September 9, 2011

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**Psychometrics**

**Standard Formulation**

Model: Score = True Score plus Error or X = T + E

Note that T and E
have implicit coefficients of 1

**Assumptions**

True score and error uncorrelated

**Example
**

**Definition of
Components
**

True score: the meaningful portion of variance or the average of all possible measures of the true score

Modern (i.e., Generalizability Theory)

The variance of the score consists of many different components. In a particular research context, some of these components are meaningful (the true score in that context) and others are not (error in that context).

Reliability

The proportion of variance due to true score or V(T)/V(X) where "V" means variance.

Do not confuse reliability with how to measure it (e.g., internal consistency or test-retest). Reliability refers not just to the measure, but to sample and context of measurement. Thus, with a different sample or context, it can change.

Use RELCOMP, a computer program for reliability computation.

Standardized Model

The path from the true score or T to measured variable or X when both variables are standardized equals the square root of the reliablity. Thus, the correlation of the measure with the true score equals the square root of the reliablity.

**Reliability Estimation
**

** Different Times
**

**Classical Assumptions
**

** Given Classical Assumptions:
Reliability = Correlation between Measures **

** Example: To measure the
reliability of mother’s or father’s report correlate the two
variables. **

**Spearman-Brown Profecy Formula**

**If the average
inter-item correlation or r is known, it can be used to forecast the reliability
of a test with k items: **

**
**

** This
formula gives Cronbach's alpha if all the measures had equal variance which
would happen if all the measures were standardized. This
formula has many uses. It can be used to determine the
reliability of a test if more or less items are used. For
instance, if a test has 12 items and an average inter-item correlation of .2,
then its reliability is .75. But if the test were to have 24
items, its reliability would be .86 and with 6 items the reliability would be
.60.
**

** Another useful formula, again derived from the Spearman-Brown, is estimate of
the average inter-item correlation if we know alpha or a and the number of items
or k. The formula is **

**The average inter-item correlation tends to be smaller than you might
think. For instance, the average inter-item correlation for
the Peabody Picture Vocabulary Test is about .08, the
Rosenberg
Self-Esteem Inventory is about .34, and the Beck Depression Inventory is about .26. **

Correction for Attenuation

**To determine what the
correlation between the two variables would be if the variables' reliabilities
were perfect, a correlation needs to be divided by the square root of the
product of the two variables' reliability. It is in essence,
a forecast of what the correlation would be if the two variables were measured
with many (i.e., infinite) items. The formula
is **

where r_{XX} and r_{YY} are the reliabilities of X and Y, respectively. So if the reliabilities of X and Y are both .8, and the correlation between X and Y is .4, then the corrected for attenuation correlation is
.5. A disattenuated correlation is not an ordinary Pearson
correlation and its range is not +1 to -1.One cannot perform
the usual significance test on these correlations. Also one
should avoid disattenuating a correlation if the reliabilities are small to
avoid wild
values.** **

**SEM
Terminology
**

Unequal loadings: Classical theory sets all factor loadings to one. But it is possible to fix one of the loadings to one and free the other loadings.

Unequal erro variances: Classical theory sets the error variances to b equal, but this assumption can be relaxed.

Estimated True Score

**where r _{XX} is the reliability of X. The estimate is said to be shrunken
or regressed to its mean.**

The Effects of Unreliability in Causal Models

Causal variable X has measurement error.

Variables X and Z are correlated.

X affects the endogenous variable.

**Correcting for the Effects due to Measurement Error
**

Instrumental Variable Estimation

Multiple Indicators (Latent Variable Models)

What has been presented above is sometimes called classical test theory (CTT). A modern alternative to CTT is item response theory which presumes a dichotomous response. Item response theory is very complicated and highly mathematical.