Page 238 - DMTH404_STATISTICS

Statistics

Notes

Note Although alpha is relatively insensitive to the relative contributions of group
and general factor, beta is very sensitive. Alpha, however, can be found from item and test
statistics, beta needs to be estimated by finding the worst split half. Such an estimate is
computationally much more difficult.

Omega, a more general estimate, based upon the factor structure of the test, allows for bette
estimate of the first factor saturation.

Generalizabilty Theory Reliability across facets:

The consistency of Individual Differences across facets may be assessed by analysing variance
components associated with each facet. i.e., what amount of variance is associated with a particular
facet across which one wants to generalize?
Facets of reliability

Across Items Domain Sampling
Internal Consistency
Across Time Temporal Stability
Across Forms Alternate Form Reliability
Across Raters Inter-rater agreement

Across Situations Situational Stability
Across “Tests” (facets unspecified) Parallel Test reliability

Generalizability theory is a decomposition of variance components to estimate sources of variance
across which one wants to generalize.
All of these conventional approaches are concerned with generalizing about individual
differences (in response to an item, time, form, rater, or situation) between people. Thus, the
emphasis is upon consistency of rank orders. Classical reliability is a function of large between
subject variability and small within subject variability. It is unable to estimate the within subject
precision.
An alternative method (Latent Response Theory or Item Response Theory) is to determine the
precision of the estimate of a particular person’s position on a latent variable.

Item Response Theory - 1

A model for item response as a function of increasing level of subject ability and increasing
levels of item difficulty. This model estimates the probability of making a particular response
(generally, correct or incorrect) as a joint function of the subject’s value on a latent attribute
dimension, and the difficulty (item endorsement rate) of a particular item.

Model 1: the Rasch model: Probability of endorsing an item given ability (ø) and difficulty (diff):
1
P(y|ø,diff) =
1+e(diff-ø)

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