Implicit in our understanding of the "Reliability of a test for a sample" is that the sample and test somehow match. Typically, we think that the test is targeted on the sample, and that the sample has a reasonably symmetric unimodal "ability" distribution covering the operational range of the test. But what if this isn't so?
Clinical samples can be highly skewed: at one end, many healthy people, at the other, a very few deathly sick. The instrument may be targeting the sick, so it will be far off-target for the majority of the sample. The reported reliability coefficients will be low, but the instrument is clearly doing its job. What to do?
Conventional raw-score reliability estimates employ empirical samples in order to estimate measurement error. Rasch methods do not need to do this. Once the test items are calibrated, the measure and standard error corresponding to every possible raw score can be estimated without further data collection. These standard errors are usually reported as the smallest possible "model" errors. In practice, these standard errors can be inflated about 10% to allow for the unmodeled noise encountered in real data.
Statistically different levels of performance
Once the standard errors have been computed, they can be used to compute how many statistically different levels of performance can be identified. See the Table below (which is based on Winsteps Table 20 or the equivalent with other software). To do this start at one end of the raw score range and work to toward the other. Advance each time by twice the joint standard error (= square-root of sum of squared standard errors) of the current starting and ending measures until there is not room for another level. In the Table, there are 4 statistically distinct levels of performance. This corresponds to a separation of at least 4, i.e,. this test has a sample-independent reliability of at least 4²/(1+4²) = 0.94. The empirical reliability of this test, reported for a highly central sample, was 0.62, a value which grossly under-reports the test's measurement effectiveness.
Benjamin D. Wright
Raw Score on Test |
Logit Measure |
Standard Error |
Minimum measure for next level (must be less than Logit Measure to start the next level) |
Statistical Level |
0 (0.3†) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 (13.7†) |
-6.17 -4.86 -3.94 -3.27 -2.64 -1.97 -1.19 -.23 .80 1.72 2.55 3.37 4.21 5.23 6.60 |
1.83 1.08 0.85 0.79 0.72 0.68 0.63 0.64 0.69 0.75 0.81 0.89 0.93 1.12 1.84 |
-6.17 -6.17 + 2√(1.83²+1.08²) = -1.92 (> -4.86) -6.17 + 2√(1.83²+0.85²) = -2.13 (> -3.94) -6.17 + 2√(1.83²+0.79²) = -2.18 (> -3.27) -6.17 + 2√(1.83²+0.72²) = -2.24 (> -2.64) -6.17 + 2√(1.83²+0.68²) = -2.27 (< -1.97) -1.97 + 2√(0.68²+0.63²) = -0.12 (> -1.19) -1.97 + 2√(0.68²+0.64²) = -0.10 (> -0.23) -1.97 + 2√(0.68²+0.69²) = -0.03 (< 0.80) 0.80 + 2√(0.69²+0.75²) = 2.84 (> 1.72) 0.80 + 2√(0.69²+0.81²) = 2.93 (> 2.55) 0.80 + 2√(0.69²+0.89²) = 3.05 (< 3.37) 3.37 + 2√(0.89²+0.93²) = 5.94 (> 4.21) 3.37 + 2√(0.89²+1.12²) = 6.23 (> 5.23) 3.37 + 2√(0.89²+1.84²) = 7.46 (> 6.60) |
1 1 1 1 1 2 2 2 3 3 3 4 4 4 4 |
† extreme scores of 0 and 14 are made more central by a Bayesian adjustment of 0.3 score-points in order to make their corresponding Rasch measures estimable. This is done automatically in Winsteps Table 20 and similar adjustments may be made by other software.
Later notes:
Another possibility would be to sample a normal distribution from the skewed distribution. Here's how to do it:
1. Divide the x-axis of your skewed distribution of persons into equal intervals (interval size depends on the size of your sample).
2. Divide the x-axis of a normal distribution of the same range into the same equal intervals
3. Compute the number of persons in each normal-distribution interval so that total numbers of persons in normal intervals matches the sample total.
4. Sample-with-replacement the persons from their skewed intervals into the corresponding normal intervals until each normal interval has the correct number of persons in it.
5. Compute the reliability from the resulting normalized sample.
In general, skew reduces reliability, so your reliability should have increased a little with the normalized sample.
There are three different Reliability/Separation/Strata options:
1. Spearman "Test" Reliability for the current sample -> Cronbach Alpha in CTT, and "Separation Reliability" for the observed person sample in Rasch.
"Separation ratio" = square-root (Spearman Reliabity / (1-Spearman Reliability)). Normal distributions are assumed.
2. Strata for the current sample. This is the separation ratio, but with the tales of the observed person distribution treated as performance levels.
"Strata reliability" = Strata^2 / ( 1+ Strata^2)
3. Wright's sample-independent method for strata. This shows the maximum number of statistically different strata the test can identify.
"Wright Strata Reliability" = Wright Strata^2 / ( 1+ Wright Strata^2). This is the maximum possible value of the Spearman "Test" reliability.
Choose the one (and make clear which one you have chosen) that makes most sense in your situation.
Separation, Reliability and Skewed Distributions: Statistically Different Sample-independent Levels of Performance. Wright B.D. … Rasch Measurement Transactions, 2001, 14:4 p.786
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