1. Item Analysis: Classical and Beyond
SCROLLA Symposium Measurement Theory and Item Analysis
Modified for EPE/EDP 711 by Kelly Bradley on January 8, 2013

2. Why is item analysis relevant?
Item analysis provides a way of measuring the quality of questions - seeing how appropriate they were for the respondents how well they measured their ability.
Item analysis also provides a way of re-using items over and over again in different instruments with prior knowledge of how they are going to perform.

3. What kinds of item analysis are there?
Classical
Latent Trait Models
Item Response theory
Rasch
IRT IRT2 IRT3 IRT4

4. Classical Test Theory
Classical analysis is the easiest and most widely used form of analysis. The statistics can be computed by generic statistical packages (or at a push by hand) and need no specialist software.
Classical analysis is performed on the survey or test instrument as a whole rather than on the item and although item statistics can be generated, they apply only to that group of students on that collection of items

5. Classical Test Theory Assumptions
Classical test theory assumes that any test score (or survey instrument sum) is comprised of a “true” value, plus randomized error.
Crucially it assumes that this error is normally distributed; uncorrelated with true score and the mean of the error is zero.
xobs = xtrue + G(0, err)

6. Classical Analysis Statistics
Difficulty (item level statistic)
Discrimination (item level statistic)
Reliability (instrument level statistic)

7. Classical Test Theory Difficulty
The difficulty of a (single response selection) question in classical analysis is simply the proportion of people who answered the question incorrectly. For multiple mark questions, it is the average mark expressed as a proportion.
Given on a scale of 0-1, the higher the proportion the greater the difficulty.

8. Classical Test Theory Discrimination
The discrimination of an item is the (Pearson) correlation between the average item mark and the average total test mark.
Being a correlation it can vary from –1 to +1 with higher values indicating (desirable) high discrimination.

9. Classical Test Theory Reliability
Reliability is a measure of how well the test or survey “holds together”. For practical reasons, internal consistency estimates are the easiest to obtain which indicate the extent to which each item correlates with every other item.
This is measured on a scale of 0-1. The greater the number the higher the reliability.

10. Classical Analysis versus Latent Trait Models
Classical analysis has the survey, or test, (not the item) as its basis. Although the statistics generated are often generalized to similar populations completing a similar survey, or taking a similar test; they only really apply to those students taking that test
Latent trait models aim to look beyond that at the underlying traits which are producing the test performance. They are measured at item level and provide sample-free measurement

11. Latent Trait Models
Latent trait models have been around since the 1940s, but were not widely used until the 1960s. Although theoretically possible, it is practically unfeasible to use these without specialist software.
They aim to measure the underlying ability (or trait) which is producing the test performance rather than measuring performance per se.
This leads to them being sample-free. As the statistics are not dependant on the test situation which generated them, they can be used more flexibly.

12. Rasch versus Item Response Theory
Mathematically, Rasch is identical to the most basic IRT model (IRT1), however there are some important differences which makes it a more viable proposition for practical testing
For instance,
In Rasch the model is superior. Data which does not fit the model is discarded (carefully and not dumped).
Rasch does not permit abilities to be estimated for extreme items and persons.

13. IRT - the generalized model
Where
ag = gradient of the ICC at the point  (item discrimination)
bg = the ability level at which ag is maximized
(item difficulty)
cg = probability of low persons correctly answering question (or endorsing) g

14. IRT - Item Characteristic Curves
An ICC is a plot of the respondents ability (likeliness to endorse) over the probability of them correctly answering the question (endorsing). The higher the ability the higher the chance that they will respond correctly.
c - intercept
b - ability at max (a)
a - gradient

15. IRT - About the Parameters Difficulty
Although there is no “correct” difficulty for any one item, it is clearly desirable that the difficulty of the test (or survey instrument) is centred around the average ability of the respondents.
The higher the “b” parameter the more difficult the question.
This is inversely proportionate to the probability of the question being answered correctly.

16. IRT - About the Parameters Discrimination
In IRT (unlike Rasch) maximal discrimination is sought.
Thus the higher the “a” parameter the more desirable the question.
Differences in the discrimination of questions can lead to differences in the difficulties of questions across the ability range.

17. IRT - About the Parameters Guessing
A high “c” parameter suggests that candidates with very little ability may choose the correct answer.
This is rarely a valid parameter outwith multiple choice testing…and the value should not vary excessively from the reciprocal of the number of choices.

18. IRT - Parameter Estimation
Before being used (in an item bank or for measurement) items must first be calibrated. That is their parameters must be estimated.
There are two main procedures - Joint Maximal Likelihood and Marginal Maximal Likelihood. JML is most common for IRT1 and 2, while MML is used more frequently for IRT3.
Bayesian estimation and estimated bounds may be imposed on the data to avoid high discrimination items being over valued.