -1, 0, 1; a = 1, 1, 1)
-3 -2 -1 0 1 2 3
0.00.20.40.60.81.0
theta
probability
45. 2PL
The 2PL assumes that there is no
guessing (c = 0.0)
Items can still differ in discrimination
This is appropriate for attitude or
psychological type data with
dichotomous responses
◦ I like recess time at school (T/F)
◦ My favorite subject is math (T/F)
46. IRT models
Three 2PL IRFs, a = 0.75, 1.5, 0.3,
b = -1.0, 0.0, 1.0
-3 -2 -1 0 1 2 3
0.00.20.40.60.81.0
theta
probability
47. 1PL
The 1PL assumes that all items are of
equal discrimination
Items only differ in terms of difficulty
The raw score is now a sufficient
statistic for the IRT score
Not the case with 2PL or 3PL; it’s not
just how many items you get right,
but which ones
10 hard items vs. 10 easy items
48. 1PL
The 1PL is also appropriate for
attitude or psychological type data,
but where there is no reason to
believe items differ substantially in
terms of discrimination
This is rarely the case
Still used: see Rasch discussion later
49. 1PL
Three 1PL IRFs: b = -1, 0, 1
-3 -2 -1 0 1 2 3
0.00.20.40.60.81.0
theta
probability
50. How to choose?
Characteristics of the items
Check with the data! (fit)
Sample size:
◦ 1PL = 100 minimum
◦ 2PL = 300 minimum
◦ 3PL = 500 minimum
Score report considerations
(sufficient statistics)
51. The Rasch Perspective
Another argument in choice
There is a group of psychometricians
(mostly from Australia and Chicago)
who believe that the 1PL is THE model
Everything else is just noise
Data should be “cleaned” to reflect
this
52. The Rasch Perspective
How to clean? A big target is to
eliminate guessing
But how do you know?
Slumdog Millionaire Effect
53. The Rasch Perspective
This group is very strong in their
belief
Why? They believe it is “objective”
measurement
Score scale centered on items, not
people, so “person-free”
Software and journals devoted just to
the Rasch idea
54. The Rasch Perspective
Should you use it?
I was trained to never use Rasch
◦ Equal discrimination assumption is
completely unrealistic… we all know
some items are better than others
◦ We all know guessing should not be
ignored
◦ Data should probably not be doctored
◦ Instead, data should drive the model
55. The Rasch Perspective
However, while some researchers
hate the Rasch model, I don’t
◦ It is very simple
◦ It works better with tiny samples
◦ It is easier to describe
◦ Score reports and sufficient statistics
◦ Discussion points from you?
◦ Nevertheless, I recommend IRT
56. Polytomous models
Polytomous models are for items that
are not scored correct/incorrect,
yes/no, etc.
Two types:
◦ Rating scale or Likert: “Rate on a scale of
1 to 5”
◦ Partial credit – very useful in
constructed-response educational items
My experience as a scorer
57. Polytomous models
Partial credit example with rubric:
◦ Open response question to “2+3(4+5)=“
0: no answer
1: 2, 3, 4, or 5 (picks one)
2: 14 (adds all)
3: 45 (does (2+3) x (4+5) )
4: 27 (everything but add 2)
5: 29 (correct)
61. Scoring
First: throw out your idea of a
“score” as the number of items
correct
We actually want something more
accurate: the precise z-score
Because the z-scores axis is called θ
in IRT, the scoring is called θ
estimation
62. Scoring
IRT utilizes the IRFs in scoring
examinees
If an examinee gets a question right,
they “get” the item’s IRF
If they get the question wrong, they
“get” the (1-IRF)
These curves are multiplied for all
items to get a final curve called the
likelihood function
66. Scoring - MLE
The score is the point on the x-axis
where the highest likelihood is
This is the maximum likelihood
estimate
In the example, 0.0 (average ability)
This obtains precise estimates on the
scale
67. Maximum likelihood
The LF is technically defined as:
Where u is a response vector of 1s
and 0s
Note what this does to the exponents
ij i j
n
u 1 u
j ij ij
i 1
L P Qu
%
68. Scoring - SEM
A quantification of just how precise
can also be calculated, called the
standard error of measurement
This is assumed to be the same for
everyone in classical test theory, but
in IRT depends on the items and the
responses, and the level of
69. Scoring - SEM
Here’s a new LF – blue has the same
MLE but is less spread out
Both are two items, blue with a = 2
70. Scoring - SEM
The first LF had an SEM ~ 1.0
The second LF had an SEM ~ 0.5
We have more certainty about the
second person’s score
This shows how much high-quality
items aid in measurement
◦ Same items and responses, except a
higher a
71. Scoring - SEM
SEM is usually used to stop CATs
General interpretation: confidence
interval
Plus or minus 1.96 (about 2) is 95%
So if the SEM in the example is 0.5,
we are 95% sure that the student’s
true ability is somewhere between
-1.0 and +1.0
72. Scoring - SEM
If a student gives aberrant responses
(cheating, not paying attention, etc.)
they will have a larger SEM
This is not enough to accuse of
cheating (they could have just dozed
off), but it can provide useful
information for research
73. Scoring - SEM
SEM CI is also used to make decisions
◦ Pass if 2 SEMs above a cutoff
74. Details on IRT Scores
Student scores are on the scale,
which is analogous to the standard
normal z scale – same interpretations!
There are four methods of scoring
◦ Maximum Likelihood (MLE)
◦ Bayesian Modal (or MAP, for maximum a
posteriori)
◦ Bayesian EAP (expectation a posteriori)
◦ Weighted MLE (less common)
77. Bayesian modal
Addresses that problem by always
multiplying the LF by a bell-shaped
curve, which forces it to have a
maximum somewhere
Still find the highest point
78. Bayesian EAP
Argues that the curve is not
symmetrical, and we should not
ignore everything except the
maximum
So it takes the “average” of the
curve by splitting it into many slices
and finding the weighted average
The slices are called quadrature
points or nodes
81. Bayesian
Why Bayesian?
◦ Nonmixed response vectors
◦ Asymmetric LF
Why not Bayesian?
◦ Biased inward – if you find the
estimates of 1000 students, the SD would
be smaller with the Bayesian estimates,
maybe 0.95
82. Newton-Raphson
Most IRT software actually uses a
somewhat different approach to MLE
and Bayesian Modal
The straightforward way is to
calculate the value of the LF at each
point in , within reason
For example, -4 to 4 at 0.001
That’s 8,000 calculations! Too much
for 1970s computers…
83. Newton-Raphson
Newton-Raphson is a shortcut method
that searches the curve iteratively
for its maximum
Why? Same 0.001 level of accuracy in
only 5 to 20 iterations
Across thousands of students, that is
a huge amount of calculations saved
But certain issues (local maxima or
minima)… maybe time to abandon?
86. The estimation problem
Estimating student given a set of
known item parameters is easy
because we have something
established
But what about the first time a test is
given?
All items are new, and there are no
established student scores
87. The estimation problem
Which came first, the chicken or the
egg?
Since we don’t know, we go back and
forth, trying one and then the other
◦ Fix “temporary” z-scores
◦ Estimate item parameters
◦ Fix the new item parameters
◦ Estimate scores
◦ Do it again until we’re satisfied
88. Calibration algorithms
There are two calibration algorithms
◦ Joint maximum likelihood (JML) – older
◦ Marginal maximum likelihood (MML) –
newer, and works better with smaller
samples… the standard
◦ Also conditional maximum likelihood, but
it only works with 1PL, so rarer
◦ New in research, but not in standard
software: Markov chain monte carlo
89. Calibration algorithms
The term maximum likelihood is used
here because we are maximizing the
likelihood of the entire data set, for
all items i and persons j
X is the data set of responses xij
b is the set of item parameters bi
is the set of examinee js
90. Calibration algorithms
This means we want to find the b and
that make that number the largest
So we set , find a good b, use it to
score students and find a new , find
a better b, etc…
◦ Marginal ML uses marginal distributions
not exact points, hence it being faster
and working better with smaller samples
of people/items
91. Calibration algorithms
Note: rather than examine the LF
(which gets incredibly small),
software examines -2*ln(LF)
IRT software tracks these iterations
because they provide information on
model fit
See output
93. Checking fit
One assumption of IRT (#2) is that our
data even follows the idea of IRT!
This is true at both the item and the
test level
Also true about examinees: they
should be getting items wrong that are
above their θ and getting items
correct that are below their θ
94. Model-data fit
Whenever fitting any mathematical
model to empirical data (not just IRT),
it is important to assess fit
Fit refers to whether the model
adequately represents the data
Alternatively, if the data is far away
from the model
95. Model-data fit
There are two types of fit important
in IRT
◦ Item (and test) - compares observed data
to the IRF
◦ Person – evaluates whether individual
students are responding according to the
model
Easy items correct, hard items incorrect
99. Model-data fit
Note that if we drew an IRF in each
of those graphs, it would be about
the same
But it is obviously less appropriate in
Graph #3 (“even worse”)
Fit analyses provide a way of
quantifying this
100. Item fit
Most basic approach is to subtract
observed frequency correct from the
expected value for each slice (g) of
This is then summarized in a chi-
square statistic
Bigger = bad fit
103. Item fit
The slices are called quadrature points
Also used for item parameter
estimation
The number of slices for chi-square
need not be the same as for
estimation, but it helps interpretation
104. Item fit
Chi-square is oversensitive to sample
size
A better way is to compute
standardized residuals
Divide a chi-square by its df = G-m
where m is the number of item
parameters
This is more interpretable because of
the well-known scale
0 is OK, examine items > 2
105. Item fit
For broad analysis of fit, use quantile
plots (Xcalibre, Iteman, or Lertap)
◦ 3 to 7 groups
◦ Can find hidden issues (My example:
social desirability in Likert #2)
See Xcalibre output
◦ Fit statistics
◦ Fit graphs (many more groups, and IRF)
106. Person fit
Is an examinee responding oddly?
Most basic measure: take the log of
the LF at the max ( estimate)
A higher number means we are more
sure of the estimate
But this is dependent on the level of
, so we need it standardized: lz
n
1i
u1
i
u
io
ii
ˆQˆPlnl
107. Person fit
lz is like a z-score for fit: z = (x-μ)/s
Less than -2 means bad fit
n
1i
2
i
i
iio
n
1i
iiiio
o
oo
z
ˆP1
ˆP
lnˆP1ˆPlVar
ˆP1lnˆP1ˆPlnˆPlE
lVar
lEl
l
108. Person fit
lz is sensitive to the distribution of
item difficulties
Works best when there is a range of
difficulty
That is, if there are no items for
high-ability examinees, none of them
will have a good estimate!
Best to evaluate groups, not
individuals
109. How is fit useful?
Throw out items?
Throw out people?
Change model used?
Bad fit can flag other possible issues
◦ Speededness: fit (and N) gets worse at
end of test
◦ Multidimensionality: certain areas
110. How is fit useful?
Note that this fits in with the
estimation process
IRT calibration is not “one-click”
Review results, then make
adjustments
◦ Remove items/people
◦ Modify par distributions
◦ Modify quadrature points
◦ Etc.
111. Summary
That was a basic intro to the
rationale of IRT
Now start talking about some
applications and uses
Also examine IRT software and output