Cronbach's Generalizability Theory (G-Theory) is "a technique for estimating the relative magnitudes of various components of error variation and for indicating the most efficient strategy for achieving desired measurement precision." (Shavelson & Webb, 1991, back cover). G-Theory is not concerned with fairness to examinees (e.g., adjusting scores for judge leniency) or with judge rating quality (e.g., idiosyncrasies, response sets). It is concerned solely with the relationship between the sought-for "true" variance in what is observed and the other unwanted main effect, interaction, or random error variances. G-Theory applies Fisher ANOVA to raw scores to estimate variance components and so compute reliability coefficients.
First, a pilot data collection is undertaken for a Generalizability Study (G-Study). Variance components for all main, interaction and random effects are estimated from these pilot data. These variances are asserted to be those that will occur in a future, more comprehensive data collection, the Decision Study (D-Study). Then a D-Study is designed to obtain a desired level of reliability based on the G-Study variance components.
Rasch-based G-Theory has several advantages. First, no preliminary G-Study is needed. The precision (standard error) of an element's measure can be predicted solely from the number of ratings (replications) in which the element participates, so long as the rating context is on target (not excessively "easy" or "hard"). The lower portion of the nomograph shown overleaf (in the printed text) illustrates this. In the Rasch method, it makes no difference to the increase in measure precision whether extra ratings result from more raters or more items.
Second, the statistical aspects of the D-Study are trivial. For any average standard error across measures and putative "true" standard deviation of the measures, the reliability can be read directly from the top half of the nomograph. The separation (shown below or to the right of the reliability) is the ratio of "true" S.D. to average S.E. and indicates how many distinct measurement levels the test design can discriminate in this measure distribution.
In practice, target groups of reasonably homogeneous examinees tend to have "true" S.D.'s of 1-2 logits. So, with no data collection at all, one can see immediately the best reliability that any data collection design can yield. Alternatively, one can read off the nomograph how many ratings of each examinee must be collected to reach the desired reliability level.
This nomograph is based on a test targeted on the persons and an effective rating scale. Perverse measure distributions or a poorly functioning scale lessen test precision and can double S.E. size. If some examinees are rated more often than others, or items have different rating scales, use average values.
Example 1: We plan to use 30 item MCQ tests with our 5th grade
children. What reliability can we expect?
Answer: Look at the bottom of the nomograph. The precision of a 30 dichotomy test is about 0.5 logits. Our 5th grade children are likely to be fairly homogeneous, with S.D. about 1 logit. Read up from S.E. of 0.5 logits, across from S.D. of 1 logit, to the diagonal line. This line is marked with a reliability of 0.8 and separation of 2. That is the best that we can expect from this design.
Example 2: How many ratings per examinee on a 5 category scale
(0-4) is the minimum required if we seek a reliability of 0.85,
when previous experience indicates the ability S.D. is 1.5
Answer: Read across from S.D. of 1.5 logits to the 0.85 reliability line. Then read down. The required average S.E. is 0.6 logits. Read down to the 5 category scale. At least 5 ratings for each examinee are needed.
Example 3: According to the test publisher, an observation
protocol has a reliability of 0.66 with four ratings per
examinee. How many ratings must be made for a reliability of
Answer: Start with the 4 ratings per examinee. Values for S.E. and S.D. are not needed. Simply read up from any "4" in the bottom half of the nomograph, say on the "Dichotomy" line, to reliability 0.66, midway between the 0.6 and 0.7 diagonal reliability lines. Read left to the 0.8 diagonal reliability line. Then read down to the answer on the "4" (Dichotomy) line. At least 9 ratings are needed.
Technical note: The nomograph was constructed using a spreadsheet and standard reliability formulas.
The top section is always the same, regardless of the analysis. It is a graphical representation of the basic reliability relationship:
Reliability = true variance / observed variance = true variance / (true variance+error variance) = true sd² / (true sd² + se²)
The bottom section pictures the estimation of the S.E. This varies somewhat depending on test targeting and the internal structure of the rating scales. To compute it, estimate the average information in a response (e.g., from a response-level residual file or from Rasch theory), and then
S.E. = 1/sqrt (response information * count of responses)
T = True S.D. (y-axis in nomograph)
E = Average person S.E. (x-axis in nomograph)
Reliability = True Variance/Observed Variance = T*T / (T*T + E*E)
Then, with reasonable person-item targeting (person ability about 1 logit higher than an item difficulty):
E = 1 / sqrt ( 0.8 * 0.2 * L) where L = number of items in the test
or, if observational data are available,
E = 1/sqrt( the sum of the "Model variance of the observed values around the expected" for the observations of interest).
Example: for a sample of persons measured with 30 items:
True S.D. = 1 logit
E = 1 / sqrt ( 0.8 * 0.2 * 30) = 0.47 logits
Reliability = 1*1 / (1*1 + 0.47*0.47) = 1/1.22 = 0.82
Linacre JM. (1993) Rasch-based Generalizability Theory. Rasch Measurement Transactions, 7:1, p. 283-284.
Shavelson RM & Webb NM (1991) Generalizability Theory. Newbury Park CA: Sage.
Rasch-based generalizability theory: Reliability and Precision (S.E.) Nomogram. Linacre JM. Rasch Measurement Transactions, 1993, 7:1 p.283
|Rasch Measurement Transactions (free, online)||Rasch Measurement research papers (free, online)||Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch||Applying the Rasch Model 3rd. Ed., Bond & Fox||Best Test Design, Wright & Stone|
|Rating Scale Analysis, Wright & Masters||Introduction to Rasch Measurement, E. Smith & R. Smith||Introduction to Many-Facet Rasch Measurement, Thomas Eckes||Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr.||Statistical Analyses for Language Testers, Rita Green|
|Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar||Journal of Applied Measurement||Rasch models for measurement, David Andrich||Constructing Measures, Mark Wilson||Rasch Analysis in the Human Sciences, Boone, Stave, Yale|
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