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# IRT for Dummies

A Crash Course in Item Response Theory

by

Tweet## Vignesh Palanimuthu

on 5 August 2010#### Transcript of IRT for Dummies

IRT for Dummies A Crash Course in Item Response Theory Estimating a Latent Trait This is the field of Psychometry

Present a series of items in the domain of interest to the person

Based on his responses, estimate his ability level in that domain Ingredients for IRT A person’s ability in a subject can be modeled by a number called the ability level

- e.g. A person with an ability level of 3 is said to have a higher

proficiency in the subject than one with a level of -0.2 Ingredients for IRT Q: How do you relate a person’s ability level to her response to an item?

A: Through an item response function that maps the ability level to the probability of answering the item correctly.

The Item Response Function Characterized by the item parameters a, b, c.

0 ≤ P(θ) ≤ 1.

Determining the Item Parameters a, b, c Ask some examinees to answer the item and record their responses

Determining a Person’s Ability θ Give a set of calibrated items to the person and record his responses

Find the probability of getting this response pattern and set θ to the value that maximizes this probability

Note: The same method can be used to calibrate items and estimate abilities simultaneously. Item Information Function This quantifies the information provided by the item at different ability levels.

The curve here provides max. info at θ ≈ 0

Test Information Function 'Sum of item information functions of the items in the test'.

Should be uniformly flat and large across the range of abilities we are interested in. IRT: A Recap Allows one to compare items in terms of their difficulty

Candidates who take different tests can be placed on a common ability scale and compared

Intelligent design of exams that test for abilities across the required range IRT: Advanced Topics How do you model items with partial credit?

Can you model the incorrect options in an item? i.e. what is the probability that a person of ability θ will choose a particular option?

What is IRT The Item Response Function P(θ) = c + (1-c)/(1+ e-a(θ – b)) = Probability of answering item correctly at ability θ. Since higher ability => higher probability of answering,

the function should be increasing. Determine the probability of getting this response pattern (in terms of a, b, c)

Choose a, b, c so that this probability is maximized

This is the maximum likelihood estimate of the item parameters

This process is called item calibration c: guessing parameter = Prob. (answering correctly at low ability levels).

b: difficulty parameter. The more difficult the item, the larger the value of b.

a: discrimination parameter. Determines the steepness of the curve. IRT stands for “Item Response Theory”.

An item is a multiple choice question.

IRT seeks to relate a person’s responses to items to

her intrinsic ability in the subject matter. Different Traits of a Human People have different traits (height, weight, IQ,…)

Some are measurable (e.g. height, weight…)

But some are not (verbal ability, mathematical skills, optimism level…)

- These are hidden or latent traits and can only be estimated The Item Response Function Effect of varying the difficulty parameter.

Note that as b increases, the probability of answering correctly

at θ = 0 decreases.

Full transcriptPresent a series of items in the domain of interest to the person

Based on his responses, estimate his ability level in that domain Ingredients for IRT A person’s ability in a subject can be modeled by a number called the ability level

- e.g. A person with an ability level of 3 is said to have a higher

proficiency in the subject than one with a level of -0.2 Ingredients for IRT Q: How do you relate a person’s ability level to her response to an item?

A: Through an item response function that maps the ability level to the probability of answering the item correctly.

The Item Response Function Characterized by the item parameters a, b, c.

0 ≤ P(θ) ≤ 1.

Determining the Item Parameters a, b, c Ask some examinees to answer the item and record their responses

Determining a Person’s Ability θ Give a set of calibrated items to the person and record his responses

Find the probability of getting this response pattern and set θ to the value that maximizes this probability

Note: The same method can be used to calibrate items and estimate abilities simultaneously. Item Information Function This quantifies the information provided by the item at different ability levels.

The curve here provides max. info at θ ≈ 0

Test Information Function 'Sum of item information functions of the items in the test'.

Should be uniformly flat and large across the range of abilities we are interested in. IRT: A Recap Allows one to compare items in terms of their difficulty

Candidates who take different tests can be placed on a common ability scale and compared

Intelligent design of exams that test for abilities across the required range IRT: Advanced Topics How do you model items with partial credit?

Can you model the incorrect options in an item? i.e. what is the probability that a person of ability θ will choose a particular option?

What is IRT The Item Response Function P(θ) = c + (1-c)/(1+ e-a(θ – b)) = Probability of answering item correctly at ability θ. Since higher ability => higher probability of answering,

the function should be increasing. Determine the probability of getting this response pattern (in terms of a, b, c)

Choose a, b, c so that this probability is maximized

This is the maximum likelihood estimate of the item parameters

This process is called item calibration c: guessing parameter = Prob. (answering correctly at low ability levels).

b: difficulty parameter. The more difficult the item, the larger the value of b.

a: discrimination parameter. Determines the steepness of the curve. IRT stands for “Item Response Theory”.

An item is a multiple choice question.

IRT seeks to relate a person’s responses to items to

her intrinsic ability in the subject matter. Different Traits of a Human People have different traits (height, weight, IQ,…)

Some are measurable (e.g. height, weight…)

But some are not (verbal ability, mathematical skills, optimism level…)

- These are hidden or latent traits and can only be estimated The Item Response Function Effect of varying the difficulty parameter.

Note that as b increases, the probability of answering correctly

at θ = 0 decreases.